Ted Draft] Logic for Philosophers

Logic for Philosophy

Theodore Sider

December 4, 2007

Preface

This book is an elementary introduction to the logic that students of contempo-

rary philosophy ought to know. It covers i) basic approaches to logic, including

proof theory and especially model theory, ii) extensions of standard logic (such

as modal logic) that are important in philosophy, and iii) some elementary

philosophy of logic. It prepares students to read the logically sophisticated

articles in today’s philosophy journals, and helps them resist bullying by symbol-

mongerers. In short, it teaches the logic necessary for being a contemporary

philosopher.

For better or for worse (I think better), the last century-or-so’s developments

in logic are part of the shared knowledge base of philosophers, and inform,

in varying degrees of directness, every area of philosophy. Logic is part of

our shared language and inheritance. The standard philosophy curriculum

therefore includes a healthy dose of logic. This is a good thing. But the advanced

logic that is part of this curriculum is usually a course in “mathematical logic”,

which usually means an intensive course in metalogic (for example, a course

based on the excellent Boolos and Jeffrey (1989).) I do believe in the value of

such a course. But if advanced undergraduate philosophy majors or beginning

graduate students are to have one advanced logic course, that course should

not, I think, be a course in metalogic. The standard metalogic course is too

mathematically demanding for the average philosophy student, and omits

material that the average student needs to know. If there is to be only one

advanced logic course, let it be a course designed to instill logical literacy.

I begin with a sketch of standard propositional and predicate logic (de-

veloped more formally than in a typical intro course.) I brie�y discuss a few

extensions and variations on each (e.g., three-valued logic, de�nite descrip-

tions). I then discuss modal logic and counterfactual conditionals in detail. I

presuppose familiarity with the contents of a typical intro logic course: the

meanings of the logical symbols of �rst-order predicate logic without identity

i

PREFACE ii

or function symbols; truth tables; translations from English into propositional

and predicate logic; some proof system (e.g., natural deduction) in propositional

and predicate logic.

I drew heavily from the following sources, which would be good for supple-

mental reading:

• Propositional logic: Mendelson (1987)

• Descriptions, multi-valued logic: Gamut (1991a)

• Sequents: Lemmon (1965)

• Further quanti�ers: Glanzberg (2006); Sher (1991, chapter 2); Wester-

ståhl (1989); Boolos and Jeffrey (1989, chapter 18)

• Modal logic: Gamut (1991b); Cresswell and Hughes (1996)

• Semantics for intuitionism : Priest (2001)

• Counterfactuals: Lewis (1973)

• Two-dimensional modal logic: Davies and Humberstone (1980)

Another source was Ed Gettier’s 1988 modal logic class at the University of

Massachusetts.

I am also deeply grateful for feedback from colleagues, and from students

in courses on this material. In particular, Marcello Antosh, Josh Armstrong,

Gabe Greenberg, Angela Harper, Sami Laine, Gregory Lavers, Alex Morgan,

Jeff Russell, Brock Sides, Jason Turner, Crystal Tychonievich, Jennifer Wang,

Brian Weatherson, and Evan Williams: thank you.

Contents

Preface i

1 Nature of Logic 11.1 Logical consequence and logical truth . . . . . . . . . . . . . . . . . 2

1.2 Form and abstraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Formal logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Correctness and application . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 The nature of logical consequence . . . . . . . . . . . . . . . . . . . 8

1.6 Extensions, deviations, variations . . . . . . . . . . . . . . . . . . . . 10

1.6.1 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6.2 Deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6.3 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.7 Metalogic, metalanguages, and formalization . . . . . . . . . . . . 12

1.8 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Propositional Logic 182.1 Grammar of PL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 The semantic approach to logic . . . . . . . . . . . . . . . . . . . . . 21

2.3 Semantics of PL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Natural deduction in PL . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.1 Sequents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.2 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.3 Sequent proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4.4 Example sequent proofs . . . . . . . . . . . . . . . . . . . . . 35

2.5 Axiomatic proofs in PL . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.5.1 Example axiomatic proofs . . . . . . . . . . . . . . . . . . . . 42

2.5.2 The deduction theorem . . . . . . . . . . . . . . . . . . . . . 47

2.6 Soundness and completeness of PL . . . . . . . . . . . . . . . . . . 47

iii

CONTENTS iv

3 Variations and Deviations from PL 523.1 Alternate connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.1.1 Symbolizing truth functions in propositional logic . . . 52

3.1.2 Inadequate connective sets . . . . . . . . . . . . . . . . . . . 56

3.1.3 Sheffer stroke . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 Polish notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3 Multi-valued logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.1 Łukasiewicz’s system . . . . . . . . . . . . . . . . . . . . . . . 60

3.3.2 Kleene’s “strong” tables . . . . . . . . . . . . . . . . . . . . . 62

3.3.3 Kleene’s “weak” tables (Bochvar’s tables) . . . . . . . . . . 63

3.3.4 Supervaluationism . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.4 Intuitionism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Predicate Logic 714.1 Grammar of predicate logic . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Semantics of predicate logic . . . . . . . . . . . . . . . . . . . . . . . 72

4.3 Establishing validity and invalidity . . . . . . . . . . . . . . . . . . . 77

5 Extensions of Predicate Logic 805.1 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.1.1 Grammar for the identity sign . . . . . . . . . . . . . . . . . 80

5.1.2 Semantics for the identity sign . . . . . . . . . . . . . . . . 81

5.1.3 Symbolizations with the identity sign . . . . . . . . . . . 81

5.2 Function symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2.1 Grammar for function symbols . . . . . . . . . . . . . . . . 85

5.2.2 Semantics for function symbols . . . . . . . . . . . . . . . . 86

5.2.3 Symbolizations with function symbols: some examples 88

5.3 De�nite descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.3.1 Grammar for ι . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.3.2 Semantics for ι . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.3.3 Eliminability of function symbols and de�nite descriptions 92

5.4 Further quanti�ers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.4.1 Generalized monadic quanti�ers . . . . . . . . . . . . . . . 96

5.4.2 Generalized binary quanti�ers . . . . . . . . . . . . . . . . . 98

5.4.3 Second-order logic . . . . . . . . . . . . . . . . . . . . . . . . 100

CONTENTS v

6 Propositional Modal Logic 1036.1 Grammar of MPL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2 Symbolizations in MPL . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.3 Semantics for MPL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.3.1 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.3.2 Kripke models . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.3.3 Semantic validity proofs . . . . . . . . . . . . . . . . . . . . . 115

6.3.4 Countermodels . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.3.5 Schemas, validity, and invalidity . . . . . . . . . . . . . . . . 135

6.4 Axiomatic systems of MPL . . . . . . . . . . . . . . . . . . . . . . . . 137

6.4.1 System K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.4.2 System D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.4.3 System T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.4.4 System B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.4.5 System S4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.4.6 System S5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

6.4.7 Substitution of equivalents and modal reduction . . . . . 153

6.5 Soundness in MPL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

6.5.1 Soundness of K . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.5.2 Soundness of T . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.5.3 Soundness of B . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.6 Completeness of MPL . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.6.1 Canonical models . . . . . . . . . . . . . . . . . . . . . . . . . 160

6.6.2 Maximal consistent sets of wffs . . . . . . . . . . . . . . . . 161

6.6.3 De�nition of canonical models . . . . . . . . . . . . . . . . 162

6.6.4 Features of maximal consistent sets . . . . . . . . . . . . . 163

6.6.5 Maximal consistent extensions . . . . . . . . . . . . . . . . . 164

6.6.6 “Mesh” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.6.7 The coincidence of truth and membership in canonical

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.6.8 Completeness of systems of MPL . . . . . . . . . . . . . . 169

7 Variations on MPL 1727.1 Propositional tense logic . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7.1.1 The metaphysics of time . . . . . . . . . . . . . . . . . . . . 172

7.1.2 Tense operators . . . . . . . . . . . . . . . . . . . . . . . . . . 174

7.1.3 Syntax of tense logic . . . . . . . . . . . . . . . . . . . . . . . 176

7.1.4 Possible worlds semantics for tense logic . . . . . . . . . . 176

CONTENTS vi

7.1.5 Formal constraints on ≤ . . . . . . . . . . . . . . . . . . . . . 178

7.2 Intuitionist propositional logic . . . . . . . . . . . . . . . . . . . . . . 180

7.2.1 Kripke semantics for intuitionist propositional logic . . 180

7.2.2 Examples and proofs . . . . . . . . . . . . . . . . . . . . . . . 183

7.2.3 Soundness and other facts about intuitionist validity . . 186

8 Counterfactuals 1908.1 Natural language counterfactuals . . . . . . . . . . . . . . . . . . . . 191

8.1.1 Not truth-functional . . . . . . . . . . . . . . . . . . . . . . . 191

8.1.2 Can be contingent . . . . . . . . . . . . . . . . . . . . . . . . . 191

8.1.3 No augmentation . . . . . . . . . . . . . . . . . . . . . . . . . 192

8.1.4 No contraposition . . . . . . . . . . . . . . . . . . . . . . . . . 193

8.1.5 Some implications . . . . . . . . . . . . . . . . . . . . . . . . . 193

8.1.6 Context dependence . . . . . . . . . . . . . . . . . . . . . . . 194

8.2 The Lewis/Stalnaker approach . . . . . . . . . . . . . . . . . . . . . 196

8.3 Stalnaker’s system (SC) . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

8.3.1 Syntax of SC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

8.3.2 Semantics of SC . . . . . . . . . . . . . . . . . . . . . . . . . . 197

8.4 Validity proofs in SC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

8.5 Countermodels in SC . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

8.6 Logical Features of SC . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

8.6.1 Not truth-functional . . . . . . . . . . . . . . . . . . . . . . . 211

8.6.2 Can be contingent . . . . . . . . . . . . . . . . . . . . . . . . . 211

8.6.3 No augmentation . . . . . . . . . . . . . . . . . . . . . . . . . 211

8.6.4 No contraposition . . . . . . . . . . . . . . . . . . . . . . . . . 211

8.6.5 Some implications . . . . . . . . . . . . . . . . . . . . . . . . . 212

8.6.6 No exportation . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

8.6.7 No importation . . . . . . . . . . . . . . . . . . . . . . . . . . 214

8.6.8 No hypothetical syllogism (transitivity) . . . . . . . . . . . 214

8.6.9 No transposition . . . . . . . . . . . . . . . . . . . . . . . . . . 215

8.7 Lewis’s criticisms of Stalnaker’s theory . . . . . . . . . . . . . . . . 216

8.8 Lewis’s system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

8.9 The problem of disjunctive antecedents . . . . . . . . . . . . . . . 222

9 Quanti�ed Modal Logic 2249.1 Grammar of QML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

9.2 Symbolizations in QML . . . . . . . . . . . . . . . . . . . . . . . . . . 224

9.3 A simple semantics for QML . . . . . . . . . . . . . . . . . . . . . . . 226

CONTENTS vii

9.4 Countermodels and validity proofs in SQML . . . . . . . . . . . . 228

9.5 Philosophical questions about SQML . . . . . . . . . . . . . . . . . 234

9.5.1 The necessity of identity . . . . . . . . . . . . . . . . . . . . 234

9.5.2 The necessity of existence . . . . . . . . . . . . . . . . . . . 236

9.5.3 Necessary existence defended . . . . . . . . . . . . . . . . . 241

9.6 Variable domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

9.6.1 Countermodels to the Barcan and related formulas . . . 246

9.6.2 Expanding, shrinking domains . . . . . . . . . . . . . . . . 248

9.6.3 Strong and weak necessity . . . . . . . . . . . . . . . . . . . 249

10 Two-dimensional modal logic 25310.1 Actuality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

10.1.1 Kripke models with actual worlds . . . . . . . . . . . . . . 254

10.1.2 Semantics for @ . . . . . . . . . . . . . . . . . . . . . . . . . . 255

10.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

10.2 × . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

10.2.1 Two-dimensional semantics for × . . . . . . . . . . . . . . 258

10.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

10.3 Fixedly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

10.3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

10.4 A philosophical application: necessity and a priority . . . . . . . . 265

References 272

Chapter 1

Nature of Logic

Since you are reading this book, you are probably already familiar with

some logic. You probably know how to translate English sentences into

symbolic notation—into propositional logic:

English Propositional logicIf snow is white then grass is green S→GEither snow is white or grass is not green S∨∼G

and into predicate logic:

English Predicate logicIf Jones is happy then someone is happy H j→∃xH xAnyone who is friends with Jones is either

insane or friends with everyone

∀x[F x j→(I x ∨∀yF xy)]

You are probably also familiar with some basic techniques for evaluating argu-

ments written out in symbolic notation. You have probably encountered truth

tables, and some form of proof theory (perhaps a “natural deduction” system;

perhaps “truth trees”.) You may have even encountered some elementary model

theory. In short: you had an introductory course in symbolic logic.

What you already have is: literacy in elementary logic. What you will

get out of this book is: literacy in the rest of logic that philosophers tend to

presuppose, plus a deeper grasp of what logic is all about.

So what is logic all about?

1

CHAPTER 1. NATURE OF LOGIC 2

1.1 Logical consequence and logical truthLogic is about logical consequence. The statement “someone is happy” is a logical

consequence of the statement “Ted is happy”. If Ted is happy, then it logicallyfollows that someone is happy. Put another way: the statement “Ted is happy”

logically implies the statement “someone is happy”. Likewise, the statement

“Ted is happy” is a logical consequence of the statements “It’s not the case that

John is happy” and “Either John is happy or Ted is happy”. The �rst statement

follows from the latter two statements. If the latter two statements are true,

the former must be true. Put another way: the argument whose premises are

the latter two statements, and whose conclusion is the former statement, is a

logically correct one.1

Relatedly, logic is about logical truth. A logical truth is a sentence that is

“true purely by virtue of logic”. Examples might include: “it’s not the case that

snow is white and also not white”, “All �sh are �sh”, and “If Ted is happy then

someone is happy”. It is plausible that logical truth and logical consequence

are related thus: a logical truth is a sentence that is a logical consequence of

any sentences whatsoever.

1.2 Form and abstractionLogicians focus on form. Consider again the following argument:

Argument A: It’s not the case that John is happy

Ted is happy or John is happy

Therefore, Ted is happy

Argument A is logically correct—its conclusion is a logical consequence of its

premises. It is customary to say that this is so in virtue of its form—in virtue of

the fact that its form is:

It’s not the case that φφ or ψTherefore ψ

1The word ‘valid’ is sometimes used for logically correct arguments, but I will reserve that

word for a different concept: that of a logical truth according to the semantic conception of

logical truth.


Likewise, we say that “it’s not the case that snow is white and snow is not white”

is a logical truth because it has the form: it’s not the case that φ and not-φ.

We need to think hard about the idea of form. Apparently, we got the

alleged form of Argument A by replacing some words with Greek letters and

leaving other words as they were. We replaced the sentences ‘John is happy’ and

‘Ted is happy’ with φ and ψ, respectively, but left the expressions ‘It’s not the

case that’ and ‘or’ as they were, resulting in the schematic form displayed above.

Let’s call that form, “Form 1”. What’s so special about Form 1? Couldn’t we

make other choices for what to leave and what to replace? For instance, if we

replace the predicate ‘is happy’ with the schematic letter α, leaving the rest

intact, we get this:

Form 2: It’s not the case that John is αTed is α or John is αTherefore, Ted is α

And if we replace the ‘or’ with the schematic letter γ and leave the rest intact,

then we get this:

Form 3: It’s not the case that John is happy

Ted is happy γ John is happy


If we think of Argument A as having Form 1, then we can think of it as being

logically correct in virtue of its form, since every “instance” of Form 1 is

logically correct. That is, no matter what sentences we substitute in for the

greek letters φ and ψ in Form 1, the result is a logically correct argument.

Now, if we think of Argument A’s form as being Form 2, we can continue to

think of Argument A as being logically correct in virtue of its form, since, like

Form 1, every instance of Form 2 is logically correct: no matter what predicate

we change α to, Form 2 becomes a logically correct argument. But if we think

of Argument A’s form as being Form 3, then we cannot think of it as being

logically correct in virtue of its form, for not every instance of Form 3 is a

logically correct argument. If we change γ to ‘if and only if’, for example, then

we get the following logically incorrect argument:

It’s not the case that John is happy

Ted is happy if and only if John is happy



So, what did we mean, when we said that Argument A is logically correct in

virtue of its form? What is Argument A’s form? Is it Form 1, Form 2, or Form

3?

There is no such thing as the form of an argument. When we assign an

argument a form, what we are doing is focusing on certain words and ignoring

others. We leave intact the words we’re focusing on, and we insert schematic

letters for the rest. Thus, in assigning Argument A Form 1, we’re focusing on

the words (phrases) ‘it is not the case that’ and ‘or’, and ignoring other words.

More generally, in (standard) propositional logic, we focus on the phrases

‘if…then’, ‘if and only if’, ‘and’, ‘or’, and so on, and ignore others. We do this

in order to investigate the relations of logical consequence that hold in virtue

of these words’ meaning. The fact that Argument A is logically correct depends

just on the meaning of the phrases ‘it is not the case that’ and ‘or’; it does not

depend on the meanings of the sentences ‘John is happy’ and ‘Ted is happy’.

We can substitute any sentences we like for ‘φ’ and ‘ψ’ in Form 1 and still get

a valid argument.

In predicate logic, on the other hand, we focus on further words: ‘all’ and

‘some’. Broadening our focus in this way allows us to capture a wider range

of logical consequences and logical truths. For example “If Ted is happy then

someone is happy” is a logical truth in virtue of the meaning of ‘someone’, but

not merely in virtue of the meanings of the characteristic words of propositional

logic.

Call the words on which we’re focusing—that is, the words that we leave

intact when we construct the forms of sentences and arguments—the logicalconstants. (We can speak of natural language logical constants—‘and’, ‘or’, etc.

for propositional logic; ‘all’ and ‘some’ in addition for predicate logic—as well

as symbolic logical constants: ∧, ∨, etc. for propositional logic; ∀ and ∃ in

addition for predicate logic.) What we’ve seen is that the forms we assign

depend on what we’re considering to be the logical constants.

We call these expressions logical constants because we interpret them in a

constant way in logic, in contrast to other terms. For example, ∧ is a logical

constant; in propositional logic, it always stands for conjunction. There are

�xed rules governing ∧, in proof systems (the rule that from P∧Q one can

infer P , for example), in the rules for constructing truth tables, and so on.

Moreover, these rules are distinctive for ∧: there are different rules for other

logical constants such as ∨. In contrast, the terms in logic that are not logical

constants do not have �xed, particular rules governing their meanings. For

example, there are no special rules governing what one can do with a P as


opposed to a Q in proofs or truth tables. That’s because P doesn’t symbolize

any sentence in particular; it can stand for any old sentence.

There isn’t anything sacred about the choices of logical constants we make

in propositional and predicate logic; and therefore, there isn’t anything sacred

about the customary forms we assign to sentences. We could treat other words

as logical constants. We could, for example, stop taking ‘or’ as a logical constant,

and instead take ‘It’s not the case that John is happy’, ‘Ted is happy’, and ‘John

is happy’ as logical constants. We would thereby view Argument A as having

Form 3. This would not be a particularly productive choice (since it would not

help to explain the correctness of Argument A), but it’s not wrong simply by

virtue of the concept of form.

More interestingly, consider the fact that every argument of the following

form is logically correct:

α is a bachelor

Therefore, α is unmarried

Accordingly, we could treat the predicates ‘is a bachelor’ and ‘is unmarried’

as logical constants, and develop a corresponding logic. We could introduce

special symbolic logical constants for these predicates, we could introduce

distinctive rules governing these predicates in proofs. (The rule of “bachelor-

elimination”, for instance, might allow one to infer “α is unmarried” from “αis a bachelor”.) As with the choices of the previous paragraph, this choice of

what to treat as a logical constant is also not ruled out by the concept of form.

And it would be more productive than the choices of the last paragraph. Still,

it would be far less productive than the usual choices of logical constants in

predicate and propositional logic. The word ‘bachelor’ doesn’t have as general

application as the words commonly treated as logical constants in propositional

and predicate logic; the latter are ubiquitous.

At least, this remark about “generality” is one idea about what should be

considered a “logical constant”, and hence one idea about the scope of what

is usually thought of as “logic”. Where to draw the boundaries of logic—and

indeed, whether the logic/nonlogic boundary is an important one to draw—is

an open philosophical question about logic. At any rate, in this course, one

thing we’ll do is study systems that expand the list of logical constants from

standard propositional and predicate logic.


1.3 Formal logicModern logic is “mathematical” or “formal” logic. This means simply that

one studies logic using mathematical techniques. More carefully: in order

to develop theories of logical consequence, and logical truth, one develops a

formal language (see below), one treats the sentences of the formal language as

mathematical objects; one uses the tools of mathematics (especially, the tools

of very abstract mathematics, such as set theory) to formulate theories about

the sentences in the formal language; and one applies mathematical standards

of rigor to these theories. Mathematical logic was originally developed to study

mathematical reasoning2, but its techniques are now applied to reasoning of all

kinds.

Think, for example, of propositional logic (this will be our �rst topic below).

The standard approach to analyzing the logical behavior of ‘and’, ‘or’, and so

on, is to develop a certain formal language, the language of propositional logic.

The sentences of this language look like this:

P

(Q→R)∨(Q→∼S)

P↔(P∧Q)

The symbols ∧, ∨, etc., are used to represent the English words ‘and’, ‘or’, and

so on (the logical constants for propositional logic), and the sentence letters

P,Q, etc., are used to represent declarative English sentences.

Why ‘formal’? Because we stipulate, in a mathematically rigorous way,

a grammar for the language; that is, we stipulate a mathematically rigorous

de�nition of the idea of a sentence of this language. Moreover, since we are

only interested in the logical behavior of the chosen logical constants ‘and’,

‘or’, and so on, we choose special symbols (∧,∨ . . . ) for these words only; we

use P,Q, R, . . . indifferently to represent any English sentence whose internal

logical structure we are willing to ignore.3

We go on, then, to study (as always, in a mathematically rigorous way) vari-

ous concepts that apply to the sentences in formal languages. In propositional

2Notes

3Natural languages like English also have a grammar, and the grammar can be studied

using mathematical techniques. But the grammar is much more complicated, and is discovered

rather than stipulated; and natural languages lack abstractions like the sentence letters.


logic, for example, one constructs a mathematically rigorous de�nition of a

tautology (“all Trues in the truth table”), and a rigorous de�nition of a prov-

able formula (e.g., in terms of a system of deduction, using rules of inference,

assumptions, and so on).

Of course, the real goal is to apply the notions of logical consequence and

logical truth to sentences of English and other natural languages. The formal

languages are merely a tool; we need to apply the tool.

1.4 Correctness and applicationTo apply the tools we develop for formal languages, we need to speak of a

formal system as being correct. What does that sort of claim mean?

As we saw, logicians use formal languages and formal structures to study

logical consequence and logical truth. And the range of structures that one

could in principle study is very wide. For example, I could introduce a new

notion of “provability” by saying “in Ted Logic, the following rule may be

used when constructing proofs: if you have P on a line, you may infer ∼P .

The annotation is ‘T’.” I could then go on to investigate the properties of

such a system. Logic can be viewed as a branch of mathematics, and we can

mathematically study any system we like, including a system (like Ted logic) in

which one can “prove” ∼P from P .

But no such formal system would shed light on genuine logical consequence

and genuine logical truth. It would be implausible to claim, for example, that

when we translate an English argument into symbols, the conclusion of the

resulting symbolic argument may be derived in Ted logic from its premises iff

the conclusion of the original English argument is a logical consequence of its

premises.

Thus, the existence of a coherent, speci�able logical system must be dis-

tinguished from its application. When we say that a logical system is correct,we have in mind some application of that system. Here’s an oversimpli�ed

account of one such correctness claim. Suppose we have developed a certain

formal system for constructing proofs in propositional logic. And suppose

we have speci�ed some translation scheme from English into the language

of propositional logic. This translation schema would translate English ‘and’

into the logical ∧ , English ‘or’ into the logical ∨, and so on. Then, the claim

that the formal system gives a correct logic of English ‘and’, ‘or’, etc. might be

taken to be the claim that one English sentence is a logical consequence of


some other English sentences in virtue of ‘and’, ‘or’, etc., iff one can prove the

translation of the former English sentence from the translations of the latter

English sentences in the formal system.

In this book I won’t spend much time on philosophical questions about

which formal systems are correct. My goal is rather to introduce those for-

malisms that are ubiquitous in philosophy, to give you the tools you need to

address such philosophical questions yourself. Still, from time to time, we’ll dip

just a bit into these philosophical questions, in order to motivate our choices

of logical systems to study.

1.5 The nature of logical consequenceThe previous section discussed what it means to say that a formal theory gives a

correct account of logical consequence (as applied to sentences of English and

other natural languages). But what is it for sentences to stand in the relation of

logical consequence? What is logical consequence?

The question here is a philosophical question, as opposed to a mathematical

one. Logicians de�ne various notions concerning sentences of formal languages:

derivability in this or that proof-system, “all trues in the truth table”, and so

on. They thereby stipulatively introduce various formal concepts. These

formal concepts are good insofar as they correctly model logical truth and

logical consequence. But in what do logical truth and logical consequence—the

intuitive concepts, as opposed to the stipulatively introduced concepts—consist?

This is one of the core questions of philosophical logic.

This book is not primarily a book in philosophical logic, so we won’t spend

much time on the question. However, I do want to make clear that the question

is indeed a question. The question is sometimes obscured by the fact that terms

like ‘logical truth’ are often stipulatively de�ned in logic books. This can lead

to the belief that there are no genuine issues concerning these notions. It is also

obscured by the fact that one philosophical theory of these notions—the model-

theoretic one—is so dominant that one can forget that it is a nontrivial theory.

Stipulative de�nitions are of course not things whose truth can be questioned;

but stipulative de�nitions of logical notions are good insofar as the stipulated

notions accurately model the real, intuitive, nonstipulated notions of logical

consequence and logical truth. Further, the stipulated de�nitions generally

concern formal languages, whereas the ultimate goal is an understanding of

correct reasoning of the sort that we actually do, using natural languages.


Let’s focus just on logical consequence. Here is a quick survey of some

competing philosophical accounts of its nature. Probably the most standard

account is the semantic, or model-theoretic one. Intuitively, a logical truth is

“true no matter what”. The model theoretic account is one way of making this

slogan precise. It says that φ is a logical consequence of the sentences in set Γif the formal translation of φ is true in every model (interpretation) in which

the formal translations of the members of Γ are true. This account needs to be

spelled out in various ways. First, “formal translations” are translations into a

formal language; but which formal language? It will be a language that has a

logical constant for each English logical expression. But that raises the question

of which expressions of English are logical expressions. In addition to ‘and’,

‘or’, ‘all’, and so on, are any of the following logical expressions?

necessarily

it will be the case that

most

it is morally wrong that

Further, the notion of translation must be de�ned; further, an appropriate

de�nition of ‘model’ must be chosen.

Similar issues of re�nement confront a second account, the proof-theoreticaccount: φ is a logical consequence of the members of Γ iff the translation of

φ is provable from the translations of the members of Γ. We must decide what

formal language to translate into, and we must decide upon an appropriate

account of provability.

A third view is Quine’s: φ is a logical consequence of the members of Γiff there is no way to (uniformly) substitute new nonlogical expressions for

nonlogical expressions in φ and the members of Γ so that the members of Γbecome true and φ becomes false.

Three other accounts should be mentioned. The �rst account is a modal

one. Say that Γ modally implies φ iff it is not possible for φ to be false while the

members of Γ are true. (What does ‘possible’ mean here? There are many kinds

of possibility one might have in mind: so-called “metaphysical possibility”,

“absolute possibility”, “idealized epistemic possibility”…. Clearly the accept-

ability of the proposal depends on the legitimacy of these notions. We discuss

modality later in the book, beginning in chapter 6.) One might then propose

thatφ is a logical consequence of the members of Γ iff Γmodally impliesφ. (An


intermediate proposal: φ is a logical consequence of the members of Γ iff, invirtue of the forms ofφ and the members of Γ, Γmodally impliesφ. More carefully:

φ is a logical consequence of the members of Γ iff Γ modally implies φ, and

moreover, whenever Γ′ and φ′ result from Γ and φ by (uniform) substitution of

nonlogical expressions, Γ′ modally implies φ′. This is like Quine’s de�nition,

but with modal implication in place of truth-preservation.) Second, there is

a primitivist account, according to which logical consequence is a primitive

notion. Third, there is a pluralist account according to which there is no one

kind of genuine logical consequence. There are, of course, the various con-

cepts proposed by each account, each of which is trying to capture genuine

logical consequence; but in fact there is no further notion of genuine logical

consequence at all; there are only the proposed construals.

As I say, this is not a book on philosophical logic, and so we will not inquire

further into which (if any) of these accounts is correct. We will, rather, focus

exclusively on two kinds of formal proposals for modeling logical consequence

and logical truth: model-theoretic and proof-theoretic proposals.

1.6 Extensions, deviations, variations4

“Standard logic” is what is usually studied in introductory logic courses. It

includes propositional logic (logical constants: ∧,∨,∼,→,↔), and predicate

logic (logical constants: ∀,∃, variables). In this book we’ll consider various

modi�cations of standard logic:

1.6.1 ExtensionsHere we add to standard logic. We add both:

• new symbols

• new cases of logical consequence and logical truth that we can

model

We do this in order to get a better representation of logical consequence. There

is more to logic than that captured by plain old standard logic.

We extended propositional logic, after all, to get predicate logic. You can

do a lot with propositional logic, but you can’t capture the obvious fact that

4See Gamut (1991a, pp. 156-158).


‘Ted is happy’ logically implies ‘someone is happy’ using propositional logic

alone. It was for this reason that we added quanti�ers, variables, predicates,

etc., to propositional logic (added symbols), and added means to deal with

these new symbols in semantics and proof theory (new cases of logical conse-

quence and logical truth). But there is no need to stop with plain old predicate

logic. We will consider adding symbols for identity, function symbols, and

de�nite descriptions to predicate logic, for example, and we’ll add a sign for

“necessarily” when we get to modal logic. And in each case, we’ll introduce

modi�cations to our formal theories that let us account for logical truths and

logical consequences involving the new symbols.

1.6.2 DeviationsHere we change, rather than add. We retain the same symbols from standard

logic, but we alter standard logic’s proof theory and semantics. We therefore

change what we say about the logical consequences and logical truths that

involve the symbols.

Why do this? Perhaps because we think that standard logicians are wrongabout what the right logic for English is. If we want to correctly model logical

consequence in English, therefore, we must construct systems that behave

differently from standard logic.

For example, in the standard semantics for propositional logic, every for-

mula is either true or false. But some have argued that natural language sen-

tences like the following are neither true nor false:

The king of the United States is bald

Sherlock Holmes weighs more than 178 pounds

Bill Clinton is tall

There will be a sea battle tomorrow

If so, perhaps we should abandon the standard semantics for propositional logic

in favor of multi-valued logic, in which formulas are allowed to be neither true

nor false.

1.6.3 VariationsHere we also change standard logic, but we change the notation without

changing the content of logic. We study alternate ways of expressing the same

thing.


For example, in intro logic we show how:

∼(P∧Q)∼P∨∼Q

are two different ways of saying the same thing. We will study other ways of

saying what those two sentences say, including:

P |Q∼∧PQ

In the �rst case, | is a new symbol for “not both”. In the second case (“Polish

notation”), the ∼ and the ∧mean what they mean in standard logic; but instead

of going between the P and the Q, the ∧ goes before P and Q. The value of

this, as we’ll see, is that we no longer will need parentheses.

1.7 Metalogic, metalanguages, and formalizationIn introductory logic, we learned how to use certain logical systems. We learned

how to do truth tables, construct derivations, and so on. But logicians do not

spend much of their time developing systems only to sit around all day doing

derivations in those systems. As soon as a logician develops a new system, he

or she will begin to ask questions about that system. For an analogy, imagine

people who make up games. They might invent a new version of chess. Now,

they might spend some time actually playing the new game. But if they were

like logicians, they would soon get bored with this and start asking questions

about the game, such as: “is the average length of this new game longer than the

average length of a game of standard chess?”. “Is there any strategy one could

pursue which will guarantee a victory?” Analogously, logicians ask questions

like: what things can be proved in such and such a system? Can you prove the

same things in this system as in system X? Proving things about logical systems

is part of “meta-logic”, which is an important part of logic.

One particularly important question of metalogic is that of soundness and

completeness. Standard textbooks introduce a pair of methods for characterizing

logical truth for the formulas of propositional logic. One is semantic: a formula

is a semantic logical truth iff the truth table for that formula has all “trues” in its

�nal column. Another is proof-theoretic: a sentence is a proof-theoretic logical

truth iff there exists a derivation of it (from no premises), where a derivation


is then appropriately de�ned. (Think: introduction- and elimination- rules,

conditional and indirect proof, and so on.) The question of soundness and

completeness is: how do these two methods for characterizing logical truth

relate to each other? The question is answered, in the case of propositional

logic, by the following metalogical results, which are proved in standard books

on metalogic:

Soundness of propositional logic: Any proof-theoretic logical

truth is a semantic logical truth

Completeness of propositional logic: Any semantic logical truth

is a proof-theoretic logical truth

These are really interesting claims! They show that the method of truth tables

and the method of constructing derivations amount to the same thing, as

applied to symbolic formulas of propositional logic. One can establish similar

results for standard predicate logic.

A couple remarks about proving things in metalogic.

First: what do we mean by “proving”? We do not mean: constructing a

derivation in the logical system we’re investigating. We’re trying to construct a

proof about the system. We do this in English, and we do it with informal (though

rigorous!) reasoning of the sort one would encounter in a mathematics book.

Logicians often distinguish the “object language” from the “metalanguage”.

The object language is the language that’s being studied—the language of

propositional logic, for example. Sentences of this object language look like

this:

P∧Q∼(P∨Q)↔R

The metalanguage is the language we use to talk about the object language.

In the case of the present book, the metalanguage is English. Here are some

example sentences of the metalanguage:

‘P∧Q’ is a sentence with three symbols, one of which is a logical

constant

Every sentence of propositional logic has the same number of left

parentheses as right parentheses


If there exists a derivation of a formula, then its truth table contains

all “trues” in its �nal column (i.e., soundness)

Thus, we formulate metalogical claims in the metalanguage, and our proofs in

metalogic take place in the metalanguage.

Second: to get anywhere in metalogic, we will have to get picky about a few

things about which one can afford to be lax in introductory logic. Let’s look at

soundness, for instance. To be able to prove this, in a mathematically rigorous

way, we’ll need to have the terms in it de�ned very carefully. In particular, we’ll

need to say exactly what we mean by ‘sentence of propositional logic’, ‘truth

tables’, and ‘derived’. De�ning these terms precisely (another thing we’ll do

using English, the metalanguage!) is known as formalizing logic. Our �rst task

will be to formalize propositional logic.

1.8 Set theory5

As mentioned above, modern logic uses mathematical techniques to study

formal languages. The mathematical techniques in question are those of “set

theory”. Only the most elementary set-theoretic concepts and assumptions will

be needed, and you are probably already familiar with them; but nevertheless,

here is a brief overview.

Sets have members. Consider, for example, the set, N, of natural numbers.

Each natural number is a member of N: 1 is a member of N, 2 is a member of N,

and so on. We use the expression “∈” for this relationship of membership; thus,

we can say: 1 ∈N, 2 ∈N, and so on. We often name a set by putting names of

its members between braces: “{1,2,3,4, . . .}” is another name of N.

Sets are not limited to sets of mathematical entities; anything can be a

member of a set. Thus, we may speak of the set of people, the set of cities,

or—to draw nearer to our intended purpose—the set of sentences in a given

language.

There is also the empty set, ∅. This is the one set with no members. That

is, for each object u, u is not a member of ∅ (i.e.: for each u, u /∈∅.)

Though the notion of a set is an intuitive one, it is deeply perplexing. This

can be seen by re�ecting on the Russell Paradox, discovered by Bertrand Russell,

the great philosopher and mathematician. Let us call R the set of all and only

5Supplementary reading: the beginning of Enderton (1977)


those sets that are not members of themselves. For short, R is the set of non-

self-members. Russell asks the following question: is R a member of itself?

There are two possibilities:

i) R /∈ R. Thus, R is a non-self-member. But R was said to

be the set of all non-self-members, and so we’d have R ∈ R.

Contradiction.

ii) R ∈ R. So R is not a non-self-member. R, by de�nition,

contains only non-self-members. So R /∈ R. Contradiction.

Thus, each possibility leads to a contradiction. But there are no remaining

possibilities—either R is a member of itself or it isn’t! So it looks like the very

idea of sets is paradoxical.

The modern discipline of axiomatic set theory arose in part to develop a

notion of sets that isn’t subject to this sort of paradox. This is done by imposing

rigid restrictions on when a given “condition” picks out a set. In the example

above, the condition “is a non-self-member” will be ruled out—there’s no set

of all and only the things satisfying this condition. The details of set theory are

beyond the scope of this course; for our purposes, we’ll help ourselves to the

existence of sets, and not worry about exactly what sets are, or how the Russell

paradox is avoided.

Various other useful set-theoretic notions can be de�ned in terms of the

notion of membership. We say that A is a subset of B (“A⊆ B”) when every

member of A is a member of B . We say that the intersection of A and B (“A∩B”)

is the set that contains all and only those things that are in both A and B , and

that the union of A and B (“A∪B”) is the set containing all and only those things

that are members of either A or B .

Suppose we want to refer to the set of the so-and-sos—that is, the set

containing all and only objects, u, that satisfy the condition “so-and-so”. We’ll

do this with the term “{u: u is a so-and-so}”. Thus, we could write: “N= {u :u is a natural number}”. And we could restate the de�nitions of ∩ and ∪ from

the previous paragraph as follows:

A∩B =df{u : u ∈A and u ∈ B}

A∪B =df{u : u ∈A or u ∈ B}

Sets have members, but they don’t contain them in any particular order. For

example, the set containing me and Bill Clinton doesn’t have a “�rst” member.

This is re�ected in the fact that “{Ted, Clinton}” and “{Clinton, Ted}” are


two different names for the same set—the set containing just Clinton and Ted.

But sometimes we need to talk about a set-like thing containing Clinton and

Ted, but in a certain order. For this purpose, logicians use ordered sets. Two-

membered ordered sets are called ordered pairs. To name the ordered pair of

Clinton and Ted, we use: “⟨Clinton, Ted⟩”. Here, the order is signi�cant, for

⟨Clinton, Ted⟩ and ⟨Ted, Clinton⟩ are not the same thing. The three-membered

ordered set of u, v, and w (in that order) is written: ⟨u, v, w⟩; and similarly for

ordered sets of any �nite size. A n-membered ordered set is called an n-tuple.Let’s even allow 1-tuples: let’s de�ne the 1-tuple ⟨u⟩ as being the object u itself.

In addition to sets, and ordered sets, we’ll need a further related concept:

that of a function. A function is a rule that “takes in” an object or objects,

and “spits out” a further object. For example, the addition function is a rule

that takes in two numbers, and spits out their sum. As with sets and ordered

sets, functions are not limited to mathematical entities: they can “take in” and

“spit out” any objects whatsoever. We can speak of the father-of function, for

example, which is a rule that takes in a person, and spits out the father of that

person. And later in this book we will be considering functions that take in

and spit out linguistic entities: sentences and parts of sentences from formal

languages.

Each function has a �xed number of “places”: a �xed number of objects

it must take in before it is ready to spit out something. You need to give

the addition function two arguments (numbers) in order to get it to spit out

something, so it is called a two-place function. You only need to give the father-

of function one object, on the other hand, to get it to spit out something, so it

is a one-place function.

The objects that the function takes in are called its arguments, and the object

it spits out is called its value. Suppose f is an n-place function, and u1 . . . un are

n of its arguments; one then writes “ f (u1 . . . un)” for the value of function f as

applied to arguments u1 . . . un. f (u1 . . . un) is the object that f spits out, if you

feed it u1 . . . un. For example, where f is the father-of function, since Ron is

my father, we can write: f (Ted) =Ron; and, where a is the addition function,

we can write: a(2,3) = 5.

There’s a trick for “reducing” talk of both ordered pairs and functions to

talk of sets. One �rst de�nes ⟨u, v⟩ as the set {{u},{u, v}}; one de�nes ⟨u, v, w⟩as ⟨u, ⟨v, w⟩⟩, and similarly for n-membered ordered sets, for each positive

integer n. And, �nally, one de�nes an n-place function as a set, f , of n+ 1-

tuples obeying the constraint that if ⟨u1, . . . , un, v⟩ and ⟨u1, . . . , un, w⟩ are both

members of f , then v = w; f (u1, . . . , un) is then de�ned as the object, v, such


that ⟨u1, . . . , un, v⟩ ∈ f . Thus, ordered sets and functions are de�ned as certain

sorts of sets. The trick of the de�nition of ordered pairs is that we put the

set together in such a way that we can look at the set and tell what the �rstmember of the ordered pair is: it’s the one that “appears twice”. Similarly, the

trick of the de�nition of a function is that we can take any arguments to the

function, look at the set that is identi�ed with the function, and �gure out

what value the function spits out for those arguments. But the technicalities of

these reductions won’t matter for us; I’ll just feel free to speak of ordered pairs,

triples, functions, etc., without de�ning them as sets.

Chapter 2

Propositional Logic

We begin with the simplest logic commonly studied: propositional logic.

Despite its simplicity, it has great power and beauty.

2.1 Grammar of PLModern logic has made great strides by treating the language of logic as a

mathematical object. To do so, grammar needs to be developed rigorously.

(Our study of a new logical system will always begin with grammar.)

If all you want to do is understand the language of logic informally, and be

able to use it effectively, you don’t really need to get so careful about grammar.

For even if you haven’t ever seen the grammar of propositional logic formalized,

you can recognize that things like this make sense:

P→QR∧(∼S↔P )

Whereas things like this do not:

→PQR∼(P∼Q∼(∨

But to make any headway in metalogic, we will need more than an intuitive

understanding of what makes sense and what does not; we will need a precise

de�nition that has the consequence that only the strings of symbols in the �rst

group “make sense”.

18

CHAPTER 2. PROPOSITIONAL LOGIC 19

Grammatical formulas (i.e., ones that “make sense”) are called well-formedformulas, or “wffs” for short. We de�ne these by �rst carefully de�ning exactly

which symbols are allowed to occur in wffs (the “primitive vocabulary”), and

second, carefully de�ning exactly which strings of these symbols count as wffs:

Primitive vocabulary:

Sentence letters: P,Q, R . . . , with or without numerical

subscripts

Connectives: →,∼Parentheses: (, )

De�nition of wff:

i) Sentence letters are wffs

ii) If φ and ψ are wffs then φ→ψ and ∼φ are also wffs

iii) Only strings than can be shown to be wffs using i) and ii) are

wffs

Notice an interesting feature of this de�nition: the very expression we

are trying to de�ne, ‘wff’, appears on the right hand side of clause ii) of the

de�nition. In a sense, we are using the expression ‘wff’ in its own de�nition.

Is that “circular”? Not in any objectionable way. This de�nition is what is

called a “recursive” de�nition, and recursive de�nitions are legitimate despite

this sort of circularity. The reason is that clause ii) de�nes the notion of a

wff for certain complex expressions (namely, ∼φ and φ→ψ)) in terms of the

notion of a wff as applied to smaller expressions (φ and ψ). These smaller

expressions may themselves be complex, and therefore may have their statuses

as wffs determined, via clause ii), in terms of yet smaller expressions, and so on.

But eventually this procedure will lead us to clause i), not clause ii). And clause

i) is not circular: in that clause, we do not appeal to the notion of a wff in its

own de�nition; we rather say directly that sentence letters are wffs. Recursive

de�nitions always “bottom out” in this way; they always include a clause (called

the “base” clause) like i).

Think of this procedure in reverse: we begin with the smallest wffs (sentence

letters), and build up complex wffs using clause ii). Example: we can use clauses

i) and ii) to show that the expression (∼P→(P→Q)) is a wff:


• P is a wff (clause i))

• so, ∼P is a wff (clause ii))

• Q is a wff (clause i))

• so, since P and Q are both wffs, (P→Q) is also a wff (clause

ii))

• so, since ∼P and (P→Q) are both wffs, (∼P→(P→Q)) is also

a wff (clause ii))

What’s the point of clause iii)? Clauses i) and ii) provide only suf�cient

conditions for being a wff, and therefore do not on their own exclude non-

sense combinations of primitive vocabulary like P∼Q∼R, or even strings like

(P∨147)→⊕ that include disallowed symbols. Clause iii) rules these strings out,

since there is no way to build up either of these strings from clauses i) and ii),

in the way that we built up the wff (∼P→(P→Q)).What happened to ∧, ∨, and↔? Our de�nition of a wff mentions only

→ and ∼; it therefore counts expressions like P∧Q, P∨Q, and P↔Q as notbeing wffs. Answer: we can de�ne the ∧, ∨, and↔ in terms of ∼ and→:

φ∧ψ =df∼(φ→∼ψ)

φ∨ψ =df∼φ→ψ

φ↔ψ =df

(φ→ψ)∧(ψ→φ)

(The symbol “=df

” means “is de�ned as meaning”.) So, whenever we subse-

quently write down an expression that includes one of the de�ned connectives,

we can regard it as being short for an expression that includes only the of�cial

primitive connectives,∼ and→. (We will show below that the above de�nitions

are good ones; in short, they are good because they generate the correct truth

tables for ∧, ∨, and↔.)

Our choice to begin with→ and∼ as our primitive connectives was arbitrary.

We could have started with ∼ and ∧, and de�ned the others as follows:

φ∨ψ =df∼(∼φ∧∼ψ)

φ→ψ =df∼(φ∧∼ψ)

φ↔ψ =df(φ→ψ)∧(ψ→φ)

And other alternate choices are possible. We’ll talk about this later.

So: → and ∼ are our primitive connectives; the others are de�ned. Why

do we choose only a small number of primitive connectives? Because, as we

will see, it makes meta-proofs easier.


2.2 The semantic approach to logicIn the next section we will introduce a “semantics” for propositional logic. A

semantics for a language is a way of assigning meanings to words and sentences

of that language. For us, the central notion of meaning will be that of truth.

Roughly speaking, our approach will be to de�ne, for each wff of propositional

logic, the circumstances in which it is true.

Philosophers disagree over how to understand the notion of meaning in

general. But the idea that the meaning of a sentence has something to do with

truth-conditions is hard to deny, and at any rate has currency within logic. On

this approach, one explains the meaning of a sentence by showing how that

sentence depends for its truth or falsity on the way the world is.

We will provide a truth-conditional semantics for a symbolic (formal) lan-

guage in two stages. First, we will de�ne mathematical models of the various

con�gurations the world could be in. Second, we will de�ne the conditions

under which a sentence of the symbolic language is true in one of these mathe-

matical con�gurations.

These de�nitions will have two main bene�ts. First, they will illuminate

meaning. In logic, the symbols of symbolic languages are typically intended to

represent bits of natural language. The PL connectives, for example, represent

‘and’, ‘or’, and so on. If the de�nitions are well-constructed, then the ways

in which the con�gurations render symbolic sentences true and false will be

parallel to the ways in which the real world renders corresponding natural

language sentences true and false. The de�nitions will therefore shed light on

the meanings of the natural language sentences represented by our symbolic

language. Second, our de�nitions will allow us to construct a precise theory (of

the semantic/model-theoretic variety) of logical consequence and logical truth.

The semantic conception is a way of making precise the idea that a logical truth

is a sentence that is “true no matter what”, and the idea that one sentence is a

logical consequence of some other sentences iff there is “no way” for the latter

sentences to be true without the former sentence being true. We will use our

de�nitions to make these rough statements more precise: we will say that one

formula is a logical consequence of others iff there is no con�guration in which

the latter formulas are true but the former is not, and that a formula is a logical

truth iff it is true in all con�gurations.


2.3 Semantics of PLOur semantics for propositional logic is really just truth tables, only presented

a little more carefully than in introductory logic books. Let’s de�ne the notion

of a propositional logic interpretation function, or PL-interpretation:

A PL-interpretation is is a function I , that assigns either 1 or 0 to

every sentence letter of PL

Think of 0 and 1 as truth values; thus a PL-interpretation assigns truth values

to sentence letters. Instead of saying “let P be false, and Q be true”, we can

say: let I be an interpretation such that I (P ) = 0 and I (Q) = 1.

Once we settle what truth values a given interpretation assigns to the sen-

tence letters, the truth values of complex sentences containing those sentence

letters are thereby �xed. The usual, informal, method for showing exactly how

those truth values are �xed is by giving truth tables. The standard truth tables

for the→ and ∼ are the following:

φ → ψ1 1 1

1 0 0

0 0 1

0 0 0

∼ φ1 0

0 1

What we will do, instead, is write out a formal de�nition of a function—the

valuation function—that assigns truth values to complex sentences as a function

of the truth values of their sentence letters—i.e., as a function of a given PL

assignment I . But the idea is the same as the truth tables: truth tables are

really just pictures of the de�nition of a valuation function:

For any PL-interpretation,I , the PL-valuation forI , VI , is de�ned

as the function that assigns to each wff either 1 or 0, and which is

such that, for any wffs φ and ψ,

i) if φ is a sentence letter then VI (φ) =I (φ)ii) VI (φ→ψ) = 1 iff either VI (φ) = 0 or VI (ψ) = 1

iii) VI (∼φ) = 1 iff VI (φ) = 0


We have another recursive de�nition: the valuation function’s values for

complex formulas are determined by its values for smaller formulas; and this

procedure bottoms out in the values for sentence letters, which are determined

directly by the interpretation function I .

Notice also that in the de�nition of a valuation function I use the English

logical connectives ‘either…or’, and ‘iff ’. I used these English connectives

rather than the logical connectives ∨ and↔, because at that point I was notwriting down wffs of the language of study (in this case, the language of propo-

sitional logic). I was rather using sentences of English—our metalanguage, the

informal language we’re using to discuss the formal language of propositional

logic—to construct my de�nition of the valuation function. My de�nition

needed to employ the logical notions of disjunction and bi-implication, the

English words for which are ‘either…or’ and ‘iff’.

One might again worry that something circular is going on. We de�ned the

symbols for disjunction and bi-implication, ∨ and↔, in terms of ∼ and→ in

section 2.1, and now we’ve de�ned the valuation function in terms of disjunction

and bi-implication. So haven’t we given a circular de�nition of disjunction and

bi-implication? No. When we de�ne the valuation function, we’re not trying to

de�ne logical concepts such as negation, conjunction, disjunction, implication,

and bi-implication, and so on, at all. A reductive de�nition of these very basic

concepts is probably impossible (though one can de�ne some of them in terms

of the others). What we are doing is i) starting with the assumption that we

already understand the logical concepts, and then ii) using those notions to

provide a formalized semantics for a logical language. This can be put in terms

of object- and meta-language: we use metalanguage connectives, such as ‘iff’

and ‘or’, which we simply take ourselves to understand, to provide a semantics

for the object language connectives ∼,→, etc.

Back to the de�nition of the valuation function. The de�nition applies

only to of�cial wffs, which can contain only the primitive connectives→ and

∼. But sentences containing ∧, ∨, and↔ are abbreviations for of�cial wffs,

and therefore they too are governed by the de�nition. In fact, given the

abbreviations de�ned in section 2.1, one can show that the de�nition assigns

the intuitively correct truth values to sentences containing ∧, ∨, and↔; one

can show that for any PL-interpretation I , and any wffs ψ and χ ,

VI (ψ∧χ ) = 1 iff VI (ψ) = 1 and VI (χ ) = 1VI (ψ∨χ ) = 1 iff either VI (ψ) = 1 or VI (χ ) = 1VI (ψ↔χ ) = 1 iff VI (ψ) =VI (χ )


I’ll show that the �rst statement is true here; the others are exercises for the

reader. I’ll write out my proof in excessive detail, to make it clear exactly how

the reasoning works:

Let ψ and χ be any wffs. The expression ψ∧χ is an abbreviation

for the expression∼(ψ→∼χ ). So we want to show that, for any PL-

interpretation I , VI (∼(ψ→∼χ )) = 1 iff VI (ψ) = 1 and VI (χ ) = 1.

Now, in order to show that a statement α holds iff a statement

β holds, we must �rst show that if α holds, then β holds (the

“forwards⇒ direction”); then we must show that if β holds then αholds (the “backwards⇐direction”):

⇒: First assume that VI (∼(ψ→∼χ )) = 1. Then, by de�nition

of the valuation function, clause for ∼, VI (ψ→∼χ ) = 0. So1,

VI (ψ→∼χ ) is not 1. But then, by the clause in the de�nition of VIfor the→, we know that it’s not the case that: either VI (ψ)=0 or

VI (∼χ )=1. That is: VI (ψ) = 1 and VI (∼χ ) = 0. From the latter, by

the clause for ∼, we know that VI (χ ) = 1. That’s what we wanted

to show—that VI (ψ) = 1 and VI (χ ) = 1.

⇐: This is sort of like undoing the previous half. Suppose that

VI (ψ) = 1 and VI (χ ) = 1. Since VI (χ ) = 1, by the clause for ∼,

VI (∼χ ) = 0; but now since VI (ψ) = 1 and VI (∼χ ) = 0, by the

clause for→ we know that VI (ψ→∼χ ) = 0; then by the clause for

∼, we know that VI (∼(ψ→∼χ )) = 1, which is what we were trying

to show.

Let’s re�ect on what we’ve done so far. We have de�ned the notion of

a PL-interpretation, which assigns 1s and 0s to sentence letters of the sym-

bolic language of propositional logic. And we have also de�ned, for any PL-

interpretation, a corresponding valuation function, which extends the inter-

pretation’s assignment of 1s and 0s to complex wffs of PL. Note that we have

been informally speaking of these assignments as assignments of truth values.That’s because the assignments of 1s and 0s accurately models the truth values

of statements in English that are represented in the obvious way by PL-wffs.

1The careful reader will note that here (and henceforth), I treat “VI (α)=0” and “VI (α) is

not 1” interchangeably (for any wff α). (Similarly for “VI (α)=1” and “VI (α) is not 0”.) This is

justi�ed as follows. First, if VI (α) is 0, then it can’t also be that VI (α) is 1—VI was stipulated

to be a function. Second, since it was stipulated that VI assigns either 0 or 1 to each wff, if

VI (α) is not 1, then VI (α) must be 0.


For example, the ∼ of propositional logic is supposed to model the English

phrase ‘it is not the case that’. Accordingly, just as an English sentence “It is

not the case that φ” is true iff φ is false, one of our valuation functions assigns

1 to ∼φ iff it assigns 0 to φ.

Semantics in logic, recall, generally de�nes two things: con�gurations and

truth-in-a-con�guration. In the propositional logic semantics we have laid

out, the con�gurations are the interpretation functions, and the valuation

function de�nes truth-in-a-con�guration. Each interpretation function gives

a complete assignment of truth values to the sentence letters. Thus, insofar

as the sentence letters are concerned, an interpretation function completely

speci�es a possible con�guration of the world. And for any interpretation

function, its corresponding valuation function speci�es, for each complex wff,

what truth value that wff has in that interpretation. Thus, for each wff (φ) and

each con�guration (I ), we have speci�ed the truth value of that wff in that

con�guration (VI (φ)).Onward. We are now in a position to de�ne the semantic versions of the

notions of logical truth and logical consequence for PL. The semantic notion

of a logical truth is that of a valid formula:

φ is valid =df

for every PL-interpretation, I , VI (φ) = 1

We write “� φ” for “φ is valid”. (The valid formulas of propositional logic

are also called tautologies.) As for logical consequence, the semantic version of

this notion is that of a single formula’s being a semantic consequence of a set of

formulas:

φ is a semantic consequence of the wffs in set Γ =df

for every PL-

interpretation, I , IF for each γ in Γ, VI (γ ) = 1, THEN VI (φ) = 1

That is, φ is a semantic consequence of Γ iff φ is true whenever each member

of Γ is true. We write “Γ �φ” for “φ is a semantic consequence of Γ”. (When

Γ has just one member, ψ, then instead of writing {ψ} �φ, let’s write simply:

ψ �φ.)

We will be discussing a number of different semantical systems throughout

this book, each with its own de�nition of validity and semantic consequence.

What we have de�ned here is validity and semantic consequence in just one sys-

tem: PL. So strictly, we should speak of PL-validity and PL-semantic-consequence,and we should subscript our symbol � thus: �

PL. But let’s omit the subscript

where there is no danger of ambiguity.


A parenthetical remark: now we can see the importance for setting up the

grammar for our system according to precise rules. If we hadn’t, the de�nition

of ‘truth value’ given here would have been impossible. In this de�nition we

de�ned truth values of complicated formulas based on their form. For example,

if a formula has the form (φ→ψ), then we assigned it an appropriate truth value

based on the truth values of φ and ψ. But suppose we had a formula in our

language that looked as follows:

P→P→P

and suppose that P has truth value 0. What is the truth value of the whole? We

can’t tell, because of the missing parentheses. For if the parentheses look like

this:

(P→P )→P

then the truth value is 0, whereas if the parentheses look like this:

P→(P→P )

then it is 1. Certain kinds of grammatical ambiguity, then, make it impossible

to assign truth values. We solve this problem in logic by pronouncing the

original string “P→P→P” as ill-formed; it is missing parentheses. Thus, the

precise rules of grammar assure us that when it comes time to do semantics, we

are able to assign semantic values (in this case, truth values) in an unambiguous

way.

Notice also a fact about validity in propositional logic: it is mechanically

“decidable”—a computer program could be written that is capable of telling,

for any given formula, whether or not that formula is valid. The program

would simply construct a complete truth table for the formula in question.

We can observe that this is possible by noting the following: every formula

contains a �nite number N of sentence letters, and so for any formula, there

are only a �nite number of different “cases” one needs to check—namely, the

2N

permutations of truth values for the contained sentence letters. But given

any assignment of truth values to the sentence letters of a formula, it’s clearly a

perfectly mechanical procedure to compute the truth value the formula takes

for those truth values—simply apply the rules for the ∼ and→ repeatedly.

It’s worth being very clear about two assumptions in this proof (which, by

the way, is our �rst bit of metatheory—our �rst proof about a logical system).

They are: that every formula has a �nite number of sentence letters, and that


the truth values of sentence letters not contained in a formula do not affect

the truth value of the formula. We need the latter assumption to be sure that

we only need to check a �nite number of cases—namely, the permutations

of truth values of the contained sentence letters—to see whether a formula is

valid. After all, there are in�nitely many interpretation functions (since there

are in�nitely many sentence letters in the language of PL), and a valid formula

must be true in each one.

These two assumptions are obviously true, but it would be good to prove

them. I’ll prove the �rst assumption here (the second may be proved by a

similar method), and take this opportunity to introduce an important technique

for metalanguage proofs: proof by induction.

Proof that every wff contains a �nite number of sentence letters: In this

sort of proof by induction, we’re trying to prove a statement of

the form: every wff has property P. The property P in this case is

having a �nite number of different sentence letters. In order to do this,

we must show two separate statements:

base case: we show that every atomic sentence has the property. This

is obvious—atomic sentences are just sentence letters, and each of

them contains one sentence letter, and thus �nitely many different

sentence letters.

induction step: we begin by assuming that if formulas φ and ψ have

the property, then so will the complex formulas one can form from

φ and ψ by the rules of formation, namely ∼φ and φ→ψ. So, we

assume that φ and ψ have �nitely many different sentence letters;

and we show that the same must hold for ∼φ and φ→ψ. That’s

obvious: ∼φ has as many different sentence letters as does φ; since

φ, by assumption, has only �nitely many, then so does ∼φ. As for

φ→ψ, by hypothesis,φ andψ have �nitely many different sentence

letters, and so φ→ψ has, at most, n +m sentence letters, where

n and m are the number of different sentence letters in φ and ψ,

respectively.

We’ve shown that every atomic formula has the property having a�nite number of different sentence letters; and we’ve shown that the

property is inherited by complex formulas built according to the

recursion rules. But every wff is either atomic, or built from atomics


by a �nite series of applications of the recursion rules. Therefore,

by induction, every wff has the property. QED.

2.4 Natural deduction in propositional logicWe have investigated a semantic conception of the notions of logical truth and

logical consequence. An alternate conception is proof-theoretic, in which the

central conception is that of proof. On this conception, logical consequence

means “provable from”, and a logical truth is a sentence that can be proved

starting from no premises at all. A “proof” procedure, informally, is a method

of reasoning one’s way, step by step, according to mechanical rules, from some

premises to a conclusion. This all is, of course, informal; we must now make it

precise.

One method for characterizing proof is called the method of natural de-duction. Any system in which one has assumptions for “conditional proof”,

assumptions for “indirect derivation”, etc. is a system of natural deduction.

This is the usual method in introductory logic books. Proofs in these systems

often look like this:

1 P→(Q→R)

2 P∧Q

3 P 2, ∧E

4 Q 2, ∧E

5 Q→R 1, 3→E

6 R 4, 5→E

7 (P∧Q)→R 2-6,→I

or like this:


1.

2.

3.

4.

5.

6.

7.

8.

P→(Q→R)show (P∧Q)→R

P∧Qshow R

PQQ→RR

Pr.

CD

As.

DD

3, ∧E

3, ∧E

1, 5→E

6, 7→E

We will implement natural deduction a little differently here, in order to reveal

what is really going on. Our derivations will therefore look a little different

from the derivations familiar from introductory books. Our version of the

above derivation will look like this:

1. P→(Q→R) ` P→(Q→R) RA

2. P∧Q ` P∧Q RA

3. P∧Q ` P 2, ∧E

4. P∧Q `Q 2, ∧E

5. P→(Q→R), P∧Q `Q→R 1,3→E

6. P→(Q→R), P∧Q ` R 4,5→E

7. P→(Q→R) ` (P∧Q)→R 5,→I

It looks different, but the underlying idea is nevertheless the same.

2.4.1 SequentsNatural deduction systems model the kind of reasoning one employs in everyday

life. How does that reasoning work? In its simplest form, one reasons in a

step-by-step fashion from premises to a conclusion, each step being sanctioned

by a rule of inference. For example, suppose that one already knows the premise

that P∧(P→Q) is true. One can then reason one’s way to the conclusion that

Q is also true, as follows:

1. P∧(P→Q) premise

2. P from line 1

3. P→Q from line 1

4. Q from lines 2 and 3


In this kind of proof, each step is a tiny, indisputably correct, logical inference.

Consider the moves from 1 to 2 and from 1 to 3, for example. These are

indisputably correct because a conjunctive statement clearly logically implies

either of its conjuncts. Likewise for the move from 2 and 3 to 4: it is clear

that a conditional statement, plus its antecedent, together imply its consequent.

Natural deduction systems consist in part of simple general principles like these

(“a conjunctive statement logically implies either of its conjuncts”); they are

known as rules of inference.In addition to rules of inference, ordinary reasoning employs a further

technique: the use of assumptions. In order to establish a conditional claim “if Pthen Q”, one would ordinarily i) assume P , ii) reason one’s way to Q, and then

iii) on that basis conclude that the conditional claim “if P then Q” is true. Once

P ’s assumption is shown to lead to Q, the conditional claim “if P then Q” may

be concluded. Another example: to establish a claim of the form “not-P”, one

would ordinarily i) assume P , ii) reason one’s way to a contradiction, and iii)

on that basis conclude that “not-P” is true. Once P ’s assumption is shown to

lead to a contradiction, “not-P” may be concluded. The �rst sort of reasoning

is called conditional proof, the second, reductio ad absurdum.

When one reasons with assumptions, one writes down statements that one

does not know to be true. When you write down P as an assumption, with the

goal of proving the conditional “if P then Q”, you do not know P to be true.

You’re merely assuming P for the sake of establishing the conditional “if Pthen Q”. Outside the context of this proof, the assumption need not hold; once

you’ve reasoned your way to Q on the basis of the assumption of P , and so

concluded that the conditional “if P then Q” is true, you stop assuming P . To

model this sort of reasoning formally, we need a way to keep track of how the

conclusions we establish depend on the assumptions we have made. Natural

deduction systems in introductory textbooks tend to do this geometrically (by

placement on the page), with special markers (e.g., ‘show’), and by drawing

lines or boxes around parts of the proof once the assumptions that led to those

parts are no longer operative. We will do it differently: we will keep track of the

dependence of conclusions on assumptions by writing down explicitly, for each

conclusion, which assumptions it depends on. We will do this by constructing

our derivations out of sequents.A sequent looks like this:

Γ `φ


Γ is a set of formulas, called the premises of the sequent. φ is a single formula,

called the conclusion of the sequent. “`” is a sign that goes between the sequent’s

premises and its conclusion, to indicate that the whole thing is a sequent. We

will introduce sequent proofs below; and when you write down the sequent

Γ ` φ in one of them, the idea is that φ is an established conclusion, but it

was established by making the assumptions in Γ. Take away those assumptions,

and φ may no longer be established. In fact, one may think of a sequent as

“meaning” that its conclusion is a logical consequence of its premises.

Thus, our proofs will be proofs of sequents. It’s a bit weird at �rst to think in

terms of proving sequents, rather than formulas, since each sequent itself asserts

a relation of logical consequence between its premises and its conclusion, but

the idea nevertheless makes sense. Let’s introduce an informal notion of logicalcorrectness for sequents: the sequent Γ `φ is logically correct if the formula φis a logical consequence of the formulas in Γ. Thus, one is entitled to conclude

the conclusion of a logically correct sequent from its premises. The idea, then,

of constructing a sequent proof of a sequent is to show that that sequent is

logically correct—to show, that is, that its consequent is a logical consequence

of its premises.

From our investigation of the semantics of propositional logic, we already

have the makings of a semantic criterion for when a sequent is logically correct:

the sequent Γ ` φ is logically correct iff φ is a semantic consequence of Γ.

What we will be doing in this section is giving a new, proof-theoretic, criterion

for the logical correctness of sequents.

2.4.2 RulesThe �rst step in developing our system is to write down sequent rules. A sequent

rule is a permission to move from certain sequents to certain other sequents. Our

goal is to construct rules with the following feature: if the “from” sequents are

all logically correct sequents, then any of the “to” sequents will be guaranteed

to be a logically correct sequent. Call such sequent rules “logical-correctness

preserving”.

Consider, as an example, the �rst rule of our system “∧ introduction”, or

“∧I” for short. We picture this sequent rule thus:

Γ `φ ∆ `ψΓ,∆ `φ∧ψ

∧I


Above the line go the “from” sequents; below the line go the “to”-sequents.

(The comma between Γ and ∆ in the “to” sequent simply means that the

premises of this sequent are all the members of Γ plus all the members of ∆.

Strictly speaking it would be more correct to write this in set-theoretic notation

as: Γ∪∆ `φ∧ψ.) Thus, ∧I permits us to move from the sequents Γ `φ and

∆ `ψ to the sequent Γ,∆ `φ∧ψ. For any sequent rule, we say that any of the

“to” sequents (Γ,∆ `φ∧ψ in this case) follows from the “from” sequents (in this

case Γ `φ and ∆ `ψ) via the rule.

It seems intuitively clear that ∧I preserves logical correctness. For if some

assumptions Γ logically imply φ, and some assumptions ∆ logically imply ψ,

then (sinceφ∧ψ intuitively follows fromφ andψ taken together) the conclusion

φ∧ψ should indeed logically follow from all the assumptions together, the ones

in Γ and the ones in ∆.

Our next sequent rule is ∧E:

Γ `φ∧ψΓ `φ Γ `ψ

∧E

This lets one move from the sequent Γ `φ∧ψ to either the sequent Γ `φ or

the sequent Γ `ψ (or both). This, too, appears to preserve logical correctness.

If the members of Γ imply the conjunction φ∧ψ, then (since φ∧ψ intuitively

implies both φ and ψ individually) it must be that the members of Γ imply φ,

and they must also imply ψ.

The rule ∧I is known as an introduction rule for ∧, since it allows us to move

to a sequent whose major connective is the ∧. Likewise, the rule ∧E is known

as an elimination rule for ∧, since it allows us to move from a sequent whose

major connective is the ∧. In fact our sequent system contains introduction and

elimination rules for the other connectives as well: ∼, ∨, and→ (let’s forget

the↔ here.) We’ll present those rules in turn.

First ∨I and ∨E:

Γ `φΓ `φ∨ψ Γ `ψ∨φ

∨I

Γ `φ∨ψ ∆1,φ ` χ ∆2,ψ ` χΓ,∆1,∆2 ` χ

∨E

Let’s think about what∨E means. Remember the intuitive meaning of a sequent:

its conclusion is a logical consequence of its premise. Another (related) way to

think of it is that Γ `φ means that one can establish that φ if one assumes the

members of Γ. So, if the sequent Γ `φ∨ψ is logically correct, that means we’ve

got the disjunction φ∨ψ, assuming the formulas in Γ. Now, suppose we can


reason to a new formula χ , assuming φ, plus perhaps some other assumptions

∆1. And suppose we can also reason to χ from ψ, plus perhaps some other

assumptions∆2. Then, since either φ or ψ (plus the assumptions in∆1 and∆2)

leads to χ , and we know thatφ∨ψ is true (conditional on the assumptions in Γ),

we ought to be able to infer χ itself, assuming the assumptions we needed along

the way (∆1 and ∆2), plus the assumptions we needed to get φ∨ψ, namely, Γ.

Next, we have double negation:

Γ `φΓ `∼∼φ

Γ `∼∼φΓ `φ

DN

In connection with negation, we also have the rule of reductio ad absurdum:

Γ,φ `ψ∧∼ψΓ `∼φ

RAA

That is, if φ (along with perhaps some other assumptions, Γ) leads to a contra-

diction, we can conclude that ∼φ is true (given the assumptions in Γ). RAA

and DN together are our introduction and elimination rules for ∼.

And �nally we have→I and→E:

Γ,φ `ψΓ `φ→ψ

Γ `φ→ψ ∆ `φΓ,∆ `ψ

→E is perfectly straightforward; it’s just the familiar rule of modus ponens.

But→I requires a bit more thought. →I is the principle of conditional proof.

Suppose you can get to ψ on the assumption that φ (plus perhaps some other

assumptions Γ.) Then, you should be able to conclude that the conditional

φ→ψ is true (assuming the formulas in Γ). Put another way: if you want to

establish the conditional φ→ψ, all you need to do is assume that φ is true, and

reason your way to ψ.

We add, �nally, one more sequent rule, the rule of assumptions

φ `φ RA

Note that this is the one sequent rule when no “from” sequent is required,

since there are no sequents above the line. The rule permits us to move from

no sequents at all to a sequent of the form φ `φ. (Strictly, this sequent should

be written “{φ} `φ”.) Call any such sequent an “assumption sequent”. Clearly,

any assumption sequent is a logically correct sequent, since clearly φ can be

proved if we assume φ itself.


2.4.3 Sequent proofsWe have assembled all the sequent rules; now we need to show how to use

those rules to provide a criterion for logically correct sequents. We do this by

�rst de�ning the notion of a “sequent proof”:

A sequent proof is a series of sequents, each of which is either an

assumption sequent, or follows from earlier sequents in the series

by some sequent rule.

So, for example, the following is a sequent proof

1. P∧Q ` P∧Q As

2. P∧Q ` P 1, ∧E

3. P∧Q `Q 1, ∧E

4. P∧Q `Q∧P 2, 3 ∧I

Though it isn’t strictly required, we write a line number to the left of each

sequent in the series, and to the right of each line we write the sequent rule

that justi�es it, together with the line or lines (if any) that contained the “from”

sequents required by the sequent rule in question. (The rule of assumptions

requires no “from” sequents, recall.)

(It’s important to distinguish what we’re now calling proofs, namely, sequent

proofs, from the kinds of informal arguments I gave in section 2.3, and will

give elsewhere in this book. Sequent proofs (and also the axiomatic proofs we

will introduce in section 2.5) are formalized object-language proofs. The sentences

in sequent proofs are sentences in the object language; they are wffs of PL.

Moreover, we gave a rigorous de�nition of what a sequent proof is. Moreover,

sequent proofs are restrictive in that only the system’s of�cial rules may be

used. For contrast, consider the argument I gave in section 2.3 that any PL-

valuation assigns 1 to φ∧ψ iff it assigns 1 to φ and 1 to ψ. That argument was

an informal metalanguage proof. The sentences in the argument were sentences

of English, and the argument used informal (i.e., not formalized) techniques of

reasoning. “Informal” doesn’t imply lack of rigor. The argument was perfectly

rigorous: it conforms to the standards of good argumentation that generally

prevail in mathematics. We’re free to use any reasonable pattern of reasoning,

for example “universal proof” (to establish something of the form “everything

is thus-and-so”, we consider an arbitrary thing and show that it is thus-and-so).

We may “skip steps” if it’s clear how the argument is supposed to go. In short,


what we must do is convince a well-informed and mathematically sophisticated

reader that the result we’re after is indeed true.)

Next we introduce the notion of a “provable sequent”. The idea is that

each sequent proof culminates in the proof of some sequent. Thus we offer the

following de�nition:

A provable sequent is a sequent that is the last line of some sequent

proof

(Note that it would be equivalent to de�ne a sequent proof as any line in any

sequent proof, because at any point in a sequent proof one may simply stop

adding lines, and the proof up until that point counts as a legal sequent proof.)

So, for example, the sequent proof given above establishes that P∧Q `Q∧P is

a provable sequent. (We call a sequent proof, whose last line is Γ `φ, a sequent

proof of Γ `φ.)

Given the notion of a provable sequent, we can now offer sequent-proof-

theoretic notions of logical truth and logical consequence:

A formula φ is a logical truth iff the sequent ∅ ` φ is a provable

sequent.

Formula φ is a logical consequence of the formulas in set Γ iff the

sequent Γ `φ is a provable sequent.

The symbol ∅ stands for the “empty set”—the set containing no members.

Thus, a logical truth here is understood as a formula that is provable from no

assumptions at all.

2.4.4 Example sequent proofsLet’s explore how to construct sequent proofs. You may �nd this a bit less

intuitive, initially, than constructing natural deduction proofs in the systems

familiar from introductory textbooks. But a little experimentation will show

that the techniques for proving things in the usual systems carry over to the

present system.

A �rst simple example: let’s return to the sequent proof of P∧Q `Q∧P :


1. P∧Q ` P∧Q As

2. P∧Q ` P 1, ∧E

3. P∧Q `Q 1, ∧E

4. P∧Q `Q∧P 2, 3 ∧I

Notice the strategy. We �rst use the rule of assumptions to enter the premise

of the sequent we’re trying to prove: P∧Q. We then use the rules of inference

to infer the consequent of that sequent: Q∧P . Since our initial assumption of

P∧Q was dependent on the formula P∧Q, our subsequent inferences remain

dependent on that same assumption, and so the �nal formula concluded, Q∧P ,

remains dependent on that assumption.

Let’s write our proofs out in a simpler way. Instead of writing out entire

sequents, let’s write out only their conclusions. We can indicate the premises

of the sequent using line numbers; the line numbers indicating the premises

of the sequent will go to the left of the number indicating the sequent itself.

Rewriting the previous proof in this way yields:

1 (1) P∧Q As

1 (2) P 1, ∧E

1 (3) Q 1, ∧E

1 (4) Q∧P 2, 3 ∧I

Next, let’s have an example to illustrate conditional proof. Let’s construct a

sequent proof of P→Q,Q→R ` P→R:

1. P→Q ` P→Q As

2. Q→R `Q→R As

3. P ` P As

4. P→Q, P `Q 1,3→E

5. P→Q,Q→R, P ` R 2,4→E

6. P→Q,Q→R ` P→R 5,→I

It can be rewritten in the simpler style as follows:


1 (1) P→Q As

2 (2) Q→R As

3 (3) P As

1,3 (4) Q 1, 3→E

1,2,3 (5) R 2, 4→E

1,2 (6) P→R 5,→I

Let’s think about this example. We’re trying to establish P→R on the ba-

sis of two formulas, P→Q and Q→R, so we start by assuming the latter two

formulas. Then, since the formula we’re trying to establish is a conditional,

we assume the antecedent of the conditional, in line 3. We then proceed, on

that basis, to reason our way to R, the consequent of the conditional we’re

trying to prove. (Notice how in lines 4 and 5, we add more line numbers on

the very left. Whenever we use→ E, we increase dependencies: when we infer

Q from P and P→Q, our conclusion Q depends on all the formulas that P and

P→Q depended on, namely, the formulas on lines 1 and 3. Look back to the

statement of the rule→ E: the conclusion ψ depends on all the formulas that φandφ→ψ depended on: Γ and∆.) That brings us to line 5. At that point, we’ve

shown that R can be proven, on the basis of various assumptions, including P .

The rule→I (that is, the rule of conditional proof) then lets us conclude that

the conditional P→R follows merely on the basis of the other assumptions;

that rule, note, lets us in line 6 drop line 3 from the list of assumptions on

which P→R depends.

Next let’s establish an instance of DeMorgan’s Law, ∼(P∨Q) `∼P∧∼Q:

1 (1) ∼(P∨Q) As

2 (2) P As (for reductio)

2 (3) P∨Q 2, ∨I

1,2 (4) (P∨Q)∧∼(P∨Q) 1, 3 ∧I

1 (5) ∼P 4, RAA

6 (6) Q As (for reductio)

6 (7) P∨Q 6, ∨I

1,6 (8) (P∨Q)∧∼(P∨Q) 1, 7 ∧I

1 (9) ∼Q 8, RAA

1 (10) ∼P∧∼Q 5, 9∧I

Next let’s establish ∅ ` P∨∼P :


1 (1) ∼(P∨∼P ) As


2 (3) P∨∼P 2, ∨I

2,1 (4) (P∨∼P )∧∼(P∨∼P ) 1, 3 ∧I

1 (5) ∼P 4, RAA

6 (6) ∼P As (for reductio)

6 (7) P∨∼P 6, ∨I

6,1 (8) (P∨∼P )∧∼(P∨∼P ) 1, 7 ∧I

1 (9) ∼∼P 8, RAA

1 (10) ∼P∧∼∼P 5, 9 ∧I

∅ (11) ∼∼(P∨∼P ) 10, RAA

∅ (12) P∨∼P 11, DN

Comment: my overall goal was to assume ∼(P∨∼P ) and then derive a con-

tradiction. And my route to the contradiction was to separately establish ∼P(lines 2-5) and ∼∼P (lines 6-9), each by reductio arguments.

Finally, let’s establish a sequent corresponding to a way that ∨ E is some-

times formulated: P∨Q,∼P `Q:

1 (1) P∨Q As

2 (2) ∼P As

3 (3) Q As (for use with ∨E)

4 (4) P As (for use with ∨E)

5 (5) ∼Q As (for reductio)

4,5 (6) ∼Q∧P 4,5 ∧I

4,5 (7) P 6, ∧E

2,4,5 (8) P∧∼P 2,7 ∧I

2,4 (9) ∼∼Q 8, RAA

2,4 (10) Q 9, DN

1,2 (11) Q 1,3,10 ∨E

The basic idea of this proof is to use ∨ E on line 1 to get Q. That calls,

in turn, for showing that each disjunct of line 1, P and Q, leads to Q. Showing

that Q leads to Q is easy; that was line 3. Showing that P leads to Q took lines

4-10; line 10 states the result of that reasoning, namely that Q follows from P(as well as line 2). I began at line 4 by assuming P . Then my strategy was to

establish Q by reductio, so I assumed ∼Q in line 5. At this point, I basically

had my contradiction: at line 2 I had ∼P and at line 4 I had P . (You might

think I had another contradiction: Q at line 3 and ∼Q at line 5. But at the end


of the proof, I don’t want my conclusion to depend on line 3, whereas I don’t

mind it depending on line 2, since that’s one of the premises of the sequent

I’m trying to establish.) So I want to put P and ∼P together, to get P∧∼P ,

and then conclude ∼∼Q by RAA. But there is a minor hitch. Look carefully at

how RAA is formulated. It says that if we have Γ,φ `ψ∧∼ψ, we can conclude

Γ `∼φ. The �rst of these two sequents includes φ in its premises. That means

that in order to conclude ∼φ, the contradiction ψ∧∼ψ needs to depend on φ.

So in the present case, in order to �nish the reductio argument and conclude

∼∼Q, the contradiction P∧∼P needs to depend on the reductio assumption

∼Q (line 5.) But if I just used ∧I to put lines 2 and 4 together, the resulting

contradiction will only depend on lines 2 and 4. To get around this, I used

a little trick. Whenever you have a sequent Γ ` φ, you can always add any

formula ψ you like to the premises on which φ depends, using the following

argument:2

Γ `φ (begin with this)

ψ `ψ As (ψ is any chosen formula)

Γ,ψ `φ∧ψ ∧I

Γ,ψ `φ ∧E

Lines 4, 6 and 7 in the proof employ this trick: initially, at line 4, P only depends

on 4, but then by line 7, P also depends on 5. That way, the move from 8 to 9

by RAA is justi�ed.

2.5 Axiomatic proofs in propositional logicNatural deduction proofs are comparatively easy to construct; that is their

great advantage. A different approach to proof theory, the axiomatic method,

offers different advantages. Like natural deduction, the axiomatic method is a

proof-theoretic approach to logic, based on the step-by-step reasoning model

in which each step is sanctioned by a rule of inference. But unlike natural

deduction systems, axiomatic systems do not allow reasoning by assumptions,

and they have very few rules of inference. Although these differences make

2Adding arbitrary dependencies is not allowed in relevance logic, where a sequent is provable

only when all of its premises are, in an intuitive sense, relevant to its conclusion. Relevant

logicians modify various rules of classical logic, including the rule of ∧E.


axiomatic proofs much harder to construct, there is a compensatory advantage

in metalogic: it is far easier to prove things about axiomatic systems.

Let’s �rst think about axiomatic systems informally. An axiomatic proof is

a series of formulas (not sequents—we no longer need them since we’re not

reasoning with assumptions), the last of which is the conclusion of the proof.

Each line in the proof must be justi�ed in one of two ways: it may be inferred

by a rule of inference from earlier lines in the proof, or it may be an axiom.

An axiom is a certain kind of formula, a formula that one is allowed to enter

into a proof without any justi�cation at all. Axioms are the “starting points” of

proofs, the foundation on which proofs rest. Since axioms are to play this role,

the axioms in a good axiomatic system ought to be indisputable logical truths.

For example, “P→P” would be a good axiom—it’s obviously a logical truth.

(As it happens, we won’t choose this particular axiom; we’ll instead choose

other axioms from which this one may be proved.) Similarly, for each rule of

inference in a good axiomatic system, there should be no question but that the

premises of the rule logically imply its conclusion.

Formally, to apply the axiomatic method, we must choose i) a set of rules,

and ii) a set of axioms. An axiom is simply any chosen sentence (though as we

saw, in a good axiomatic system the axioms will be clear logical truths.) A rule

is simply a permission to infer one sort of sentence from other sentences. For

example, the rule modus ponens can be stated thus: “From φ→ψ and φ you mayinfer ψ”, and pictured as follows:

MP φ→ψφ

ψ

(Modus ponens is the analog of the sequent rule→E.)

Given any chosen axioms and rules, we can de�ne the following concepts:

A proof is a �nite sequence of wffs, each of which either i) is an

axiom, or ii) follows from earlier wffs in the sequence via a rule.

Where Γ is a set of wffs, a proof from Γ is a �nite sequence of wffs,

each of which either i) is an axiom, ii) is a member of Γ, or iii)

follows from earlier wffs in the sequence via a rule.

A theorem is the last line of any proof (we write “` φ” for “φ is a

theorem”).


Wff φ is provable from set of wffs Γ (“Γ `φ”) iff φ is the last line of

some proof from Γ.

The symbol ` may be subscripted with the name of the system in question.

Thus, for our axiom system for PL below, we may write: `PL

. (We’ll omit this

subscript when it’s clear which axiomatic system is at issue.)

Here is an axiomatic system for propositional logic:3

Axiomatic system for PL

Rule: modus ponens

Axioms: Where φ, ψ, and χ are wffs, anything that comes from

the following schemas are axioms

(A1) φ→(ψ→φ)(A2) (φ→(ψ→χ ))→((φ→ψ)→(φ→χ ))(A3) (∼ψ→∼φ)→((∼ψ→φ)→ψ)

Thus, a PL-theorem is any formula that is the last line of a sequence of formulas,

each of which is either an A1, A2, or A3 axiom, or follows from earlier formulas

in the sequence by modus ponens. And

The axiom “schemas” A1-A3 are not themselves axioms. They are, rather,

“recipes” for constructing axioms. Take A1, for example:

φ→(ψ→φ)

This string of symbols isn’t itself an axiom because it isn’t a wff; it isn’t a wff

because it contains Greek letters, which aren’t allowed in wffs (since they’re

not on the list of PL primitive vocabulary). φ and ψ are variables of our

metalanguage; you only get an axiom when you replace these variables with

wffs. P→(Q→P ), for example, is an axiom; it results from A1 by replacing φwith P and ψ with Q. (Note: since you can put in any wff for these variables,

and there are in�nitely many wffs, there are in�nitely many axioms.)

A few points of clari�cation about how to construct axioms from schemas.

First point: you can stick in the same wff for two different Greek letters. Thus

you can let both φ and ψ in A1 be P , and construct the axiom P→(P→P ).(But of course, you don’t have to stick in the same thing for φ as for ψ.) Sec-

ond point: you can stick in complex formulas for the Greek letters. Thus,

3See Mendelson (1987, p. 29).


(P→Q)→(∼(R→S)→(P→Q)) is an axiom (I put in P→Q for φ and ∼(R→S)forψ in A1). Third point: within a single axiom, you cannot substitute different

wffs for a single Greek letter. For example, P→(Q→R) is not an axiom; you

can’t let the �rst φ in A1 be P and the second φ be R. Final point: even though

you can’t substitute different wffs for a single Greek letter within a single axiom,

you can let a Greek letter become one wff when making one axiom, and let

it become a different wff when making another axiom; and you can use each

of these axioms within a single axiomatic proof. For example, each of the

following is an instance of A1, and one could include both in a single axiomatic

proof:

P→(Q→P )∼P→((Q→R)→∼P )

In the �rst case, I made φ be P and ψ be Q; in the second case I made φ be

∼P and ψ be Q→R. This is �ne because I kept φ and ψ constant within each

axiom.

The de�nitions we have given in this section constitute a formal way of

making precise the proof-theoretic conception of the core logical notions, as

applied to propositional logic. A logical truth, on this conception, is a PL-

theorem; one formula is a logical consequence of others iff it is provable from

them in PL.

2.5.1 Example axiomatic proofsAxiomatic proofs are much harder than natural deduction proofs. Some are

easy, of course. Here is a proof of (P→Q)→(P→P ):

1. P→(Q→P ) (A1)

2. P→(Q→P ))→((P→Q)→(P→P )) (A2)

3. (P→Q)→(P→P ) 1,2 MP

The existence of this proof shows that (P→Q)→(P→P ) is a theorem. And,

building on the previous proof, we can construct a proof of P→P from {P→Q}:

1. P→(Q→P ) (A1)

2. (P→(Q→P ))→((P→Q)→(P→P )) (A2)

3. (P→Q)→(P→P ) 1,2 MP

4. P→Q member of {(P→Q)}5. P→P 3, 4 MP


Thus, we have shown that {(P→Q)} ` (P→P ).The next example is a little harder: (R→P )→(R→(Q→P ))

1. [R→(P→(Q→P ))]→[(R→P )→(R→(Q→P ))] A2

2. P→(Q→P ) A1

3. [P→(Q→P )]→[R→(P→(Q→P ))] A1

4. R→(P→(Q→P )) 2,3 MP

5. (R→P )→(R→(Q→P )) 1,4 MP

Here’s how I approached this problem. What I was trying to prove, namely

(R→P )→(R→(Q→P )), is a conditional whose antecedent and consequent both

begin: (R→. That looks like the consequent of A2. So I wrote out an instance

of A2 whose consequent was the formula I was trying to prove; that gave me

line 1 of the proof. Then I tried to �gure out a way to get the antecedent of

line 1; namely, R→(P→(Q→P )). And that turned out to be pretty easy. The

consequent of this formula, P→(Q→P ) is an axiom (line 2 of the proof). And

if you can get a formula φ, then you choose anything you like—say, R,—and

then get R→φ, by using A1 and modus ponens; that’s what I did in lines 3 and

4. In fact, we’ll want to make a move like this in many proofs, whenever we

have φ on its own, and we want to move to ψ→φ. Let’s call this move “adding

an antecedent”; this is how it is done:

1. φ (from earlier lines)

2. φ→(ψ→φ) A1

3. ψ→φ 1, 2 MP

In future proofs, instead of repeating such steps, let’s just move directly from φto ψ→φ, with the justi�cation “adding an antecedent”.

This proof was a bit tricky, and most proofs are trickier still. Moreover, the

proofs quickly get very long. Practically speaking, the best way to make progress

in an axiomatic system like this is by building up a toolkit. The toolkit consists

of theorems and techniques for doing bits of proofs which are applicable in a

wide range of situations. Then, one approaching a new problem, one can look

to see whether the problem can be reduced to a few chunks, each of which can

be accomplished by using the toolkit. Further, one can cut down on writing by

citing bits of the toolkit, rather than writing down entire proofs.

So far, we have just one tool in our toolkit: “adding an antecedent”. Let’s

add another: the “MP technique”. Here’s what the technique will let us do.

Suppose we can separately prove φ→ψ and φ→(ψ→χ ). The MP technique


then shows us how to construct a proof of φ→χ . I call this the MP technique

because its effect is that you can do modus ponens “within the consequent of

the conditional φ→”. Here’s how the MP technique works:

1. φ→ψ from earlier lines

2. φ→(ψ→χ ) from earlier lines

3. (φ→(ψ→χ ))→((φ→ψ)→(φ→χ )) A2

4. (φ→ψ)→(φ→χ ) 2,3 MP

5. φ→χ 1,4 MP

Note that the lines in this “proof schema” are schemas (they contain Greek

letters), rather than wffs. It therefore isn’t a proof at all; rather, it becomes a

proof once you �ll in wffs for the φ,ψ, and χ . We constructed a proof schema

because we want the MP technique to be applicable whenever one wants to

move from formulas of the form φ→ψ and φ→(ψ→χ ), to a formula of the

form φ→χ , no matter what φ,ψ, and χ may be.

And while we’re on the topic of proof schemas, note also that whenever

one constructs a proof of a formula containing sentence letters, one could just

as well have constructed a similar proof schema. Corresponding to the proof

of (R→P )→(R→(Q→P )), for example, there is this proof schema:

1. [φ→(ψ→(χ→ψ))]→[(φ→ψ)→(φ→(χ→ψ))] A2

2. ψ→(χ→ψ) A1

3. [ψ→(χ→ψ)]→[φ→(ψ→(χ→ψ))] A1

4. φ→(ψ→(χ→ψ)) 2,3 MP

5. (φ→ψ)→(φ→(χ→ψ)) 1,4 MP

It’s usually more useful to think in terms of proof schemas, rather than

proofs, because they can go into our toolkit, if they have general applicability.

The proof schema we just constructed, for example, shows that anything of the

form (φ→ψ)→(φ→(χ→ψ)) is a theorem. As it happens, this is a fairly intuitive

theorem schema. Think of it as the principle of “weakening the consequent”:

χ→ψ is logically weaker than ψ, so if φ leads to ψ, φ must also lead to χ→ψ.

That sounds like a pattern that might well recur, so let’s put it into the toolkit,

under the label “weakening the consequent”. If we’re ever in the midst of a

proof and could really use a line of the form (φ→ψ)→(φ→(χ→ψ)), then we

can simply write that line down, and annotate on the right “weakening the

consequent”. Given the proof sketch above, we know that we could always


in principle insert a �ve-line proof of line; to save writing we simply won’t

bother. (Note that once we do this—omitting those �ve lines—the proofs we

are constructing will cease to be of�cial proofs, since not every line will be

either an axiom or a line that follows from earlier lines by MP. They will be

instead proof sketches, which are in essence metalanguage arguments to the

effect that there exists some proof or other of the desired type. An ambitious

reader could always construct an of�cial proof on the basis of the proof sketch,

by taking each of the bits, �lling in the details using the toolkit, and assembling

the results into one proof.

Next I want to add to our toolkit the principle of “strengthening the an-

tecedent”: [(φ→ψ)→χ ]→(ψ→χ ). The intuitive idea is that if φ→ψ leads to

χ , then ψ ought to lead to χ , since ψ is logically stronger than φ→ψ. This

proof will be harder still; we’ll need to break it into bits and use the toolkit to

complete it. Here’s a sketch of the overall proof:

a. [(φ→ψ)→χ ]→[ψ→(φ→ψ)] see below

b. [(φ→ψ)→χ ]→[(ψ→(φ→ψ))→(ψ→χ )] see below

c. [(φ→ψ)→χ ]→[ψ→χ ] a,b MP method

All that remains is to supply separate proofs of lines a and b. Step a is pretty

easy. Its consequent, ψ→(φ→ψ) is an instance of A1, so we can prove it in one

line, then use “adding an antecedent” to get a.

Line b is a bit harder. It has the form: (α→β)→[(γ→α)→(γ→β)]. Call

this “adding antecedents”, since it lets you add the same antecedent (γ ) to both

the antecedent and consequent of a conditional (α→β). The following proof

sketch for adding antecedents uses the MP technique again!

1. [γ→(α→β)]→[(γ→α)→(γ→β)] A2

2. (α→β)→[γ→(α→β)] A1

3. (α→β)→[γ→(α→β)]→[(γ→α)→(γ→β)] adding an antecedent to

line 1.

4. (α→β)→[(γ→α)→(γ→β)] 2, 3, MP method

(For this use of the MP method, we let φ = α→β, ψ = γ→(α→β), and

χ =(γ→α)→(γ→β).) Since we’ve now provided proof sketches for parts a. and

b., we’re �nished our proof sketch for strengthening the antecedent.

Next let’s add a tool to our toolkit, to the effect that conditionals are “tran-

sitive”. Here’s a proof sketch for ` (φ→ψ)→[(ψ→χ )→(φ→χ )]:


1. (ψ→χ )→[(φ→ψ)→(φ→χ )] adding antecedents

2. {(ψ→χ )→[(φ→ψ)→(φ→χ )]}→{[(ψ→χ )→(φ→ψ)]→[(ψ→χ )→(φ→χ )]}

A2

3. [(ψ→χ )→(φ→ψ)]→[(ψ→χ )→(φ→χ )] 1, 2 MP

4. {[(ψ→χ )→(φ→ψ)]→[(ψ→χ )→(φ→χ )]}→{(φ→ψ)→[(ψ→χ )→(φ→χ )]}

strengthening the an-

tecedent

5. (φ→ψ)→[(ψ→χ )→(φ→χ )] 3, 4 MP

Given this theorem, we can always move from φ→ψ and ψ→χ to φ→χthus:

1. φ→ψ from earlier lines

2. ψ→χ from earlier lines

3. (φ→ψ)→[(ψ→χ )→(φ→χ )] theorem just proved

4. (ψ→χ )→(φ→χ ) 1, 3 MP

5. φ→χ 2, 4 MP

So, such moves may henceforth be justi�ed by appeal to “transitivity”.

With transitivity in our toolkit, we can really get moving:

` [φ→(ψ→χ )]→[ψ→(φ→χ )] (“swapping antecedents”):

1. [φ→(ψ→χ )]→[(φ→ψ)→(φ→χ )] A2

2. [(φ→ψ)→(φ→χ )]→[ψ→(φ→χ )] strengthening the antecedent

3. [φ→(ψ→χ )]→[ψ→(φ→χ )] 1,2 transitivity

` (∼ψ→∼φ)→(φ→ψ) (“contraposition”):

1. (∼ψ→∼φ)→[(∼ψ→φ)→ψ] A3

2. [(∼ψ→φ)→ψ]→(φ→ψ) strengthening the antecedent

3. (∼ψ→∼φ)→(φ→ψ) 1, 2 transitivity

`∼φ→(φ→ψ) (“ex falso quodlibet”):

1. ∼φ→(∼ψ→∼φ) A1

2. (∼ψ→∼φ)→(φ→ψ) contraposition

3. ∼φ→(φ→ψ) 1, 2 transitivity


2.5.2 The deduction theoremAs we saw, once we added some tools to our toolkit, we could construct ax-

iomatic proofs of some interesting theorems. But it certainly wasn’t easy! Think

of all the work that went into proving, for example, such a simple theorem as

the strengthening of antecedents. That same theorem is much easier to prove

in the the natural deduction system, because in that system one can reason

using assumptions; in particular, one can use conditional proof.

While conditional proof is not allowed within axiomatic proofs, one can

prove the the following metalogical theorem about our axiomatic system:

Deduction theorem: If Γ∪{φ} `ψ, then Γ ` (φ→ψ)

That is: whenever there exists a proof from (Γ and)φ to ψ, then there also exists

a proof of φ→ψ (from Γ). That is not to say that one is allowed to assume φin a proof of φ→ψ. But since someone has proved the deduction theorem (I

won’t prove it here, but you can look up a proof in Mendelson (1987)) then if

one can prove ψ after assuming φ, one is entitled to conclude that some proof

of φ→ψ must exist.

2.6 Soundness and completeness of PLIn this chapter we have discussed two approaches to propositional logic: the

proof-theoretic approach and the semantic approach. In each case, we in-

troduced formal notions of logical truth and logical consequence. For the

semantic approach, these notions involved truth in PL-interpretations. For the

proof-theoretic approach, we considered two formal de�nitions, one involving

sequent-proofs, the other involving axiomatic proofs.

An embarrassment of riches! We have multiple formal accounts of our

logical notions. But in fact, it can be shown that each of our de�nitions yieldsexactly the same results. That is, whether you de�ne a logical truth (for example)

as a formula that is true in all PL-interpretations, or as a formula that is the

last line of some PL axiomatic proof, or as a formula φ for which the sequent

∅ `φ is a provable sequent, exactly the same formulas turn out logical truths.

Proving this is a major accomplishment of meta-logic. We won’t prove this

here, not in full anyway, but we will discuss the issues a bit, to show how such

metalogical proofs proceed.

Let’s focus, for the moment, on just two of our notions, the notion of

a theorem (last line of an axiomatic proof) and the notion of a valid formula


(true in all PL-interpretations). An important accomplishment of metalogic is

the establishment of the following two important connections between these

notions:

Soundness: every theorem is valid (If `PLφ then �

PLφ)

Completeness: every valid wff is a theorem (If �PLφ then `

PLφ)

It’s pretty easy to establish soundness:

Soundness proof for PL: We’re going to prove this by induction as

well. But our inductive proof here is slightly different. We’re not

trying to prove something of the form “Every wff has property

P”. Instead, we’re trying to prove something of the form “Every

theorem has property P”. In this case, the property P is: being a validformula.

Here’s how induction works in this case. A theorem is the last line

of any proof. So, to show that every theorem has a certain property

P, all we need to do is show that every time one adds another line to

a proof, that line has property P. Now, there are two ways one can

add to a proof. First, one can add an axiom. The base case of the

inductive proof must show that adding axioms always means adding

a line with property P. Second, one can add a formula that follows

from earlier lines by a rule. The inductive step of the inductive

proof must show that in this case, too, one adds a line with property

P, provided all the preceding lines have property P. OK, here goes:

base case: here we need to show that every PL-axiom is valid. This

is tedious but straightforward. Take A1, for example. Suppose

for reductio that some instance of A1 is invalid, i.e., for some PL-

interpretationI , VI (φ→(ψ→φ))=0. Thus, VI (φ)=1 and VI (ψ→φ)=0.

Given the latter, VI (φ)=0—contradiction. Analogous proofs can

be given that instances of A2 and A3 are also valid.

induction step: here we begin by assuming that every line in a proof

up to a certain point is valid (this is the “inductive hypothesis”); we

then show that if one adds another line that follows from earlier

lines by the rule modus ponens, that line must be valid too. I.e.,

we’re trying to show that “modus ponens preserves validity”. So,

assume the inductive hypothesis: that all the earlier lines in the


proof are valid. And now, consider the result of applying modus

ponens. That means that the new line we’ve added to the proof

is some formula ψ, which we’ve inferred from two earlier lines

that have the forms φ→ψ and φ. We must show that ψ is a valid

formula, i.e., that VI (ψ)=1 for every PL-interpretation I . By the

inductive hypothesis, all earlier lines in the proof are valid, and

hence bothφ→ψ andφ are valid. Thus, VI (φ)=1 and VI (φ→ψ)=1.

But if VI (φ)=1 then VI (ψ) can’t be 0, for if it were, then VI (φ→ψ)

would be 0, and it isn’t. Thus, VI (ψ)=1.

We’ve shown that axioms are valid, and that modus ponens pre-

serves validity. So, by induction, every time one adds to a proof,

one adds a valid formula. So the last line in a proof is always a valid

formula. Thus, every theorem is valid. QED.

Notice the general structure of this proof: we �rst showed that every axiom

has a certain property, and then we showed that the rule of inference preserves

the property. Given the de�nition of ‘theorem’, it followed that every theorem

has the property. We chose our de�nition of a theorem with just this sort of

proof in mind.

Remember that this is a proof in the metalanguage, about propositional

logic. It isn’t a proof in any system of derivation.

Completeness is harder to prove, and I won’t prove it here.

One nice thing about soundness is that it lets us establish facts of unprov-ability. Soundness says “if ` φ then � φ”. Equivalently, it says: “if 2 φ then

0 φ”. So, to show that something isn’t a theorem, it suf�ces to show that it

isn’t valid. Consider, for example, the formula (P→Q)→(Q→P ). There exist

PL-interpretations in which the formula is false, namely, PL-interpretations in

which P is 0 and Q is 1. So, (P→Q)→(Q→P ) is not valid (since it’s not true

in all PL-interpretations.) But then soundness tells us that it isn’t a theorem

either. In general: if we’ve established soundness, then in order to show that a

formula isn’t a theorem, all we need to do is �nd an interpretation in which it

isn’t true.

One could also prove soundness and completeness for the natural deduction

system of section 2.4.4

The soundness proof, for instance, would proceed by

4Note: if we allow sequents with in�nite premise-sets, then to secure completeness we will

need to add a rule for adding dependencies: from Γ `φ conclude Γ∪∆ `φ, where ∆ is any set

of wffs. The trick described above for adding dependencies (using RA, ∧I and ∧E) only allows


proving by induction that whenever sequent Γ `φ is provable, φ is a semantic

consequence of Γ. This would proceed by showing that each rule of inference

(As, RAA, ∧I, ∧E, etc.) preserves semantic consequence. But note how much

more involved this proof would be, since there are so many rules of inference.

The paucity of rules in the axiomatic system made the construction of proofs

within that system a real pain in the neck, but now we see how it makes

metalogical life easier.

Before we leave this chapter, let me summarize and clarify the nature of

proofs by induction. Induction is the method of proof to use whenever one is

trying to prove that every member of an in�nite class of entities has a certain

feature F, where each member of that in�nite class of entities is generated from

certain “starting points” by a �nite number of successive “operations”. To do

this, one establishes two things: a) that the starting points have feature F, and b)

that the operations preserve feature F—i.e., that if the inputs to the operations

have feature F then the output also has feature F.

In logic, it is important to distinguish two different cases where proofs by

induction are needed. One case is where one is establishing a fact of the form:

every theorem has a certain feature F. (The proof of the soundness theorem

is an example of this case.) Here’s why induction is applicable: a theorem is

de�ned as the last line of a proof. So the fact to be established is that every

line in every proof has feature F. Now, a proof is de�ned as a �nite sequence,

where each member is either an axiom or follows from earlier lines by the rule

modus ponens. The axioms are the “starting points” and modus ponens is the

“operation”. So if we want to show that every line in every proof has feature F,

all we need to do is show that a) the axioms all have feature F, and b) show that

if you start with formulas that have feature F, and you apply modus ponens,

then what you get is something with feature F. More carefully, b) means: if φhas feature F, and φ→ψ has feature F, then ψ has feature F. Once a) and b)

are established, one can conclude by induction that all lines in all proofs have

feature F.

A second case in which induction may be used is when one is trying to

establish a fact of the form: every formula has a certain feature F. (The proof

that every wff has a �nite number of sentence letter is an example of this

case.) Here’s why induction is applicable: all formulas are built out of sentence

letters (the “starting points”) by successive applications of the rules of formation

us to add one wff at a time to the premise set of a given sequent, whereas this new rule allows

adding in�nitely many.


(“operations”) (the rules of formation, recall, say that if φ and ψ are formulas,

then so are (φ→ψ) and ∼φ.) So, to show that all formulas have feature F, we

must merely show that a) all the sentence letters have feature F, and b) show

that if φ and ψ both have feature F, then both (φ→ψ) and ∼φ also will have

feature F.

If you’re ever proving something by induction, it’s important to identify

which sort of inductive proof you’re constructing.

Chapter 3

Variations and Deviations fromStandard Propositional Logic

As promised, we will not stop with the standard logics familiar from in-

troductory textbooks. In this chapter we examine some philosophically

important variations and deviations from standard propositional logic.

3.1 Alternate connectives

3.1.1 Symbolizing truth functions in propositional logicOur propositional logic is in a sense “expressively complete”. To get at this

sense, let’s introduce the idea of a truth function.

A truth function is a (�nite-placed) function or rule that maps truth values

(i.e., 0s and 1s ) to truth values. For example, here is a truth function, f :

f (1) = 0f (0) = 1

This is a called one-place function because it takes only one truth value as

input.

In fact, we have a name for this truth function: negation. And we express

that truth function with our symbol ∼. So the negation truth function is one

we can symbolize in propositional logic. More carefully, what we mean by

saying “We can symbolize truth function f in propositional logic” is this:

52

CHAPTER 3. VARIATIONS AND DEVIATIONS FROM PL 53

there is some sentence of propositional logic, φ, containing a single

sentence letter, P , which has the following feature: whenever P has

a truth value t , then the whole sentence φ has the truth value f (t ).

The sentence φ is in fact ∼P .

Here’s another truth function, g . g is a two-place truth function, which

means that it takes two truth values as inputs:

g (1,1) = 1g (1,0) = 0g (0,1) = 0g (0,0) = 0

In fact, this is the conjunction truth function. And we have a symbol for this

truth function: ∧. And as before, we can symbolize function g in propositional

logic; this means that:

There is some sentence of propositional logic, φ, containing two

sentence letters, P and Q, which has the following feature: when-

ever P has a truth value, t , and Q has a truth value, t ′, then the

whole sentence φ has the truth value g (t , t ′)

One such sentence φ is in fact: P∧Q.

Notice that the sentence φ that symbolizes1

the function g had to have two

sentence letters rather than one. That’s because the function g was a two-place

function. In general, the de�nition of “can be symbolized” is this:

n-place truth function h can be symbolized in propositionallogic iff there is some sentence of propositional logic, φ, contain-

ing n sentence letters, P1 . . . Pn, which has the following feature:

whenever P1 has a truth value, t1, and, …, and Pn has a truth value,

tn, then the whole sentence φ has the truth value h(t1 . . . tn)

Here’s another truth function:

i(1,1) = 0i(1,0) = 1i(0,1) = 1i(0,0) = 1

1Strictly speaking, we should speak of a sentence symbolizing a truth function relative to

an ordering of its sentence letters.


Think of this truth function as “not both”. Unlike the negation and conjunc-

tion truth functions, we don’t have a single symbol for this truth function.

Nevertheless, it can be symbolized in propositional logic: by the following

sentence:

∼(P∧Q)

Now, it’s not too hard to prove that:

All truth functions (of any �nite number of places) can be symbol-

ized in propositional logic using just the ∧,∨, and ∼

I’ll begin by illustrating the idea of the proof with an example. Suppose we

want to symbolize the following three-place truth-function:

f ( 1, 1, 1 ) = 0f ( 1, 1, 0 ) = 1f ( 1, 0, 1 ) = 0f ( 1, 0, 0 ) = 1f ( 0, 1, 1 ) = 0f ( 0, 1, 0 ) = 0f ( 0, 0, 1 ) = 1f ( 0, 0, 0 ) = 0

Since this truth function returns the value 1 in just three cases (rows two,

four, and seven), what we want is a sentence containing three sentence letters,

P1, P2, P3, that is true in exactly those three cases: (a) when P1, P2, P3 take on the

three truth values in the second row (i.e., 1, 1, 0) , (b) when P1, P2, P3 take on

the three truth values in the fourth row (1, 0, 0) , and (c) when P1, P2, P3 take

on the three truth values in the seventh row (0, 0, 1) . Now, we can construct a

sentence that is true in case (a) and false otherwise: P1∧P2∧∼P3. We can also

construct a sentence that’s true in case (b) and false otherwise: P1∧∼P2∧∼P3.

And we can also construct a sentence that’s true in case (c) and false otherwise:

∼P1∧∼P2∧P3. But then we can simply disjoin these three sentences to get the

sentence we want:

(P1∧P2∧∼P3) ∨ (P1∧∼P2∧∼P3) ∨ (∼P1∧∼P2∧P3)


(Strictly speaking the three-way conjunctions, and the three-way disjunction,

need parentheses added, but since it doesn’t matter where they’re added—

conjunction and disjunction are associative—I’ve left them off.)

This strategy is in fact purely general. Any n-place truth function, f , can

be represented by a chart like the one above. Each row in the chart consists of

a certain combination of n truth values, followed by the truth value returned

by f for those n inputs. For each such row, construct a conjunction whose

i thconjunct is Pi if the i th

truth value in the row is 1, and ∼Pi if the i thtruth

value in the row is 0. Notice that the conjunction just constructed is true if and

only if its sentence letters have the truth values corresponding to the row in

question. The desired formula is then simply the disjunction of all and only

the conjunctions for rows where the function f returns the value 1.2

Since the

conjunction for a given row is true iff its sentence letters have the truth values

corresponding to the row in question, the resulting disjunction is true iff its

sentence letters have truth values corresponding to one of the rows where freturns the value true, which is what we want.

Say that a set of connectives is adequate iff one can symbolize all the truth

functions using a sentence containing only those connectives. What we just

showed was that the set {∧,∨,∼} is adequate. We can then use this fact to

prove that other sets of connectives are adequate. For example, it is easy to

prove that φ∨ψ has the same truth table as (is true relative to exactly the same

PL-assignments as)∼(∼φ∧∼ψ). But that means that for any sentence χ whose

only connectives are ∧,∨, and ∼, we can construct another sentence χ ′ with

the same truth table but whose only connectives are ∧ and ∼: simply begin

with χ and use the equivalence between φ∨ψ and ∼(∼φ∧∼ψ) to eliminate all

occurrences of ∨ in favor of occurrences of ∧ and ∼. But now consider any

truth function f . Since {∧,∨,∼} is adequate, f can be symbolized by some

sentence χ ; but χ has the same truth table as some sentence χ ′ whose only

connectives are ∧ , and ∼; hence f can be symbolized by χ ′ as well. So {∧,∼}is adequate.

Similar arguments can be given to show that other connective sets are

adequate as well. For example, the ∧ can be eliminated in favor of the→ and

the ∼ (since φ∧ψ has the same truth table as ∼(φ→∼ψ)); therefore, since

{∧,∼} is adequate, {→, ∼} is also adequate.

2Special case: if there are no such rows—i.e., if the function returns 0 for all inputs—

then let the formula be simply any logically false formula containing P1 . . . Pn , for example

P1∧∼P1∧P2∧P3∧· · ·∧Pn .


3.1.2 Inadequate connective setsCan we show that certain sets of connectives are not adequate?

We can quickly answer yes, for a trivial reason. The set {∼} isn’t adequate,

for the simple reason that, since ∼ is a one-place connective, no sentence with

more than one sentence letter can be built using just ∼. So there’s no hope of

symbolizing all the n-place truth functions, for any n > 1, using just the ∼.

More interestingly, we can show that there are inadequate connective sets

containing two-place connectives. Let’s prove that {∧,→} is not an adequate set

of connectives. We’ll do this by proving that if those were our only connectives,

we couldn’t symbolize the negation truth function. And we’ll demonstrate thatby proving the following fact:

For any sentence, φ, containing just sentence letter P and the

connectives ∧ and→, if P is true then so is φ.

We’ll again use the method of induction. We want to show that the assertion is

true for all sentences. So we �rst prove that the assertion is true for all sentence

with no connectives (i.e., for sentences containing just sentence letters.) This is

the base case, and is very easy here, since ifφ has no connectives, then obviously

φ is just the sentence letter P itself, in which case, clearly, if P is true then so is

φ. Next we assume the inductive hypothesis:

Inductive hypothesis: the assertion holds true for sentences φ and

ψ

And we try to show, on the basis of this assumption, that:

The assertion holds true for φ∧ψ and φ→ψ

This is easy to show. We want to show that the assertion holds for φ∧ψ—that

is, if P is true then so is φ∧ψ. But we know by the inductive hypothesis that

the assertion holds for φ and ψ individually. So we know that if P is true, then

both φ and ψ are true. But then, we know from the truth table for ∧ that

φ∧ψ is also true in this case. The reasoning is exactly parallel for φ→ψ: the

inductive hypothesis tells us that whenever P is true, so are φ and ψ, and then

we know that in this case φ→ψ must also then be true, by the truth table for

→. Therefore, by induction, the result is proved.


3.1.3 Sheffer strokeWe’ve seen how we can choose alternate sets of connectives. Some of these

choices are adequate (i.e., allow symbolization of all truth functions), others

are not.

As we saw, there are some truth functions that can be symbolized in propo-

sitional logic, but not by a single connective (e.g., the not-both function idiscussed above.)

We could change this, by adding a new connective. Let’s use a new connec-

tive, the “Sheffer stroke”, |, to symbolize not-both. φ|ψ is to mean that not

both φ and ψ are true, so let’s stipulate that φ|ψ will have the same truth table

as ∼(φ∧ψ), i.e:

φ | ψ1 0 11 1 00 1 10 1 0

Now here’s an exciting thing about |: it’s an adequate connective all on its own.

You can symbolize all the truth functions using just |!Here’s how we can prove this. We showed above that {→, ∼} is adequate;

so all we need to do is show how to de�ne the→ and the ∼ using just the |.De�ning ∼ is easy; φ|φ has the same truth table as ∼φ. As for φ→ψ, think of

it this way. φ→ψ is equivalent to ∼(φ∧∼ψ), i.e., φ|∼ψ. But given the method

just given for de�ning ∼ in terms of |, we know that ∼ψ is equivalent to ψ|ψ.

Thus, φ→ψ has the same truth table as: φ|(ψ|ψ).

3.2 Polish notationAlternate connectives, like the Sheffer stroke, are called “variations” of standard

logic because they don’t really change what we’re saying with propositional

logic; it’s just a change in notation.

Another fun change in notation is polish notation. The basic idea of polish

notation is that the connectives all go before the sentences they connect. Instead

of writing P∧Q, we write ∧PQ. Instead of writing P∨Q we write ∨PQ.

Formally, here is the de�nition of a wff:

i) sentence letters are wffs


ii) if φ and ψ are wffs, then so are:

∧φψ∨φψ→φψ↔φψ∼φ

What’s the point? This notation eliminates the need for parentheses. With the

usual notation, in which we put the connectives between the sentences they

connect, we need parentheses to distinguish, e.g.:

(P∧Q)→RP∧(Q→R)

But with polish notation, these are distinguished without parentheses; they

become:

→∧PQR∧P→QR

respectively.

3.3 Multi-valued logic3

Logicians have considered adding a third truth value to the usual two. In these

new systems, in addition to truth (1) and falsity (0) , we have a third truth value,

#. There are a number of things one could take # to mean (e.g., “meaningless”,

or “unde�ned”, or “unknown”).

Standard logic is “bivalent”—that means that there are no more than two

truth values. So, moving from standard logic to a system that admits a third

truth value is called “denying bivalence”. One could deny bivalence, and go

even further, and admit four, �ve, or even in�nitely many truth values. But

we’ll only discuss trivalent systems—i.e., systems with only three truth values.

Why would one want to admit a third truth value? There are various

philosophical reasons one might give. One concerns vagueness. A person with

one dollar is not rich. A person with a million dollars is rich. Somewhere in

the middle, there are some people that are hard to classify. Perhaps a person

3See Gamut (1991a, pp. 173-183).


with $100,000 is such a person. They seem neither de�nitely rich nor de�nitely

not rich. So there’s pressure to say that the statement “this person is rich” is

capable of being neither de�nitely true nor de�nitely false. It’s vague.Others say we need a third truth value for statements about the future. If

it is in some sense “not yet determined” whether there will be a sea battle

tomorrow, then (it is argued) the sentence:

There will be a sea battle tomorrow

is neither true nor false. In general, statements about the future are neither

true nor false if there is nothing about the present that determines their truth

value one way or the other.4

Yet another case in which some have claimed that bivalence fails concerns

failed presupposition. Consider this sentence:

Ted stopped beating his dog.

In fact, I have never beaten a dog. I don’t even have a dog. So is it true that

I stopped beating my dog? Obviously not. But on the other hand, is this

statement false? Certainly no one would want to assert its negation: “Ted has

not stopped beating his dog”. The sentence presupposes that I was beating a dog;

since this presupposition is false, the question of the sentence’s truth does not

arise: the sentence is neither true nor false.

For a �nal challenge to bivalence, consider the sentence:

Sherlock Holmes has a mole on his left leg

‘Sherlock Holmes’ doesn’t refer to a real entity. Further, Sir Arthur Conan

Doyle does not specify in his Sherlock Holmes stories whether Holmes has

such a mole. Either of these reasons might be argued to result in a truth value

gap for the displayed sentence.

It’s an interesting philosophical question whether any of these arguments

for bivalence’s failing are any good. But we won’t take up that question. Instead,

we’ll look at the formal result of giving up bivalence. That is, we’ll introduce

some non-bivalent formal systems. We won’t ask whether these systems really

model English correctly.

4An alternate view preserves the “openness of the future” as well as bivalence: both ‘There

will be a sea battle tomorrow’ and ‘There will fail to be a sea battle tomorrow’ are false. This

combination is not contradictory, provided one rejects the equivalence of “It will be the case

tomorrow that ∼φ” and “∼ it will be the case tomorrow that φ”.


These systems all give different truth tables for the Boolean connectives.

The original truth tables give you the truth values of complex formulas based

on whether their sentence letters are true or false (1 or 0) . The new truth

tables need to take into account cases where the sentence letters are # (neither

1 nor 0) .

3.3.1 Łukasiewicz’s systemHere are the new truth tables (let’s skip the↔):

∼1 00 1# #

∧ 1 0 #1 1 0 #

0 0 0 0# # 0 #

∨ 1 0 #1 1 1 10 1 0 #

# 1 # #

→ 1 0 #1 1 0 #

0 1 1 1# 1 # 1

Using these truth tables, one can calculate truth values of wholes based on

truth values of parts.

Example: Where P is 1, Q is 0 and R is #, calculate the truth value of (P∨Q)→∼(R→Q).

First, what is R→Q? Answer, from the truth table for→: #. Next,

what is ∼(R→Q)? From the truth table for ∼, we know that the

negation of a # is a #. So, ∼(R→Q) is #. Next, P∨Q: that’s 1 ∨0—i.e., 0. Finally, the whole thing: 0→#, i.e., 1.

We can formalize this a bit more by de�ning up new notions of an inter-

pretation, and of truth relative to an interpretation:

A trivalent interpretation is a function, I , that assigns to each sen-

tence letter exactly one of the values: 1, 0, #.

For any trivalent interpretation, I , the valuation for I , VI , is

de�ned as the function that assigns to each wff either 1, 0, or #, and

which is such that, for any wffs φ and ψ,

i) if φ is a sentence letter then VI (φ) =I (φ)

ii) VI (φ∧ψ) =

1 if VI (φ) = 1 and VI (ψ) = 10 if VI (φ) = 0 or VI (ψ) = 0# otherwise


iii) VI (φ∨ψ) =

1 if VI (φ) = 1 or VI (ψ) = 10 if VI (φ) = 0 and VI (ψ) = 0# otherwise

iv) VI (φ→ψ) =

1 if either: VI (φ) = 0, or VI (ψ) = 1,

or VI (φ) = # and VI (ψ) = #0 VI (φ) = 1 and VI (ψ) = 0# otherwise

v) VI (∼φ) =

1 if VI (φ) = 00 if VI (φ) = 1# otherwise

Let’s de�ne validity and semantic consequence for Łukasiewicz’s system

much like we did for standard PL:

φ is Łukasiewicz-valid (“�Łφ”) iff every trivalent interpretation I ,

VI (φ) = 1

φ is a Łukasiewicz-semantic-consequence of Γ (“Γ �Łφ”) iff for every

trivalent interpretation, I , if VI (γ ) = 1 for each γ ∈ Γ, then

VI (φ) = 1

Notice that there are now two ways a formula can fail to be valid. It can be

0 under some PL-valuation, or it can be # under some PL-valuation. “Valid”

(under this de�nition) means always true; it does not mean never false. (A formula

can be never false and still not always be true, if it sometimes is #.)

Example: is P ∨∼P valid?Answer: no, it isn’t. Suppose P is #. Then ∼P is #; but then the

whole thing is # (since #∨# is #.)

Example: is P→P valid?Answer: yes. P could be either 1, 0 or #. From the truth table for

→, we see that P→P is 1 in all three cases.


3.3.2 Kleene’s “strong” tablesThis system is like Łukasiewicz’s system, except that the truth table for the→is different:

→ 1 0 #1 1 0 #

0 1 1 1# 1 # #

As with Łukasiewicz’s system, let’s continue to understand validity as truth in

all trivalent interpretations, and semantic consequence as the preservation of

truth in a given trivalent interpretation.

Here is the intuitive idea behind the Kleene tables. Let’s call the truth values

0 and 1 the “classical” truth values. If a formula’s halves have only classical

truth values, then the truth value of the whole formula is just the classical

truth value determined by the classical truth values of the halves. But if one or

both halves are #, then we must consider the result of turning each # into one

of the classical truth values. If the entire formula would sometimes be 1 and

sometimes be 0 after doing this, then the entire formula is #. But if the entire

formula always takes the same truth value, X, no matter which classical truth

value any #s are turned into, then the entire formula gets this truth value X.

Intuitively: if there is “enough information” in the classical truth values of a

formula’s parts to settle on one particular classical truth value, then that truth

value is the formula’s truth value.

Take the truth table for φ→ψ, for example. When φ is 0 and ψ is #, the

whole formula is 1—because the false antecedent is suf�cient to make the whole

formula true, no matter what classical truth value we convert ψ to. On the

other hand, when φ is 1 and ψ is #, then the whole formula is #. The reason is

that what classical truth value we substitute in for ψ’s # affects the truth value of

the whole. If the # becomes a 0 then the whole thing is 0; but if the # becomes

a 1 then the whole thing is 1.

There are two important differences between Łukasiewicz’s and Kleene’s

systems. The �rst is that, unlike Łukasiewicz’s system, Kleene’s system makes

the formula P→P invalid. The reason is that in Kleene’s system, #→# is #;

thus, P→P isn’t true in all valuations (it is # in the valuation where P is #.)

In fact, it’s easy to show that there are no valid formulas in Kleene’s system.

Consider the valuation that makes every sentence letter #. Here’s an inductive

proof that every wff is # in this interpretation. Base case: all the sentence letters


are # in this interpretation. (That’s obvious.) Inductive step: assume that φand ψ are both # in this interpretation. We need now to show that φ∧ψ, φ∨ψ,

and φ→ψ are all # in this interpretation. But that’s easy—just look at the truth

tables for ∧,∨ and→. #∧# is #, #∨# is #, and #→# is #. QED.

Even though there are no valid formulas in Kleene’s system, there are still

cases of semantic consequence. Semantic consequence for Kleene’s system is

de�ned as truth-preservation: Γ �Kleene

φ iff φ is true whenever every member

of Γ is true, given Kleene’s truth tables. Then P∧Q �Kleene

P , since the only

way for P∧Q to be true is for P to be true and Q to be true.

The second (related) difference is that in Kleene’s system, → is interde-

�nable with the ∼ and ∨, in that φ→ψ has exactly the same truth table as

∼φ∨ψ. (Look at the truth tables to verify that this is true.) But that’s not true

for Łukasiewicz’s system. In Łukasiewicz’s system, when φ and ψ are both #,

then φ→ψ is 1, but ∼φ∨ψ is #.

3.3.3 Kleene’s “weak” tables (Bochvar’s tables)This �nal system is based on a very different intuitive idea: that # is “infectious”.

That is, if any formula has a part that is #, then the entire formula is #. Thus,

the tables are as follows:

∼1 00 1# #

∧ 1 0 #1 1 0 #

0 0 0 #

# # # #

∨ 1 0 #1 1 1 #

0 1 0 #

# # # #

→ 1 0 #1 1 0 #

0 1 1 #

# # # #

So basically, the classical bit of each truth table is what you’d expect; but

everything gets boring if any constituent formula is a #.

One way to think about these tables is to think of the # as indicating nonsense.The sentence “The sun is purple and blevledgekl;rz”, one might naturally think,

is neither true nor false because it is nonsense. It is nonsense even though it

has a part that isn’t nonsense.

3.3.4 SupervaluationismRecall the guiding thought behind the strong Kleene tables: if a formula’s

classical truth values �x a particular truth value, then that is the value that the

formula takes on. There is a way to take this idea a step further, which results

in a new and interesting way of thinking about three-valued logic.


According to the strong Kleene tables, we get a classical truth value for

φ©ψ, where © is any connective, only when we have “enough classical

information” in the truth values of φ and ψ to �x a classical truth value for

φ© ψ. Consider φ∧ψ for example: if either φ or ψ is false, then since

falsehood of a conjunct is classically suf�cient for the falsehood of the whole

conjunction, the entire formula is false. But if, on the other hand, both φ and

ψ are #, then neither φ nor ψ has a classical truth value, we do not have enough

classical information to settle on a classical truth value for φ∧ψ, and so the

whole formula is #.

But now consider a special case of the situation just considered, where φ is

P , ψ is ∼P , and P is #. According to the strong Kleene tables, the conjunction

P∧∼P is #, since it is the conjunction of two formulas that are #. But there is a

way of thinking about truth values of complex sentences according to which

the truth value ought to be 0, not #: no matter what classical truth value P were

to take on, the whole sentence P∧∼P would be 0—therefore, one might think,

P∧∼P ought to be 0. If P were 0 then P∧∼P would be 0∧∼0—that is 0; and if

P were 1 then P∧∼P would be 1∧∼1—0 again.

The general thought here is this: suppose a sentence φ contains some

sentence letters P1 . . . Pn that are #. If φ would be false no matter how we assign

classical truth values to P1 . . . Pn—that is, no matter how we precisi�ed φ—then

φ is in fact false. Further, if φ would be true no matter how we precisi�ed it,

then φ is in fact true. But if precisifying φ would sometimes make it true and

sometimes make it false, then φ in fact is #.

The idea here can be thought of as an extension of the idea behind the

strong Kleene tables. Consider a formula φ©ψ, where© is any connective.

If there is enough classical information in the truth values of φ and ψ to �x on a

particular classical truth value, then the strong Kleene tables assign φ©ψ that

truth value. Our new idea goes further, and says: if there is enough classical

information within φ and ψ to �x a particular classical truth value, then φ©ψgets that truth value. Information “within” φ and ψ includes, not only the

truth values of φ and ψ, but also a certain sort of information about sentence

letters that occur in both φ and ψ. For example, in P∧∼P , when P is #, there

is insuf�cient classical information in the truth values of P and of ∼P to settle

on a truth value for the whole formula P∧∼P (since each is #). But when we

look inside P and ∼P , we get more classical information: we can use the fact

that P occurs in each to reason as we did above: whenever we turn P to 0, we

turn ∼P to 1, and so P∧∼P becomes 0; and whenever we turn P to 1 we turn


∼P to 0, and so again, P∧∼P becomes 0.

This new idea—that a formula has a classical truth value iff every way of

precisifying it results in that truth value—is known as supervaluationism. Let us

lay out this idea formally.

As above, a trivalent interpretation is a function that assigns to each sentence

letter one of the values 1, 0, #. Where I is a trivalent interpretation and Cis a PL-interpretation (i.e., a bivalent interpretation in the sense of section

2.3), say that C is a precisi�cation of I iff: whenever I assigns a sentence letter

a classical truth value (i.e., 1 or 0), C assigns that sentence letter the same

classical value. Thus, precisi�cations of I agree with I on the classical truth

values, but in addition—being PL-interpretations—they also assign classical

truth values to sentence letters to which I assigns #. A given precisi�cation of

I “decides” all of I ’s #s in a certain way.

We can now say how the supervaluationist assigns truth values to complex

formulas relative to a trivalent interpretation. When φ is any wff and I is a

trivalent interpretation, let us de�ne SI (φ)—the “supervaluation of φ relative

to I—thus:

SI (φ) = 1 if VC (φ) = 1 for every precisi�cation, C , of ISI (φ) = 0 if VC (φ) = 0 for every precisi�cation, C , of ISI (φ) = # otherwise

Here VC is the valuation for PL-interpretation C , as de�ned in section 2.3.

Some common terminology: when SI (φ) = 1, we say thatφ is supertrue inI ,

and when SI (φ) = 0, we say that φ is superfalse in I . For the supervaluationist,

a formula is true when it is supertrue (i.e., true in all precisi�cations of I ), false

when it is superfalse (i.e., false in all precisi�cations of I ), and # when it is

neither supertrue nor superfalse (i.e., when it is true in some precisi�cations of

I but false in others.)

Supervaluational notions of validity and semantic consequence may be

de�ned thus:

φ is supervaluationally valid (“�SVφ”) iff φ is supertrue in every

trivalent interpretation

φ is a supervaluational semantic consequence of Γ (“Γ �SVφ”) iff φ is

supertrue in each trivalent assignment in which every member of Γis supertrue


To return to the example considered above: the supervaluationist assigns a

different truth value to P∧∼P , when P is #, than do the strong Kleene tables

(and indeed, than do all the other tables we have considered.) The strong

Kleene tables say that P∧∼P is # in this case. But the supervaluationist says

that it is 0: each precisi�cation of any trivalent interpretation that assigns P #

is by de�nition a PL-interpretation, and P∧∼P is 0 in each PL-interpretation.

Let us note a few facts about supervaluationism.

First, note that every tautology (PL-valid formula) turns out to be superval-

uationally valid. For let φ be a tautology; and consider any trivalent interpreta-

tion I , and any precisi�cationC of I . Precisi�cations are PL-interpretations;

so, since φ is a tautology, φ is true in C .

Second, note that according to supervaluationism, some formulas are nei-

ther true nor false in some trivalent interpretations. For instance, take the

formula P∧Q, in any trivalent interpretation I in which P is 1 and Q is #. Any

precisi�cation of I must continue to assign P 1. But some precisi�cations of

I will assign 1 to Q, whereas others will assign 0 to Q. Any of the former

precisi�cations will assign 1 to P∧Q, whereas any of the latter will assign 0 to

P∧Q. Hence P∧Q is neither supertrue nor superfalse in I : SI (P∧Q) = #.

Finally, notice that the propositional connectives are not truth functional

according to supervaluationism. To say that a connective© is truth functional

is to say that the truth value5

of a complex statement whose major connective

is© is a function of the truth values of the immediate constituents of that

formula—that is, any two such formulas whose immediate constituents have

the same true values must themselves have the same truth value. According to

all of the truth tables for three-valued logic we considered earlier (Łukasiewicz,

Kleene strong and weak), the propositional connectives are truth-functional.

Indeed, this is in a way trivial: if a connective isn’t truth-functional then one

can’t give it a truth table at all: what a truth table does is specify how a connective

determines the truth value of entire sentences as a function of the truth values

of its parts. But supervaluationism renders the truth functional connectives

not truth functional. Consider the following pair of sentences, in a trivalent

interpretation in which P and Q are both #:

P∧Q

P∧∼P5I am counting #, in addition to 1 and 0, as a “truth value”.


As we have seen, in such trivalent interpretations, the �rst formula is # and

the second formula is 0 (since it is superfalse). But each of these formulas is a

conjunction, each of whose conjuncts is #: the truth values of φ and ψ do not

determine the truth value of φ∧ψ.

Similarly, the truth values of other complex formulas are not determined

by the truth values of their parts, as the following pairs of formulas show (the

indicated truth values, again, are relative to a trivalent interpretation in which

P and Q are #):

P∨Q #

P∨∼P 1P→Q #

P→P 1

3.4 IntuitionismIntuitionism in the philosophy of mathematics is a view according to which

there are no mind-independent mathematical facts. Rather, mathematical facts

and entities are mental constructs that owe their existence to the mental activity

of mathematicians constructing proofs. This philosophy of mathematics leads

intuitionists to a distinctive form of logic: intuitionist logic.

Let P be the statement: The sequence 0123456789 occurs somewhere in thedecimal expansion of π. How should we think about its meaning? For the classicalmathematician, the answer is straightforward. P is a statement about a part of

mathematical reality, namely, the in�nite decimal expansion of π. Either the

sequence 0123456789 occurs somewhere in that expansion, in which case P is

true, or it does not, in which case P is false and ∼P is true.

For the intuitionist, this whole picture is mistaken, premised as it is on the

reality of an in�nite decimal expansion of π. Our minds are �nite, and so only

the �nite initial segment of π’s decimal expansion that we have constructed so

far is real. The intuitionist’s alternate picture of P ’s meaning, and indeed of

meaning generally (for mathematical statements) is a radical one.6

The classical mathematician, comfortable with the idea of a realm of mind-

independent entities, thinks of meaning in terms of truth and falsity. As we saw,

she thinks of P as being true or false depending on the facts about π’s decimal

expansion. Further, she explains the meanings of the propositional connectives

in truth-theoretic terms: a conjunction is true iff each of its conjuncts are true;

6One intuitionist picture, anyway, on which see Dummett (1973). What follows is a crude

sketch. It does not do justice to the actual intuitionist position, which is, as they say, subtle.


a negation is true iff the negated formula is false; and so on. Intuitionists, on

the other hand, reject the centrality of truth to meaning, since truth is tied

up with the rejected picture of mind-independent mathematical reality. For

them, the central semantic concept is that of proof. They simply do not think

in terms of truth and falsity; in matters of meaning, they think in terms of the

conditions under which formulas have been proved.

Take P , for example. Intuitionists advise us: don’t think in terms of what

it would take for P to be true. Think, rather, in terms of what it would take

to prove P . And the answer is clear: we would need to actually continue our

construction of the decimal expansion of π to a point where we found the

sequence 0123456789.

What, now, of ∼P? Again, thinking in terms of proof, not truth: what

would it take for ∼P to be proved? The answer here is less straightforward.

Since P said that there exists a number of a certain sort, it was clear how it

would have to be proved: by actually exhibiting (calculating) some particular

number of that sort. But ∼P says that there is no number of a certain sort; how

do we prove something like that? The intuitionist’s answer: by proving that theassumption that there is a number of that sort leads to a contradiction. In general, a

negation, ∼φ, is proved by proving that φ leads to a contradiction.7

Similarly for the other connectives: the intuitionist explicates their meanings

by their role in generating proof conditions, rather than truth conditions. φ∧ψis proved by separately giving a proof of φ and a proof of ψ; φ∨ψ is proved

by giving either a proof of φ or a proof of ψ; φ→ψ is proved by exhibiting a

construction whereby any proof of φ can be converted into a proof of ψ.

Likewise, the intuitionist thinks of logical consequence as the preservation

of provability, not the preservation of truth. For example,φ∧ψ logically implies

φ because if one has a proof of φ∧ψ, then one has a proof of φ; and conversely,

if one has proofs of φ and ψ separately, then one has the materials for a proof

of φ∧ψ. So far, so classical. But ∼∼φ does not logically imply φ, for the

intuitionist. Simply having a proof of ∼∼P—a proof that the assumption that

0123456789 occurs nowhere in π’s decimal expansion leads to a contradiction—

wouldn’t give us a proof of P , since proving P would require exhibiting a

particular place in π’s decimal expansion where 0123456789 occurs.

Likewise, intuitionists do not accept the law of the excluded middle, φ∨∼φ,

as a logical truth. To be a logical truth, according to an intuitionist, a sentence

7Given the contrast with the classical conception of negation, a different symbol (often

“¬”) is sometimes used for intuitionist negation.


should be provable from no premises whatsoever. But to prove P∨∼P , for

example, would require either exhibiting a case of 0123456789 in π’s decimal

expansion, or proving that the assumption that 0123456789 occurs inπ’s decimal

expansion leads to a contradiction. We’re not in a position to do either.

Though we won’t consider intuitionist predicate logic, one of its most

striking features is easy to grasp informally. Intuitionists say that an existentially

quanti�ed sentence is proved iff one of its instances has been proved. Therefore

they reject the inference from ∼∀xF x to ∃x∼F x, for one might be able to

prove a contradiction from the assumption of ∀xF x without being able to

prove any instance of ∃x∼F x.

We have so far been considering a putative philosophical justi�cation for

intuitionist propositional logic. That justi�cation has been rough and ready;

but intuitionist propositional logic itself is easy to present, perfectly precise,

and is a coherent system regardless of what one thinks of its philosophical

underpinnings. Two simple modi�cations to the natural deduction system of

section 2.4 generate a natural deduction system for intuitionistic propositional

logic. First, we drop one half of the double-negation rule, namely what we

might call “double-negation elimination” (DNE):

Γ `∼∼φΓ `φ

Second, we add the rule “ex falso” (EF):

Γ `φ∧∼φΓ `ψ

Note that EF can be proved in the original system of chapter 2.4: simply use

RAA and then DNE. So, intuitionist logic results from a system for classical

logic by simply dropping one rule (DNE) and adding another rule that was pre-

viously provable (EF). It follows that every intuitionistically provable sequent

is also classically provable (because every intuitionistic proof can be converted

to a classical proof).

Notice how dropping DNE blocks proofs of various classical theorems the

intuitionist wants to avoid. The proof of∅ ` P∨∼P in chapter 2.4, for instance,

used DNE. Of course, for all we’ve said so far, there might be some other way

to prove this sequent. Only when we have a semantics for intuitionistic logic,

and a soundness proof relative to that semantics, can we show that this sequent


cannot be proven without DNE. We will discuss a semantics for intuitionism

in section 7.2.

It is interesting to note that even though intuitionists reject the inference

from ∼∼P to P , they accept the inference from ∼∼∼P to ∼P , since its proof

only requires the half of DN that they accept, namely the inference from P to

∼∼P :

1 (1) ∼∼∼P As


2 (3) ∼∼P 2, DN (accepted version)

1,2 (4) ∼∼P ∧∼∼∼P 1,3 ∧I

1 (5) ∼P 4, RAA

Note that you can’t use this sort of proof to establish ∼∼P ` P . Given the

way RAA is stated, its application always results in a formula beginning with

the ∼.

Chapter 4

Predicate Logic

Let’s now turn from propositional logic to the “predicate calculus” (PC),

as it is sometimes called. As with propositional logic, we’re going to

formalize predicate logic. We’ll �rst do grammar, and then move to semantics.

We won’t consider proof theory at all.1

4.1 Grammar of predicate logicAs before, we start by specifying the kinds of symbols that may be used in

sentences of predicate logic—primitive vocabulary—and then go on to de�ne

the well formed formulas as strings of primitive vocabulary that have the right

form.

Primitive vocabulary (of PC):

i) logical: →, ∼, ∀ii) nonlogical:

a) for each n > 0, n-place predicates F ,G . . ., with or without

subscripts

b) variables x, y . . . with or without subscripts

c) individual constants (names) a, b . . ., with or without sub-

scripts

1Proof systems for predicate logic, in both axiomatic and natural-deduction form, are

straightforward, and can be found in standard logic textbooks.

71

CHAPTER 4. PREDICATE LOGIC 72

iii) parentheses

No symbol of one type is a symbol of any other type. Let’s call any variable or

constant a term.

De�nition of (PC-) wff:

i) ifΠ is an n-place predicate andα1 . . .αn are terms, thenΠα1 . . .αnis a wff

ii) ifφ, ψ are wffs, and α is a variable, then∼φ, (φ→ψ), and ∀αφare wffs

iii) nothing else is a wff.

We’ll call formulas that are wffs in virtue of clause i) “atomic” formulas. When

a formula has no free variables, we’ll say that it is a closed formula, or sentence;

otherwise it is an open formula.

We have the same de�ned logical terms: ∧,∨,↔. We also add the following

de�nition of the existential quanti�er:

∃vφ=df∼∀α∼φ (where α is a variable and φ is a wff).

4.2 Semantics of predicate logicRecall from section 2.2 the truth-conditional conception of semantics. Se-

mantics is about meaning; meaning is about the way truth is determined by

the world; and the way we represent the dependence of truth on the world in

logic consists of i) de�ning certain abstract con�gurations, which represent

different ways the world could be, and ii) de�ning the notion of truth for

formulas in these con�gurations. We thereby shed light on meaning, and we

are thereby able to de�ne precise versions of the notions of logical truth and

logical consequence.

In propositional logic, the con�gurations were the PL-interpretations:

assignments of truth or falsity (1 or 0) to sentence letters; and valuation functions

de�ned truth in a con�guration. This procedure needs to get more complicated

for predicate logic. The reason is that the method of truth tables assumes that

we can calculate the truth value of a complex formula by looking at the truth

values of its parts. But take the sentence ∃x(F x∧Gx). You can’t calculate its

truth value by looking at the truth values of F x and Gx, since sentences like


F x don’t have truth values at all. The variable ‘x’ doesn’t stand for any one

thing, and so ‘F x’ doesn’t have a truth value.

The solution to this problem is due to the Polish logician Alfred Tarski. It

begins with a new conception of a con�guration, that of a model:

A (PC-) model is an ordered pair ⟨D,I ⟩, such that:

i) D is a non-empty set (“the domain”)

ii) I is a function (“the interpretation function”) obeying the

following constraints:

a) if α is a constant then I (α) ∈Db) if Π is an n-place predicate, then I (Π) = some set of

n-tuples of members of D.

(Recall the notion of an n-tuple from section 1.8.)

Models, as we have de�ned them, seem like good ways to represent con-

�gurations of the world. The domain, D, contains, intuitively, the individuals

that exist in the con�guration. I , the interpretation function, tells us what the

non-logical constants (names and predicates) mean in the con�guration. Iassigns to each name a member of the domain—its denotation. For example,

if the domain is the set of persons, then the name ‘a’ might be assigned me.

One-place predicates get assigned sets of 1-tuples of D—that is, just sets of

members ofD. So, a one-place predicate ‘F ’ might get assigned a set of persons.

That set is called the “extension” of the predicate—if the extension is the set

of males, then the predicate ‘F ’ might be thought of as symbolizing “is male”.

Two-place predicates get assigned sets of ordered pairs of members of D—that

is, binary relations over the domain. If a two place predicate ‘R’ is assigned

the set of persons ⟨u, v⟩ such that u is taller than v, we might think of ‘R’ as

symbolizing “is taller than”. Similarly, three-place predicates get assigned sets

of ordered triples…

Relative to any PC-model ⟨D,I ⟩, we want to de�ne the notion of truth in

a model—the corresponding valuation function. But we’ll need some apparatus

�rst. It’s pretty easy to see what truth value a sentence like F a should have. Iassigns a member of the domain to a—call that member u. I also assigns a

subset of the domain to F —let’s call that subset S . The sentence F a should be

true iff u ∈ S—that is, iff the referent of a is a member of the extension of F .

That is, F a should be true iff I (a) ∈ I (F ). Similarly, Rab should be true iff

⟨I (a),I (b )⟩ ∈ I (R). And so on.


As before, we can give recursive clauses for the truth values of negations

and conditionals. φ→ψ, for example, will be true iff either φ is false or ψ is

true.

But this becomes tricky when we try to specify the truth value of ∀xF x. It

should, intuitively, be true if and only if ‘F x’ is true, no matter what we put

in in place of ‘x’. But this is vague. Do we mean “whatever name (constant)

we put in place of ‘x”’? No, because we don’t want to assume that we’ve got a

name for everything in the domain, and what if F x is true for all the objects we

have names for, but false for one of the nameless things! Do we mean, “true no

matter what object from the domain we put in place of ‘x”’? No; objects from

the domain aren’t part of our primitive vocabulary, so the result of replacing ‘x’

with an object from the domain won’t be a formula!2

Tarski’s solution to this problem goes as follows. Initially, we don’t consider

truth values of formulas absolutely. Rather, we let the variables refer to certain

things in the domain temporarily. Then, we’ll say that ∀xF x will be true iff

for all objects u in the domain D: F x is true while x temporarily refers to u.

We implement this idea of temporary reference by de�ning the notion of a

variable assignment:

g is a variable assignment for model ⟨D,I ⟩ iff g is a function that

assigns to each variable some object in D.

The variable assignments give the “temporary” meanings to the variables; when

g (x) = u, then u is the temporary denotation of x.

We need a further bit of notation. Let u be some object in D, let g be some

variable assignment, and let α be a variable. We then de�ne “gu/α” to be the

variable assignment that is just like g , except that it assigns u to α. (If g already

assigns u to α then gu/α will be the same function as g .)

Note the following important fact about variable assignments: gu/α, when

applied to α, must give the value u. (Work through the de�nitions to see that

this is so.) That is:

gu/α(α) = u

One more bit of apparatus. Given any model M (=⟨D,I⟩), and given any

variable assignment, g , and given any term (i.e., variable or name) α, we de�ne

the denotation of α, relative toM and g , “[α]M ,g ” as follows:

2Unless the domain happens to contain members of our primitive vocabulary!


[α]M ,g =I(α) if α is a constant, and

[α]M ,g = g (α) if α is a variable.

The subscriptsM and g on [ ] indicate that denotations are assigned relative

to a model (M ), and relative to a variable assignment (g ).

Now we are ready to de�ne the valuation function, which assigns truth

values relative to variable assignments. (Relativization to assignments is necessary

because, as we noticed before, F x doesn’t have a truth value absolutely. It only

has a truth value relative to an assigned value to the variable x—i.e., relative to

a choice of an arbitrary denotation for x.)

The valuation function, VM ,g , for modelM (= ⟨D,I ⟩) and variable

assignment g , is de�ned as the function that assigns to each wff

either 0 or 1 subject to the following constraints:

i) for any n-place predicateΠ and any termsα1…αn, VM ,g (Πα1…αn)=1 iff ⟨[α1]M ,g …[αn]M ,g ⟩ ∈ I (Π)

ii) for any wffs φ, ψ, and any variable α:

a) VM ,g (∼φ)=1 iff VM ,g (φ)=0

b) VM ,g (φ→ψ)=1 iff either VM ,g (φ)=0 or VM ,g (ψ)=1

c) VM ,g (∀αφ)=1 iff for every u ∈D, VM ,gu/α(φ)=1

(In understanding clause i), recall that the one tuple containing just u, ⟨u⟩ is

just u itself. Thus, in the case where Π is F , some one place predicate, clause i)

says that VM ,g (Fα)=1 iff [α]M ,g ∈I (F ).)

So far we have de�ned the notion of truth in a model relative to a variableassignment. But what we really want is a notion of truth in a model, period—that

is, absolute truth in a model. (We want this because we want to de�ne, e.g., a

valid formula as one that is true in all models.) So, let’s de�ne absolute truth in

a model in this way:

φ is true inM iff VM ,g (φ)=1, for each variable assignment g

It might seem that this is too strict a requirement—why must φ be true relative

to each variable assignment? But in fact, it’s not too strict at all. The kinds of

formulas we’re really interested in are formulas without free variables (we’re

interested in formulas like F a, ∀xF x, ∀x(F x→Gx); not formulas like F x,


∀xRxy, etc.). And if a formula has no free variables, then if there’s even a

single variable assignment relative to which it is true, then it is true relative to

every variable assignment. (And so, we could just as well have de�ned truth in

a model as truth relative to some variable assignment.) I won’t prove this fact,

but it’s not too hard to prove; one would simply need to prove (by induction)

that, for any wff φ, if variable assignments g and h agree on all variables free

in φ, then VM ,g (φ)=VM ,h(φ).

Now we can give semantic de�nitions of the core logical notions:

φ is PC-valid (“�PCφ”) iff φ is true in all PC-models

φ is a PC-semantic consequence of Γ (“Γ �PCφ”) iff for every PC-

modelM , if each member of Γ is true inM then φ is also true in

M

Since our new de�nition of the valuation function treats the propositional

connectives→ and ∼ in the same way as the propositional logic valuation did,

it’s easy to prove that it also treats the de�ned connectives ∧, ∨, and↔ in the

same way:

VM ,g (φ∧ψ)=1 iff VM ,g (φ)=1 andVM ,g (ψ)=1

VM ,g (φ∨ψ)=1 iff VM ,g (φ)=1 or VM ,g (ψ)=1

VM ,g (φ↔ψ)=1 iff VM ,g (φ) = VM ,g (ψ)

Moreover, we can also prove that the valuation function treats ∃ as it should

(given its intended meaning):

VM ,g (∃αφ)=1 iff there is some u ∈D such that VM ,gu/α(φ)=1

This can be established as follows:

The de�nition of ∃αφ is: ∼∀α∼φ. So, we must show that for

any model, and any variable assignment g based on that model,

VM ,g (∼∀α∼φ)=1 iff there is some u ∈ D such that VM ,gu/α(φ)=1.

(In arguments like these, I’ll sometimes stop writing the subscript

M in order to reduce clutter. It should be obvious from the context

what the relevant model is.) Here’s the argument:

• Vg (∼∀α∼φ)=1 iff Vg (∀α∼φ)=0 (given the clause for ∼ in the

de�nition of the valuation function)


• But, Vg (∀α∼φ)=0 iff for some u ∈D, Vgu/α(∼φ)=0

• Given the clause for ∼, this can be rewritten as:

… iff for some u ∈D, Vgu/α(φ)=1

4.3 Establishing validity and invalidityGiven our de�nitions, we can establish that particular formulas are valid. For

example, let’s show that ∀xF x→F a is valid; that is, let’s show that for any model

⟨D,I ⟩, and any variable assignment g , Vg (∀xF x→F a) = 1.

i) Suppose otherwise; then Vg (∀xF x) = 1 and Vg (F a) = 0.

ii) Given the latter, that means that [a]g /∈I (F )—that is, I (a) /∈I (F ).

iii) Given the former, for any u ∈D, Vgu/x(F x) = 1.

iv) I (a) ∈D, and so VgI (a)/x(F x) = 1.

v) By the truth condition for atomics, [x]gI (a)/x∈I (F ).

vi) By the de�nition of the denotation of a variable, [x]gI (a)/x=

gI (a)/x(x)

vii) but gI (a)/x(x) = I (a). Thus, I (a) ∈I (F ). Contradiction

The claim in step iv) that I (a) ∈ D comes from the de�nition of an inter-

pretation function: the interpretation of a constant is always a member of the

domain. Notice that “I (a)” is a term of our metalanguage; that’s why, when

I’m given that “for any u ∈D” in step iii), I can set u equal to I (a) to obtain

step iv).

One further example; let’s establish � ∀x∀yRxy→∀xRx x:

i) Suppose for reductio that (for some assignment g in some

model), Vg (∀x∀yRxy→∀xRx x) = 0. Then Vg (∀x∀yRxy) =1 and …

ii) …Vg (∀xRx x) = 0

iii) Given ii), for some u ∈D,Vgu/x(Rx x) = 0, and so ⟨[x]gu/x

,[x]gu/x⟩ /∈

I (R)


iv) [x]gu/xis gu/x(x), i.e., u. So ⟨u, u⟩ /∈I (R)

v) Given i), for every member of D, and so for u in particular,

Vgu/x(∀yRxy) = 1

vi) given v), for every member of D, and so for u in particular,

Vgu/x u/y(Rxy) = 1

vii) given vi), ⟨[x]gu/x u/y,[y]gu/x u/y

⟩ ∈ I (R)

viii) But [x]gu/x u/yand [y]gu/x u/y

are each just u. Hence ⟨u, u⟩ ∈ I (R),contradicting iv).

We’ve seen how to establish that particular formulas are valid. How do

we show that a formula is invalid? We need to simply exhibit a single model

in which the formula is false. (The de�nition of validity speci�es that a valid

formula is true in all models; therefore, it only takes one model in which a

formula is false to make that formula invalid.) So let’s take one example; let’s

show that the formula (∃xF x∧∃xGx)→∃x(F x∧Gx) isn’t valid. To do this, we

must produce a model in which this formula is false. All we need is a single

model, since in order for the formula to be valid, it must be true in all models.

My model will contain letters in its domain:

D = {u,v}I (F ) = {u}I (G) = {v}

It is intuitively clear that the formula is false in this model. In this model,

something has F (namely, u), and something has G (namely, v), but nothing

has both.

One further example: let’s show that ∀x∃yRxy 2 ∃y∀xRxy. We must show

that the �rst formula does not semantically imply the second. So we must come

up with a model in which the �rst formula is true and the second is false. It helps

to think about natural language sentences that these formulas might represent.

If R symbolizes “respects”, then the �rst formula says that “everyone respects

someone or other”, and the second says that “there is someone whom everyone

respects”. Clearly, the �rst can be true while the second is false: suppose that

each person respects a different person, so that no one person is respected by

everyone. A simple case of this occurs when there are just two people, each of

whom respects the other, but neither of whom respects him/herself:


• (( •hh

Here is a model based on this idea:

D = {u,v}I (R) = {⟨u,v⟩, ⟨v,u⟩}

Chapter 5

Extensions of Predicate Logic

The predicate logic we considered in the previous chapter is powerful.

Much natural language discourse can be represented using it, in a way

that reveals logical structure. Nevertheless, it has its limitations. In this chapter

we consider some of its limitations, and corresponding additions to predicate

logic.

5.1 Identity“Standard” predicate logic is usually taken to include the identity sign (“=”).

“a=b” means that a and b are one and the same thing.

5.1.1 Grammar for the identity signWe �rst need to expand our de�nition of a well-formed formula. We simply

add the following clause:

If α and β are terms, then α=β is a wff

We need to beware of a potential source of confusion. We’re now using the

symbol ‘=’ as the object-language symbol for identity. But I’ve also been using

‘=’ as the metalanguage symbol for identity, for instance when I write things

like “V(φ) = 1”. This shouldn’t generally cause confusion, but if there’s a

danger of misunderstanding, I’ll clarify by writing things like: “…= (i.e., is the

same object as)…”.

80

CHAPTER 5. EXTENSIONS OF PREDICATE LOGIC 81

5.1.2 Semantics for the identity signThis is easy. We keep the notion of a PC-model from the last chapter, and

simply add to our de�nition of truth-in-a-PC-model. All we need to add is a

clause to the de�nition of a valuation function telling it what truth values to

give to sentences containing the = sign. Here is the clause:

VM ,g (α=β) = 1 iff [α]M ,g = (i.e., is the same object as) [β]M ,g

That is, the sentence α = β is true iff the terms α and β refer to the same

object.

As an example, let’s show that the formula ∀x∃y x=y is valid. We need

to show that in any model, and any variable assignment g in that model,

Vg (∀x∃y x=y) = 1. So:

i) So, suppose for reductio that for some g in some model,

Vg (∀x∃y x=y) = 0.

ii) Given the clause for ∀, for some object in the domain, call it

“u”, Vgu/x(∃y x=y) = 0.

iii) Given the clause for ∃, for every v in the domain, Vgu/x v/y(x=y)

=0.

iv) Letting v in iii) be u, we have: Vgu/x u/y(x=y) = 0.

v) So, given the clause for “=”, [x]gu/x u/yis not the same object as

[y]gu/x u/y

vi) but [x]gu/x u/yand [y]gu/x u/y

are the same object. [x]gu/x u/yis gu/x u/y(x),

i.e., u; and [y]gu/x u/yis gu/x u/y(y), i.e., u.

5.1.3 Symbolizations with the identity signWhy do we ever add anything to our list of logical constants? Why not stick

with the tried and true logical constants of propositional and predicate logic?

We generally add a logical constant when it has a distinctive inferential and

semantic role, and when it has very general application—when, that is, it occurs

in a wide range of linguistic contexts. We studied the distinctive semantic role

of ‘=’ in the previous section. In this section, we’ll look at the range of linguistic

contexts that can be symbolized using ‘=’.


The most obvious sentences that may be symbolized with ‘=’ are those that

explicitly concern identity, such as:

Mark Twain is identical to Samuel Clemens t=cEvery man fails to be identical to George Sand ∀x(M x→∼x=s)

(It will be convenient to abbreviate ∼α=β as α 6=β. Thus, the second symbol-

ization can be rewritten as: ∀x(M x→x 6=s).) But many other sentences involve

the concept of identity in subtler ways.

Consider, for example, “Every lawyer hates every other lawyer”. The ‘other’

signi�es nonidentity; we have, therefore:

∀x(Lx→∀y[(Ly∧x 6=y)→H xy])

Consider next “Only Ted can change grades”. This means: “no one other

than Ted can change grades”, and may therefore be symbolized as:

∼∃x(x 6=t∧C x)

(letting ‘C x’ symbolize “x can change grades”.)

Another interesting class of sentences concerns number. We cannot sym-

bolize “There are at least two dinosaurs” as: “∃x∃y(D x∧Dy)”, since this would

be true even if there were only one dinosaur: x and y could be assigned the

same dinosaur. The identity sign to the rescue:

∃x∃y(D x∧Dy ∧ x 6=y)

This says that there are two different objects, x and y, each of which are di-

nosaurs. To say “There are at least three dinosaurs” we say:

∃x∃y∃z(D x∧Dy∧D z∧ x 6=y ∧ x 6=z ∧ y 6=z)

Indeed, for any n, one can construct a sentence φn that symbolizes “there are

at least n F s”:

φn : ∃x1 . . .∃xn(F x1∧· · ·∧F xn ∧δ)


where δ is the conjunction of all sentences “xi 6=x j ” where i and j are integers

between 1 and n (inclusive) and i < j . (The sentence δ says in effect that no

two of the variables x1 . . . xn stand for the same object.)

Since we can construct eachφn, we can symbolize other sentences involving

number as well. To say that there are at most n F s, we write: ∼φn+1. To say

that there are between n and m F s (where m > n), we write: φn∧∼φm+1. To

say that there are exactly n F s, we write: φn∧∼φn+1.

These methods for constructing sentences involving number will always

work; but one can often construct shorter numerical symbolizations by other

methods. For example, to say “there are exactly two dinosaurs”, instead of

saying “there are at least two dinosaurs, and it’s not the case that there are at

least three dinosaurs”, we could say instead:

∃x∃y(D x∧Dy ∧ x 6=y ∧∀z[D z→(z=x∨z=y)])

5.2 Function symbolsA singular term, such as ‘Ted’, ‘New York City’, ‘George W. Bush’s father’,

or ‘the sum of 1 and 2’, is a term that purports to refer to a single entity.

Notice that some of these have semantically signi�cant structure. ‘George

W. Bush’s father’, for example, means what it does because of the meaning

of ‘George W. Bush’ and the meaning of ‘father’. But standard predicate

logic’s only (constant) singular terms are its names: a, b , c . . . , which do nothave semantically signi�cant parts. Thus, using predicate logic’s names to

symbolize semantically complex English singular terms leads to an inadequate

representation.

Suppose, for example, that we give the following symbolizations:

3 is the sum of 1 and 2

a = b

George W. Bush’s father was a politician

P c

By symbolizing ‘the sum of 1 and 2’ as simply ‘b ’, the �rst symbolization

ignores the fact that ‘1’, 2’, and ‘sum’ are semantically signi�cant constituents

of ‘the sum of 1 and 2’; and by symbolizing “George W. Bush’s father” as ‘c ’,

we ignore the semantically signi�cant occurrences of ‘George W. Bush’ and


‘father’. This is a bad idea. We ought, rather, to produce symbolizations of

these terms that take account of their semantic complexity. The symbolizations

ought to account for the distinctive logical behavior of sentences containing

the complex terms. For example, the sentence


logically implies the sentence

Someone’s father was a politician

This ought to be re�ected in the symbolizations; the �rst sentence’s symboliza-

tion ought to semantically imply the second sentence’s symbolization.

One way of doing this is via an extension of predicate logic: we add functionsymbols to its primitive vocabulary. Think of “George W. Bush’s father” as

the result of plugging “George W. Bush” into the blank in “ ’s father”. “ ’s

father” is an English function symbol. Function symbols are like predicates in

some ways. The predicate “ is happy” has a blank in it, in which you can put

a name. “ ’s father” is similar in that you can put a name into its blank. But

there is a difference: when you put a name into the blank of a predicate, you

get a complete sentence, whereas when you put a name into the blank of “ ’s

father”, you get a noun phrase, such as “George W. Bush’s father”.

Corresponding to English function symbols, we’ll add logical function

symbols. We’ll symbolize “ ’s father” as f ( ). We can put names into the

blank here. Thus, we’ll symbolize “George W. Bush’s father” as “ f (a)”, where

“a” stands for “George W. Bush”.

We need to add two more complications. First, what goes into the blank

doesn’t have to be a name—it could be something that itself contains a function

symbol. E.g., in English you can say: “George W. Bush’s father’s father”. We’d

symbolize this as: f ( f (a)).Second, just as we have multi-place predicates, we have multi-place function

symbols. “The sum of 1 and 2” contains the function symbol “the sum of

and —”. When you �ll in the blanks with the names “1” and “2”, you get the

noun phrase “the sum of 1 and 2”. So, we symbolize this using the two-place

function symbol, “s( ,—). If we let “a” symbolize “1” and “b” symbolize “2”,

then “the sum of 1 and 2” becomes: s(a, b ).The result of plugging names into function symbols in English is a noun

phrase. Noun phrases combine with predicates to form complete sentences.


Function symbols function analogously in logic. Once you combine a function

symbol with a name, you can take the whole thing, apply a predicate to it, and

get a complete sentence. Thus, the sentence:


Becomes:

P f (a)

And

3 is the sum of 1 and 2

becomes

c = s(a, b )

(here “c” symbolizes “3”). We can put variables into the blanks of function

symbols, too. Thus, we can symbolize

Someone’s father was a politician

As

∃xP f (x)

5.2.1 Grammar for function symbolsWe need to update our de�nition of a wff to allow for function symbols. First,

we need to add to our vocabulary. So, the new de�nition starts like this (the

new bit is in boldface):


i) logical: →, ∼, ∀ii) nonlogical:

a) for each n > 0, n-place predicates F ,G, . . ., with or with-

out subscripts

b) for each n > 0, n-place function symbols f , g ,…, withor without subscripts

c) variables x, y, . . ., with or without subscripts


d) individual constants (names) a, b , . . ., with or without sub-

scripts

iii) parentheses

The de�nition of a wff, actually, stays the same; all that needs to change is the

de�nition of a “term”. Before, terms were just names or variables. Now, we

need to allow for f (a), f ( f (a)), etc., to be terms. This is done by the following

recursive de�nition of a term:

De�nition of terms

i) names and variables are terms

ii) if f is an n-place function symbol, and α1…αn are terms,then f (α1,…,αn) is a term

iii) nothing else is a term

5.2.2 Semantics for function symbolsWe now need to update our de�nition of a PC-model by saying what the

interpretation of a function symbol is. That’s easy: the interpretation of an

n-place function symbol ought to be an n-place function de�ned on the model’s

domain—i.e., a rule that maps any n members of the model’s domain to another

member of the model’s domain. For example, in a model in which the one-place

function symbol f ( ) is to represent “ ’s father”, the interpretation of f will

be the function that assigns to any member of the domain that object’s father.

Here’s the new general de�nition of a model (a “PC+FS-model”, for “predicate

calculus plus function symbols”):

A PC+FS-Model is de�ned as an ordered pair ⟨D,I ⟩, such that:

i) D is a non-empty set

ii) I is a function such that:


n-tuples of members of D.

c) If f is an n-place function symbol, then I (f ) is ann-place (total) function de�ned on D.


(“Total” simply means that the function must yield an output for any n members

of D.)

The de�nition of a valuation function stays the same; all we need to do is

update the de�nition of denotation to accommodate our new complex terms.

Since we now can have arbitrarily long terms (not just names or variables), we

need a recursive de�nition:

De�nition of denotation in a modelM (= ⟨D,I ⟩):

i) if α is a constant then [α]M ,g =I (α)ii) if α is a variable then [α]M ,g = g (α)

iii) if α is a complex term f (α1,…,αn), then [α]M ,g =I (f )([α1]M ,g ,…,[αn]M ,g )

Note the recursive nature of this de�nition: the denotation of a complex term

is de�ned in terms of the denotations of its smaller parts. Let’s think carefully

about what clause iii) says. It says that, in order to calculate the denotation of

the complex term f (α1,…,αn) (relative to assignment g ), we must �rst �gure

out what I ( f ) is—that is, what the interpretation function I assigns to the

function symbol f . This object, the new de�nition of a model tells us, is an

n-place function on the domain. We then take this function, I ( f ), and apply

it to n arguments: namely, the denotations (relative to g ) of the terms α1 . . .αn.

The result is our desired denotation of f (α1,…,αn).It may help to think about a simple case. Suppose that f is a one-place

function symbol; suppose our domain consists of the set of natural numbers;

suppose that the name a denotes the number 3 in this model (i.e., I (a) = 3),

and suppose that f denotes the successor function (i.e., I ( f ) is the function,

successor, that assigns to any natural number n the number n+ 1.) In that case,

the de�nition tells us that:

[ f (a)]g =I ( f )([a]g )

=I ( f )(I (a))= successor(3)= 4

Here’s a sample metalanguage proof that makes use of the new de�nitions.

As mentioned earlier, ‘George W. Bush’s father was a politician’ logically


implies ‘ ‘Someone’s father was a politician’. Let’s show that these sentences’

symbolizations stand in the relation of semantic implication. That is, let’s show

that P f (c) � ∃xP f (x)—i.e., that in any model in which P f (c) is true, ∃xP f (x)is true:

i) Suppose that P f (c) is true in a model ⟨I ,D⟩—i.e., Vg (P f (c)) =1 (where V is the valuation for this model), for each variable

assignment g .

ii) Suppose for reductio that ∃xP f (x) is false in this model; i.e.,

for some variable assignment g , Vg (∃xP f (x)) = 0

iii) By line i), Vg (P f (c)) = 1, and so [ f (c)]g ∈ I (P ). [ f (c)]g is

just I ( f )([c]g ), and [c]g is just I (c). So I ( f )(I (c)) ∈I (P ).iv) By ii), for every object u ∈D,Vgu/x

(P f (x)) = 0.

v) I (c) ∈ D. So, by line iv), VgI (c))/x(P f (x)) = 0, and hence,

[ f (x)]gI (c)/x/∈I (P )

vi) [ f (x)]gI (c)/xis justI ( f )([x]gI (c))/x

), and [x]gI (c)/xis just gI (c)/x(x)—

i.e., I (c).vii) So I ( f )(I (c)) /∈I (P ), which contradicts line iii)

5.2.3 Symbolizations with function symbols: some exam-ples

Here are some example symbolizations involving function symbols:

Everyone loves his or her father

∀xLx f (x)No one’s father is also his or her mother

∼∃x f (x)=m(x)No one is his or her own father

∼∃x x= f (x)A person’s maternal grandfather hates that person’s paternal grand-

mother

∀x H f (m(x)) m( f (x))Every even number is the sum of two prime numbers

∀x(E x→∃y∃z(P y∧P z∧x=s(y, z)))


5.3 De�nite descriptionsOur logic has gotten more powerful with the addition of function symbols,

but it still isn’t perfect. Function symbols let us “break up” certain complex

singular terms—e.g., “Bush’s father”. But there are others we still can’t break

up—e.g., “The black cat”. Even with function symbols, the only candidate for

a direct symbolization of this phrase into the language of predicate logic is a

simple name, “a” for example. But this symbolization ignores the fact that “the

black cat” contains “black” and “cat” as semantically signi�cant constituents.

It therefore fails to provide a good model of this term’s distinctively logical

behavior. For example, ‘The black cat is happy’ logically implies ‘Some cat

is happy’. But the simple-minded symbolization of the �rst sentence, H a,

obviously does not semantically imply the obvious symbolization of the second:

∃x(C x∧H x).One response is to introduce another extension of predicate logic. We

introduce a new symbol, ι, to stand for “the”. The grammatical function of

“the” in English is to turn predicates into noun phrases. “Black cat” is a predicate

of English; “the black cat” is a noun phrase that refers to the thing that satis�es

the predicate “black cat”. Similarly, in logic, given a predicate F , we’ll let ιxF xbe a term that means: the thing that is F .

We’ll want to let ιx attach to complex predicates, not just simple predi-

cates. To symbolize “the black cat”—i.e., the thing that is both black and a

cat—we want to write: ιx(B x∧C x). In fact, we’ll let ιx attach to wffs with

arbitrary complexity. To symbolize “the �reman who saved someone”, we’ll

write: ιx(F x∧∃yS xy).

5.3.1 Grammar for ιJust as with function symbols, we need to add a bit to the primitive vocabulary,

and revise the de�nition of a term. Here’s the new grammar:


i) logical: →, ∼, ∀, ι

ii) nonlogical:

a) for each n > 0, n-place predicates F ,G, . . ., with or with-

out subscripts

b) variables x, y, . . ., with or without subscripts


c) individual constants (names) a, b , . . ., with or without sub-

scripts

De�nition of terms and wffs:

i) names and variables are terms

ii) if φ is a wff and α is a variable then ιαφ is a term

iii) ifΠ is an n-place predicate andα1…αn are terms, thenΠα1…αnis a wff

iv) if φ, ψ are wffs, and α is a variable, then ∼φ, (φ→ψ), and

∀αφ are wffs

v) nothing else is a wff or term

Notice how we needed to combine the recursive de�nitions of term and wff

into a single recursive de�nition of wffs and terms together. The reason is that

we need the notion of a wff to de�ne what counts as a term containing the ιoperator (clause ii); but we need the notion of a term to de�ne what counts as

a wff (clause iii). The way we accomplish this is not circular. The reason it isn’t

is that we can always decide, using these rules, whether a given string counts as

a wff or term by looking at whether smaller strings count as wffs or terms. And

the smallest strings are said to be wffs or terms in non-circular ways.

5.3.2 Semantics for ιWe need to update the de�nition of denotation so that ιxφ will denote the one

and only thing in the domain that is φ. This is a little tricky, though. What is

there is no such thing? Suppose that ‘K ’ symbolizes “king of” and ‘a’ symbolizes

“USA”. Then, what should ‘ιxK xa’ denote? It is trying to denote the king of

the USA, but there is no such thing. Further, what if more than one thing

satis�es the predicate? In short, what do we say about “empty descriptions”?

One approach would be to say that every atomic sentence with an empty

description is false. One way to do this is to include in each model an “emptiness

marker”, E , which is an object we assign as the denotation for each empty de-

scription. The emptiness marker shouldn’t be thought of as a “real” denotation;

when we assign it as the denotation of a description, this just marks the fact that

the description has no real denotation. We will stipulate that the emptiness

marker is not in the domain; this ensures that it is not in the extension of any


predicate, and hence that atomic sentences containing empty descriptions are

always false. Here’s how the semantics looks (“PC+DD”—“predicate calculus

plus de�nite descriptions”):

A PC+DD-Model is de�ned as an ordered triple ⟨D,I ,E ⟩, such

that:

i) D is a non-empty set

ii) E /∈ Diii) I is a function such that:


n-tuples of members of D

De�nition of denotation and valuation:

The denotation and valuation functions, [ ]M ,g and VM ,g , forPC+DD-model M (=⟨D,I ⟩) and variable assignment g , arede�ned as the functions that satisfy the following constraints:

i) VM ,g assigns to each wff either 0 or 1

ii) []M ,g assigns to each term either E , or a member of Diii) if α is a constant then [α]M ,g is I (α)iv) if α is a variable then [α]M ,g is g (α)

v) if α is a complex term ιβφ, whereβ is a variable andφ is awff, then [α]M ,g = the unique u ∈ D such that Vgu/β

(φ) = 1if there is a unique such u; otherwise [α]M ,g = E

vi) for any n-place predicate Π.and any terms α1, . . . ,αn,

VM ,g (Πα1 . . .αn) = 1 iff ⟨[α1]M ,g . . .[αn]M ,g ⟩ ∈ I (Π)vii) for any wffs φ,ψ, and any variable α:

a) VM ,g (∼φ) = 1 iff VM ,g (φ) = 0b) VM ,g (φ→ψ) = 1 iff either VM ,g (φ) = 0 or VM ,g (ψ) = 1c) VM ,g (∀αφ) = 1 iff for every u ∈D, VM ,gu/α

(φ) = 1


(As with the grammar, we need to mix together the de�nition of denotation

and the de�nition of the valuation function. The reason is that we need to

de�ne the denotation of de�nite descriptions using the valuation function (in

clause v), but we need to de�ne the valuation function using the concept of

denotation in clause vi. As before, this is not circular.)

An alternate approach would appeal to three-valued logic. We could leave

the denotation of ιxφ unde�ned if there is no object in the domain such that φ.

We could then treat any atomic sentence that contains a denotationless term as

being neither true nor false—i.e., #. We would then need to update the other

clauses to allow for #s, perhaps using the Kleene tables, perhaps some other

truth tables. I won’t explore this further.

5.3.3 Eliminability of function symbols and de�nite descrip-tions

In a sense, we don’t really need function symbols or the ι. Let’s return to

the English singular term ‘the black cat’. Introducing the ι gave us a way

to symbolize this singular term in a way that takes into account its semantic

structure (namely: ιx(B x∧C x).) But even without the ι, there is a way to

symbolize whole sentences containing ‘the black cat’, using just standard predicate

plus identity. We could, for example, symbolize

The black cat is happy

as:

∃x[ (B x∧C x)∧∀y[(By∧C y)→y=x]∧H x]

That is, “there is something such that: i) it is a black cat, ii) nothing else is a

black cat, and iii) it is happy”.

This method for symbolizing sentences containing ‘the’ is called “Russell’s

theory of descriptions”, in honor of its inventor Bertrand Russell, the 19th

and

20th

century philosopher and logician.1

The general idea is to symbolize: “the

φ is ψ” as “∃x[φ(x)∧∀y(φ(y)→x=y)∧ψ(x)]. This method can be iterated so

as to apply to sentences with two or more de�nite descriptions, such as:

The 8-foot tall man drove the 20-foot long limousine.

1See Russell (1905).


which becomes, letting ‘E ’ stand for ‘is eight feet tall’ and ‘T ’ stand for ‘is

twenty feet long’:

∃x[E x∧M x ∧∀z([E z∧M z]→x=z)∧∃y[T y∧Ly ∧∀z([T z∧Lz]→y=z)∧D xy]]

An interesting problem arises with negations of sentences involving de�nite

descriptions, when we use Russell’s method:

The president is not bald.

Does this mean:

The president is such that he’s non-bald.

which is symbolized as follows:

∃x[P x∧∀y(P y→x=y)∧∼B x]

? Or does it mean:

It is not the case that the President is bald

which is symbolized thus:

∼∃x[P x∧∀y(P y→x=y)∧B x]

? According to Russell, the original sentence is simply ambiguous. Symbolizing

it the �rst way is called “giving the description wide scope (relative to the ∼)”,

since the ∼ is inside the scope of the ∃x. Symbolizing it in the second way is

called “giving the description narrow scope (relative to the ∼)”, because the ∃xis inside the scope of the ∼.

What is the difference in meaning between these two symbolizations? The

�rst says that there really is a unique president, and adds that he is not bald.

So the �rst implies that there’s a unique president. The second merely denies

that there is a unique president, who is bald. That doesn’t imply that there’s a

unique president. It would be true if there’s a unique president who is not bald,

but it would also be true in two other cases:

i) there’s no president

ii) there is more than one president


A similar ambiguity arises with the following sentence:

The round square does not exist.

We might think to symbolize it:

∃x[Rx∧S x∧∀y([Ry∧Sy]→x=y)∧∼E x]

letting “E” stands for “exists”. In other words, we might give the description

wide scope. But this is wrong, because it says there is a certain round square that

doesn’t exist, and that’s a contradiction. This way of symbolizing the sentence

corresponds to reading the sentence as saying:

The thing that is a round square is such that it does not exist

But that isn’t the most natural way to read the sentence. The sentence would

usually be interpreted to mean:

It is not true that the round square exists.

— that is, as the negation of “the round square exists”:

∼∃x[Rx∧S x∧∀y([Ry∧Sy]→x=y)∧E x]

with the∼ out in front. Here we’ve given the description narrow scope. Notice

also that saying that x exists at the end is redundant, so we could simplify to:

∼∃x[Rx∧S x∧∀y([Ry∧Sy]→x=y)]

Again, notice the moral of these last two examples: if a de�nite description

occurs in a sentence with a ‘not’, the sentence may be ambiguous: does the

‘not’ apply to the entire rest of the sentence, or merely to the predicate?

If we are willing to use Russell’s method for translating de�nite descriptions,

we can drop ι from our language. We would, in effect, not be treating “the F ”

as a referring phrase. We would instead be paraphrasing sentences that contain

“the F ” into sentences that don’t. “The black cat is happy” got paraphrased

as: “there is something that is a black cat, is such that nothing else is a black

cat, and is happy”. See?—no occurrence of “the black cat” in the paraphrased

sentence.

In fact, once we use Russell’s method, we can get rid of function symbols too.

Given function symbols, we treated “father” as a function symbol, symbolized

it with “ f ”, and symbolized the sentence “George W. Bush’s father was a

politician” as P f (b ). But instead, we could treat ‘father of’ as a two-place

predicate, F , and regard the whole sentence as meaning: “The father of George

W. Bush was a politician.” Given the ι, this could be symbolized as:


P ιxF x b

But given Russell’s method, we can symbolize the whole thing without using

either function symbols or the ι:

∃x(F x b ∧∀y(F y b→y=x)∧P x)

We can get rid of all function symbols this way, if we want. Here’s the method:

• Take any n-place function symbol f

• Introduce a corresponding n+ 1-place predicate R

• In any sentence containing the term “ f (α1 . . .αn)”, replace each occur-

rence of this term with “the x such that R(x,α1 . . .αn)”.

• Finally, symbolize the resulting sentence using Russell’s theory of de-

scriptions

For example, let’s go back to:

Every even number is the sum of two prime numbers

Instead of introducing a function symbol s(x, y) for “the sum of x and y”, let’s

introduce a predicate letter R(z, x, y) for “z is a sum of x and y”. We then use

Russell’s method to symbolize the whole sentence thus:

∀x(E x→∃y∃z[P y∧P z∧∃w(Rwy z∧∀w1(Rw1y z→w1=w)∧x=w)])

The end of the formula (beginning with ∃w) says “the product of y and z is

identical to x”—that is, that there exists some w such that w is a product of yand z, and there is no other product of y and z other than w, and w = x.

5.4 Further quanti�ersStandard logic contains just the quanti�ers ∀ and ∃. As we have seen, using just

these quanti�ers, plus the rest of standard predicate logic, one can represent

the truth conditions of a great many sentences of natural language. But not all.

For instance, there is no way to symbolize the following sentences in predicate

logic:


Most things are massive

Most men are brutes

There are in�nitely many numbers

Some critics admire only one another

Like those sentences that are representable in standard logic, these sentences

involve quanti�cational notions: most things, some critics, and so on. In this

section we examine broader notions of quanti�ers that allow us to symbolize

these sentences.

5.4.1 Generalized monadic quanti�ersWe will generalize the idea behind the standard quanti�ers ∃ and ∀ in two

ways. To approach the �rst, think about the clauses in the de�nition of truth in

a PC-model,M , with domain D, for ∀ and ∃:

VM ,g (∀αφ) = 1 iff for every u ∈D, VM ,gu/α(φ) = 1

VM ,g (∃αφ) = 1 iff for some u ∈D, VM ,gu/α(φ) = 1

Let’s introduce the following bit of terminology. For any PC-model,M (=

⟨D,I ⟩), and wff, φ, let’s introduce a name for (roughly speaking) the set of

members ofM ’s domain of which φ is true:

φM ,g ,α =df{u : u ∈D and VM ,gu/α

(φ) = 1}

Thus, if we begin with any variable assignment g , then φM ,g ,αis the set of

things u inD such that φ is true, relative to variable assignment g (u/α). Given

this terminology, we can rewrite the clauses for ∀ and ∃ as follows:

VM ,g (∀αφ) = 1 iff φM ,g ,α =D

VM ,g (∃αφ) = 1 iff φM ,g ,α 6=∅

But if we can rewrite the semantic clauses for the familiar quanti�ers ∀ and

∃ in this way—as conditions on φM ,g ,α—then why not introduce new symbols

of the same grammatical type as ∀ and ∃, whose semantics is parallel to ∀ and

∃ except in laying down different conditions on φM ,g ,α? These would be new

kinds of quanti�ers. For instance, for any integer n, we could introduce a

quanti�er ∃n, to be read as “there exists at least n”. That is, ∃nφ means: “there

are at least n φs.” The de�nitions of a wff, and of truth in a model, would be

updated with the following clauses:


Grammar: if α is a variable and φ is a wff, then ∃nαφ is a wff

Semantics: VM ,g (∃nαφ) = 1 iff |φM ,g ,α| ≥ n

The expression |A| stands for the “cardinality” of set A—i.e., the number of

members of A. Thus, this de�nition says that ∃nαφ is true iff the cardinality of

φM ,g ,αis greater than or equal to n—i.e., this set has at least n members.

Now, the introduction of the symbols ∃n do not increase the expressive

power of predicate logic, for as we saw in section 5.1.3, we can symbolize

“there are at least n F s” using just standard predicate logic (plus “=”). The

new notation is merely a space-saver. But other such additions are not mere

space-savers. For example, by analogy with the symbols ∃n, we can introduce a

symbol ∃∞, meaning “there are in�nitely many”:

Grammar: if α is a variable and φ is a wff, then “∃∞αφ” is a wff

Semantics: VM ,g (∃∞αφ) = 1 iff |φM ,g ,α| is in�nite

As it turns out (though I won’t prove it here), the addition of ∃∞ genuinely

enhances predicate logic: no sentence of standard (�rst-order) predicate logic

has the same truth condition as does ∃∞xF x.

Another generalized quanti�er that is not symbolizable using standard

predicate logic is most:

Grammar: If α is a variable and φ is a wff, then “most αφ” is a wff

Semantics: VM ,g (most αφ) = 1 iff |φM ,g ,α|> |D −φM ,g ,α|

The minus-sign in the second clause is the symbol for set-theoretic difference:

A−B is the set of things that are in A but not in B . Thus, the de�nition says

that most αφ is true iff more things in the domain D are φ than are not φ.

One could add all sorts of additional “quanti�ers” Q in this way. Each

would be, grammatically, just like ∀ and ∃, in that each would combine with

a variable, α, and then attach to a sentence φ, to form a new sentence Qαφ.

Each of these new quanti�ers, Q, would be associated with a relation between

sets, RQ , such that Qαφ would be true in a PC-model,M , with domain D,

relative to variable assignment g , iff φM ,g ,αbears RQ to D.

If such an added symbol Q is to count as a quanti�er in any intuitive sense,

then the relation RQ can’t be just any relation between sets. It should be a

relation concerning the relative “quantities” of its relata. It shouldn’t, for

instance, “concern particular objects” in the way that the following symbol,

∃Ted-loved

, concerns particular objects:


VM ,g (∃Ted-lovedαφ) = 1 iff φM ,g ,α∩{u : u ∈D and Ted loves u} 6=∅

So we should require the following of RQ . Consider any set, D, and any one-

one function, f , from D onto another set D ′. Then, if a subset X of D bears

RQ to D, the set f [X ] must bear RQ to D ′. ( f [X ] is the image of X under

function f —i.e., {u : u ∈D ′ and u = f (v), for some v ∈D}. It is the subset of

D ′ onto which f “projects” X .)

5.4.2 Generalized binary quanti�ersWe have seen how the standard quanti�ers ∀ and ∃ can be generalized in

one way: syntactically similar symbols may be introduced and associated with

different relations between sets. Our second way of generalizing the standard

quanti�ers is to allow two-place, or binary quanti�ers. ∀ and φ are monadic

in that ∀α and ∃α attach to a single open sentence φ. Compare the natural

language monadic quanti�ers ‘everything’ and ‘something’:

Everything is material

Something is spiritual

Here, the predicates (verb phrases) ‘is material’ and ‘is spiritual’ correspond

to the open sentences of logic; it is to these that ‘everything’ and ‘something’

attach.

But in fact, monadic quanti�ers in natural language are atypical. ‘Every’

and ‘some’ typically occur as follows:

Every student is happy

Some �sh are tasty

The quanti�ers ‘every’ and ‘some’ attach to two predicates. In the �rst, ‘every’

attaches to ‘[is a] student’ and ‘is happy’; in the second, ‘some’ attaches to ‘[is

a] �sh’ and ‘[is] tasty’. In these sentences, we may think of ‘every’ and ‘some’ as

binary quanti�ers. (Indeed, one might think of ‘everything’ and ‘something’ as

the result of applying the binary quanti�ers ‘every’ and ‘some’ to the predicate

‘is a thing’.) A logical notation can be introduced which exhibits a parallel

structure, in which ∀ and ∃ attach to two open sentences. In this notation, the

form of quanti�ed sentences is:


(∀α:φ)ψ(∃α:φ)ψ

The �rst is to be read: “all φs are ψ”; the second is to be read “there is a φ that

is a ψ”. The clauses for these new binary quanti�ers in the de�nition of the

valuation function for a PC-model are:

VM ,g ((∀α:φ)ψ) = 1 iff φM ,g ,α ⊆ψM ,g ,α

VM ,g ((∃α:φ)ψ) = 1 iff φM ,g ,α ∩ψM ,g ,α 6=∅

A further important binary quanti�er is the:

Grammar: if φ and ψ are wffs and α is a variable, then (theα:φ)ψis a wff

Semantics: VM ,g ((theα:φ)ψ) = 1 iff |φM ,g ,α| = 1 and φM ,g ,α ⊆ψM ,g ,α

That is, (theα:φ)ψ is true iff i) there is exactly one φ, and ii) every φ is a

ψ. This truth condition, notice, is exactly the truth condition for Russell’s

symbolization of “the φ is a ψ”; hence the name the.

As with the introduction of the monadic quanti�ers ∃n, the introduction of

the binary existential and universal quanti�ers, and of the, does not increase the

expressive power of �rst order logic, for the same effect can be achieved with

monadic quanti�ers. (∀α:φ)ψ, (∃α:φ)ψ, and (theα:φ)ψ become, respectively:

∀α(φ→ψ)

∃α(φ∧ψ)

∃α(φ∧∀β(φβ→β=α)∧ψ)

(whereφβ isφwith free αs changed toβs.) But, as with the monadic quanti�ers

∃∞ and most, there are binary quanti�ers one can introduce that genuinely

increase expressive power. For example, most occurrences of ‘most’ in English

are binary, e.g.:

Most �sh swim

To symbolize such sentences, we can introduce a binary quanti�er most2. The

sentence (most2α:φ)ψ is to be read “most φs are ψs”. The semantic clause for

most2 is:


VM ,g ((most2α:φ)ψ) = 1 iff |φM ,g ,α ∩ψM ,g ,α|> |φM ,g ,α−ψM ,g ,α|

The binary most2 increases our expressive power, even relative to the monadic

most: not every sentence expressible with the former is equivalent to a sentence

expressible with the latter.2

5.4.3 Second-order logicAll the predicate logic we have considered so far is known as �rst-order. We’ll

now brie�y look at second-order predicate logic, a powerful extension to �rst-

order predicate logic. The distinction has to do with how variables behave, and

has syntactic and semantic aspects.

The syntactic part of the idea concerns the grammar of variables. All

the variables in �rst-order logic are grammatical terms. That is, they behave

grammatically like names: you can combine them with a predicate to get a

wff; you cannot combine them solely with other terms to get a wff; etc. In

second-order logic, on the other hand, variables can occupy predicate position.

Thus, each of the following sentences is a well-formed formula in second-order

logic:

∃X X a

∃X∃yXy

Here we say the variable X occupying predicate position. Predicate variables,

like the normal predicates of standard �rst-order logic, can be one-place, two-

place, three place, etc.

The semantic part of the idea concerns the interpretation of variables. In

�rst-order logic, a variable-assignment assigns to each variable a member of the

domain. A variable assignment in second-order logic assigns to each standard

(�rst-order) variable α a member of the domain, as before, but assigns to each

n-place predicate variable a set of n-tuples drawn from the domain. (This is

what one would expect: the semantic value of a n-place predicate is its extension,

a set of n-tuples, and variable assignments assign temporary semantic values.)

Then, the following clauses to the de�nition of truth in a PC-model must be

added:

If π is an n-place predicate variable and α1…αn are terms, then

VM ,g (πα1…αn) = 1 iff ⟨[α1]M ,g …[αn]M ,g ⟩ ∈ g (π)

2Westerståhl (1989, p. 29).


If π is a predicate variable and φ is a wff, then VM ,g (∀πφ) = 1 iff

for every set U of n-tuples from D, VM ,gU/π(φ) = 1

(where gU/π is the variable assignment just like g except in assigning U to π.)

Notice that, as with the generalized monadic quanti�ers, no alteration to the

de�nition of a PC-model is needed. All we need to do is change grammar and

the de�nition of the valuation function.

Second-order logic is different from �rst-order logic in many ways. For

instance, one can de�ne the identity predicate in second-order logic:

x=y =df∀X (X x↔Xy)

This can be seen to work correctly as follows. A one-place second-order variable

X gets assigned a set of things. Thus, the atomic sentence X x says that the

object (currently assigned to) x is a member of the set (currently assigned to)

X . Thus, ∀X (X x↔Xy) says that x and y are members of exactly the same

sets. But since x and only x is a member of {x} (i.e., x’s unit set), that means

that the only way for this to be true is for y to be identical to x.

More importantly, the metalogical properties of second-order logic are dra-

matically different from those of �rst-order logic. For instance, the axiomatic

method cannot be fully applied to second order logic. One cannot write down a

set of axioms for second-order logic that are both sound and complete—unless,

that is, one resorts to cheap tricks like saying “let every valid wff be an axiom”.

This trick is “cheap” because one would have no way of telling what an axiom

is. (More precisely, the resulting set of axioms would fail to be “recursive”.3)

Second-order logic also allows us to express claims that cannot be expressed

in �rst-order logic. Consider the “Geach-Kaplan sentence”:4

(GK) Some critics admire only one another

It can be shown that there is no way to symbolize (one reading of) the sentence

using just �rst-order logic and predicates for ‘is a critic’ and ‘admires’. The

sentence (on the desired reading) says that there is a group of critics such that

the members of that group admire only other members of the group, but one

cannot say this in �rst-order logic. However, the sentence can be symbolized

in second-order logic:

3For a rigorous statement and proof of this and other metalogical results about second-order

logic, see Boolos and Jeffrey (1989, chapter 18).

4The sentence and its signi�cance were discovered by Peter Geach and David Kaplan. See

Boolos (1984).


(GK2) ∃X [∃xX x ∧∀x(X x→C x)∧∀x∀y([X x∧Axy]→[Xy∧x 6=y)]

(GK2) “symbolizes” (GK) in the sense that it contains no predicates other than

C and A, and for every model ⟨D,I ⟩, the following is true:

(*) (GK2) is true in ⟨D,I ⟩ iff D has a nonempty subset, X , such

that i) X ⊆I (C ), and ii) whenever ⟨u, v⟩ ∈ I (A) and u ∈X ,

then v ∈X as well and v is not u.

No �rst-order sentence symbolizes the Geach-Kaplan sentence in this sense.

However, one can in a sense symbolize the Geach-Kaplan sentence using a �rst-

order sentence, provided the sentence employs, in addition to the predicates Cand A, a predicate ∈ for set-membership:

(GK1) ∃z[∃x x∈z ∧∀x(x∈z→C x)∧∀x∀y([x∈z∧Axy]→[y∈z∧x 6=y)]

(GK1) doesn’t “symbolize” (GK) in the sense of satisfying (*) in every model,

for in some models the two-place predicate ∈ doesn’t mean set-membership.

Nevertheless, if we just restrict our attention to models ⟨D,I ⟩ in which ∈ doesmean set-membership (restricted to the model’s domain, of course—that is,

I (∈) = {⟨u, v⟩ : u, v ∈D and u ∈ v}), and in which D contains each subset of

I (C ) as a member, then (GK1) will indeed satisfy (*). In essence, the difference

between (GK1) and (GK

2) is that it is hard-wired into the de�nition of truth in

a model that second-order predications Xy express set-membership, whereas

this is not hard-wired into the de�nition of the �rst-order predication y ∈ z.5

5For more on second-order logic, see Boolos (1975, 1984, 1985).

Chapter 6

Propositional Modal Logic

Modal logic is the logic of necessity and possibility. In it we treat words

like “necessary”, “could be”, “must be”, etc. as logical constants. Here

are our new symbols:

2φ: “It is necessary that φ”, “Necessarily, φ”, “It must be that φ”

3φ: “It is possible that φ”, “Possibly, φ”, “It could be that φ”, “It

can be that φ”, “It might be that φ”

The phrase “φ is possible” is sometimes used in the following sense: “φcould be true, but then again, φ could be false”. For example, if one says “it

might rain tomorrow”, one might intend to say not only that there is a possibility

of rain, but also that there is a possibility that there will be no rain. This is notthe sense of ‘possible’ that we symbolize with the 3. In our intended sense,

“possiblyφ” does not imply “possibly not-φ”. To get into the spirit of this sense,

note the naturalness of saying the following: “well of course 2+2 can equal 4,

since it does equal 4”. Here, ‘can’ is used in our intended sense: it is presumably

not possible for 2+2 to fail to be 4, and so in this case, ‘it can be the case that

2+2 equals 4’ does not imply ‘it can be the case that 2+2 does not equal 4’.

It is helpful to think of the 2 and the 3 in terms of possible worlds. A

possible world is a complete and possible scenario. Calling a scenario “possible”

means simply that it’s possible that the scenario happen, i.e., be actual. This

requirement disquali�es scenarios in which, for example, it is both raining and

also not raining (at the same time and place)—such a thing couldn’t happen, and

so doesn’t happen in any possible world. But within this limit, we can imagine

all sorts of possible worlds: possible worlds with talking donkeys, possible

103

CHAPTER 6. PROPOSITIONAL MODAL LOGIC 104

worlds in which I am ten feet tall, and so on. “Complete” means simply that no

detail is left out—possible worlds are completely speci�c scenarios. There is no

possible world in which I am “somewhere between 10 and 11 feet tall” without

being some particular height.1

Likewise, in any possible world in which I am

exactly 10 feet, six inches tall (say), I must have some particular weight, must

live in some particular place, and so on. One of these possible worlds is the

actual world—this is the complete and possible scenario that in fact obtains.

The rest of them are merely possible—they do not obtain, but would have

obtained if things had gone differently.

In terms of possible worlds, we can think of our modal operators thus:

“2φ” is true iff φ is true in all possible worlds

“3φ” is true iff φ is true in at least one possible world

It is necessarily true that all bachelors are male; in every possible world, every

bachelor is male. There might have existed a talking donkey; some possible

world contains a talking donkey. Possible worlds provide, at the very least,

a vivid way to think about necessity and possibility. How much more than a

vivid guide they provide is an open philosophical question. Some maintain that

possible worlds are the key to the metaphysics of modality, that what it is for a

proposition to be necessarily true is for it to be true in all possible worlds.2

Whether this view is defensible is a question beyond the scope of this book;

what is important for present purposes is that we distinguish possible worlds as

a vivid heuristic from possible worlds as a concern in serious metaphysics.

Our �rst topic in modal logic is the addition of the 2 and the 3 to proposi-

tional logic; the result is modal propositional logic (“MPL”). A further step will be

be modal predicate logic (chapter 9).

6.1 Grammar of MPLWe need a new language: the language of propositional modal logic. The

grammar of this language is just like the grammar of propositional logic, except

that we add the 2 as a new one-place sentence connective:


1This is not to say that possible worlds exclude vagueness.

2Sider (2003) presents an overview of this topic.


Sentence letters: P,Q, R . . . , with or without numerical

subscripts

Connectives: →,∼,2Parentheses: (,)

De�nition of wff:


ii) If φ, ψ are wffs then φ→ψ, ∼φ, and 2φ are wffs

iii) nothing else is a wff

The 2 is the only new primitive connective. But just as we were able to

de�ne ∧, ∨, and↔, we can de�ne new nonprimitive modal connectives:

3φ =df∼2∼φ “Possibly φ”

φ⇒ψ =df

2(φ→ψ) “φ strictly implies ψ”

6.2 Symbolizations in MPLModal logic allows us to symbolize a number of sentences we couldn’t symbolize

before. The most obvious cases are sentences that overtly involve “necessarily”,

“possibly”, or equivalent expressions:

Necessarily, if snow is white, then snow is white or grass is green

2[S→(S∨G)]

I’ll go if I must

2G→G

It is possible that Bush will lose the election

3L

Snow might have been either green or blue

3(G∨B)

If snow could have been green, then grass could have been white

3G→3W

‘Impossible’ and related expressions signify the lack of possibility:


It is impossible for snow to be both white and not white

∼3(W∧∼W )

If grass cannot be clever then snow cannot be furry

∼3C→∼3F

God’s being merciful is inconsistent with imperfection’s being in-

compatible with your going to heaven.

∼3(M∧∼3(I∧H ))

(M=“God is merciful”, I = “You are imperfect”, H=“You go to

heaven”)

As for the strict conditional, it arguably does a decent job of representing

certain English conditional constructions:

Snow is a necessary condition for skiing

∼W⇒∼K

Food and water are required for survival

∼(F∧W )⇒∼S

Thunder implies lightning

T⇒L

Once we add modal operators, we can expose an important ambiguity in

certain English sentences. The surface grammar of a sentence like “if Ted is a

bachelor, then he must be unmarried” is misleading: it suggests the symboliza-

tion:

B→2U

But since I am in fact a bachelor, it would follow from this symbolization that

the proposition that I am unmarried is necessarily true. But clearly I am not

necessarily a bachelor—I could have been married! The sentence is not saying

that if I am in fact a bachelor, then the following is a necessary truth: I am

married. It is rather saying that, necessarily, if I am a bachelor then I am

married:

2(B→U )

It is the relationship between my being a bachelor and my being unmarried that

is necessary. Think of this in terms of possible worlds: the �rst symbolization


says that if I am a bachelor in the actual world, then I am unmarried in every

possible world (which is absurd); whereas the second one says that in each

possible world, w, if I am a bachelor in w, then I am unmarried in w (which

is quite sensible). The distinction between φ→2ψ and 2(φ→ψ) is called

the distinction between the “necessity of the consequent” (�rst sentence) and

the “necessity of the consequence” (second sentence). It is important to keep

the distinction in mind, because of the fact that English surface structure is

misleading.

English modal words are ambiguous in a systematic way. For example,

suppose I say that I can’t attend a certain conference in Cleveland. What is the

force of “can’t” here? Probably I’m saying that my attending the conference

is inconsistent with honoring other commitments I’ve made at that time. But

notice that another sentence I might utter is: “I could attend the conference;

but I would have to cancel my class, and I don’t want to do that.” Now I’ve

said that I can attend the conference; have I contradicted my earlier assertion

that I cannot attend the conference? No—what I mean now is perhaps that I

have the means to get to Cleveland on that date. I have shifted what I mean by

“can”.

In fact, there are a lot of things one could mean by a modal word like ‘can’.

Examples:

I can come to the party, but I can’t stay late. (“can” = “is not inconve-

nient”)

Humans can travel to the moon, but not Mars. (“can” = “is achievable

with current technology”)

Objects can move almost as fast as the speed of light, but nothing can travelfaster than light. (“can” = “is consistent with the laws of nature”)

Objects could have traveled faster than the speed of light (if the lawsof nature had been different), but no matter what the laws had been,nothing could have traveled faster than itself. (“can” = “metaphysical

possibility”)

You can borrow but you can’t steal. (“can” = “morally acceptable”)

So when representing English sentences using the 2 and the 3, one should

keep in mind that these expressions can be used to express different strengths

of necessity and possibility. (Though we won’t do this, one could introduce

different symbols for different sorts of possibility and necessity.)


The different strengths of possibility and necessity can be made vivid by

thinking, again, in terms of possible worlds. As we saw, we can think of the 2

and the 3 as quanti�ers over possible worlds (the former a universal quanti�er,

the latter an existential quanti�er). The very broad sort of possibility and

necessity, metaphysical possibility and necessity, can be thought of as a completely

unrestricted quanti�er: a statement is necessarily true iff it is true in all possible

worlds whatsoever. The other kinds of possibility and necessity can be thought

of as resulting from various restrictions on the quanti�ers over possible worlds.

Thus, when ‘can’ signi�es achievability given current technology, it means:

true in some possible world in which technology has not progressed beyond where it hasprogressed in fact at the current time; when ‘can’ means moral acceptability, it

means: true in some possible world in which nothing morally forbidden occurs; and so

on.

6.3 Semantics for MPLAs usual, let’s consider semantics �rst. As always, our goal is to model how

statements involving the 2 and 3 are made true by the world, in order to shed

light on the meaning of these connectives, and in order to provide semantic

de�nitions of the notions of logical truth and logical consequence.

In constructing a semantics for MPL, we face two main challenges, one

philosophical, the other technical. The philosophical challenge is simply that

it isn’t wholly clear which formulas of MPL are indeed logical truths. It’s hard

to construct an engine to spit out logical truths if you don’t know which logical

truths you want it to spit out. With a few exceptions, there is widespread

agreement over which formulas of nonmodal propositional and predicate logic

are logical truths. But for modal logic, this is less clear, especially for sentences

that contain iterations of modal operators. Is 2P→22P a logical truth? It’s

hard to say.

A quick peek at the history of modal logic is in order. Modal logic arose

from dissatisfaction with the material conditional→ of standard propositional

logic. The material conditional φ→ψ is true whenever φ is false or ψ is true;

but in expressing the conditionality of ψ on φ, we sometimes want to require a

tighter relationship: we want it not to be a mere accident that either φ is false or

ψ is true. To express this tighter relationship, C. I. Lewis introduced the strict

conditional φ⇒ψ, which he de�ned, as above, as 2(φ→ψ).3 Thus de�ned,

3See Lewis (1918); Lewis and Langford (1932).


φ⇒ψ isn’t automatically true just because φ is false or ψ is true. It must be

necessarily true that either φ is false or ψ is true.

Lewis then asked: what principles govern this new symbol 2? Certain

principles seemed clearly appropriate, for instance: 2(φ→ψ)→(2φ→2ψ).Others were less clear. Is 2φ→22φ a logical truth? What about 32φ→φ?

Lewis’s solution to this problem was not to choose. Instead, he formulated

several different modal systems. He did this axiomatically, by formulating differ-

ent systems that differed from one another by containing different axioms and

hence different theorems.

We will follow Lewis’s approach, and construct several different modal

systems. Unlike Lewis, we’ll do this semantically at �rst (the semantics for

modal logic we will study was published by Saul Kripke in the 1950s, long

after Lewis was writing), by constructing different de�nitions of a model for

modal logic. The de�nitions will differ from one another in ways that result

in different sets of valid formulas. In section 6.4 we’ll study Lewis’s axiomatic

systems, and in sections 6.5 and 6.6 we’ll discuss the relationship between the

semantics and the axiom systems.

Formulating multiple systems does not answer the philosophical question

of which formulas of modal logic are logically true; it merely postpones it.

The question re-arises when we want to apply Lewis’s systems; when we ask

which system is the correct system—i.e., which one correctly mirrors the logical

properties of the English words ‘possibly’ and ‘necessarily’? (Note that since

there are different sorts of necessity and possibility, different systems might

correctly represent different sorts of necessity.) But we won’t try to address

such philosophical questions here.

The technical challenge to constructing a semantics for MPL is that the

modal operators 2 and 3 are not truth functional. A (sentential) connective is an

expression that combines with sentences to make new sentences. A one-place

connective combines with one sentence to form a new sentence. ‘It is not the

case that’ is a one-place connective of English—the ∼ is a one-place connective

in the language of PL. A connective is truth-functional iff whenever it combines

with sentences to form a new sentence, the truth value of the resulting sentence

is determined by the truth value of the component sentences. Many think

that ‘and’ is truth-functional, since they think that an English sentence of the

form “φ and ψ” is true iff φ and ψ are both true. But ‘necessarily’ is not truth-

functional. Suppose I tell you the truth value of φ; will you be able to tell me

the truth value of this sentence? Well, if φ is false then presumably you can (it

is false), but if φ is true, then you still don’t know. If φ is “Ted is a philosopher”


then “Necessarily φ” is false, but if φ is “Either Ted is a philosopher or he isn’t

a philosopher” then “Necessarily φ” is true. So the truth value of “Necessarily

φ” isn’t determined by the truth value of φ. Similarly, ‘possibly’ isn’t truth-

functional either: ‘I might have been six feet tall’ is true, whereas ‘I might have

been a round square’ is false, despite the fact that ‘I am six feet tall’ and ‘I am a

round square’ each have the same truth value (they’re both false.)

Since the 2 and the 3 are supposed to represent ‘necessarily’ and ‘possibly’,

respectively, and since the latter aren’t truth-functional, we can’t use the method

of truth tables to construct the semantics for the 2 and the 3. For the method

of truth tables assumes truth-functionality. Truth tables are just pictures of truth

functions: they specify what truth value a complex sentence has as a function of

what truth values its parts have. Imagine trying to construct a truth table for

the 2. It’s presumably clear (though see the discussion of systems K, D, and T

below) that 2φ should be false if φ is false, but what about when φ is true?:

2 φ? 1

0 0

There’s nothing we can put in this slot in the truth table, since when φ is true,

sometimes 2φ is true and sometimes it is false.

Our challenge is clear: we need a semantics for the 2 and the 3 other than

the method of truth tables.

6.3.1 RelationsBefore we investigate how to overcome this challenge, a digression is necessary.

Recall our discussion of ordered sets from section 1.8. In addition to their use

in constructing models, ordered sets are also useful for constructing relations.We take a binary (2-place) relation to be a set of ordered pairs. For example,

the taller-than relation may be taken to be the set of ordered pairs ⟨u, v⟩ such

that u is taller than v. The less-than relation for positive integers is the set

of ordered pairs ⟨m, n⟩ such that m is a positive integer less than n, another

positive integer. That is, it is the following set:

{⟨1,2⟩, ⟨1,3⟩, ⟨1,4⟩ . . . ⟨2,3⟩, ⟨2,4⟩…}

When ⟨u, v⟩ is a member of relation R, we say that u and v stand in R, or

that u bears R to v. Most simply, we write “Ruv”. (This notation is like that


of predicate logic; but here I’m speaking the metalanguage, not displaying

sentences of a formalized language.)

Some de�nitions. Let R be any binary relation.

• The domain of R (“dom(R)”) is the set {u: for some v, Ruv}• The range of R (“ran(R)”) is the set {u: for some v, Rv u}• R is over A iff dom(R)⊆A and ran(R)⊆A

In other words, the domain of R is the set of all things that bear R to something;

the range is the set of all things that something bears R to; and R is over A iff

the members of the ’tuples in R are all drawn from A.

Let R be any binary relation over A. Then we de�ne the following notions

with respect to A:

• R is serial (in A) iff for every u ∈A, there is some v ∈A such

that Ruv.

• R is re�exive (in A) iff for every u ∈A, Ru u

• R is symmetric iff for all u, v, if Ruv then Rv u

• R is transitive iff for any u, v, w, if Ruv and Rvw then Ruw

• R is an equivalence relation (in A) iff R is symmetric, transitive,

and re�exive (in A)

• R is total (in A) iff for every u, v ∈A, Ruv

Notice that we relativize some of these notions to a given set A. We de�ne

the notion of re�exivity relative to a set, for example. We do this because the

alternative would be to say that a relation is re�exive simpliciter if everythingbears R to itself; but that would require the domain and range of any re�exive

relation to be the set of absolutely all objects. It’s better to introduce the notion

of being re�exive relative to a set, which is applicable to relations with smaller

domains and ranges. (I will sometimes omit the quali�er ‘in A’ when it is clear

which set that is.) Why don’t symmetry and transitivity have to be relativized

to a set?—because they only say what must happen if R holds among certain

things. Symmetry, for example, says merely that if R holds between u and v,

then it must also hold between v and u, and so we can say that a relation is

symmetric absolutely, without implying that everything is in its domain and

range.


6.3.2 Kripke modelsNow we’re ready to introduce a semantics for MPL. As we saw, we can’t

construct truth tables for the 2 or the 3. Instead, we will pursue an approach

called possible-worlds semantics. The intuitive idea is to count 2φ as being true

iff φ is true in all possible worlds, and 3φ as being true iff φ is true in some

possible worlds. More carefully: we are going to develop models for modal

propositional logic. These models will contain objects we will call “possible

worlds”. And formulas are going to be true or false “at” these worlds—that

is, we are going to assign truth values to formulas in these models relative to

possible worlds, rather than absolutely. Truth values of propositional-logic

compound formulas—that is, negations and conditionals—will be determined

by truth tables within each world; ∼φ, for example, will be true at a world iff φis false at that world. But the truth value of 2φ at a world won’t be determined

by the truth value of φ at that world; the truth value of φ at other worlds will

also be relevant.

Speci�cally, 2φ will count as true at a world iff φ is true at every world that

is “accessible” from the �rst world. What does “accessible” mean? Each model

will come equipped with a binary relation, R , that holds between possible

worlds; we will say that world v is “accessible from” world w whenRwv . The

intuitive idea is thatRwv if and only if v is possible relative to w. That is, if you

live in world w, then from your perspective, the events in world v are possible.

The idea that what is possible might vary depending on what possible

world you live in might at �rst seem strange, but it isn’t really. “It is physically

impossible to travel faster than the speed of light” is true in the actual world,

but false in worlds where the laws of nature allow faster-than-light travel.

On to the semantics. We �rst de�ne a general notion of a MPL model,

which we’ll then use to give a semantics for each of our systems:

An MPL-model is an ordered triple, ⟨W ,R ,I ⟩, such that:

i) W is a non-empty set of objects (“possible worlds”)

ii) R is a binary relation overW (“accessibility relation”)

iii) I is a two-place function (the “interpretation” function) that

assigns a 0 or 1 to each sentence letter, relative to (“at”, or

“in”) each world—that is, for any sentence letter α, and any

w ∈W ,I (α, w) is either 0 or 1.


Each MPL-model contains a setW of possible worlds, and an accessibility

relationR . ⟨W ,R⟩ is sometimes called the model’s frame. Think of the frame

as a map of the “structure” of the model’s space of possible worlds: it contains

information about how many worlds there are, and which worlds are accessible

from which. In addition to a frame, each model also contains an interpretation

function I , which assigns truth values to sentence letters.

A model’s interpretation function assigns truth values only to sentence

letters. But the sum total of all the truth values of sentence letters relative to

worlds determines the truth values of all complex wffs, again relative to worlds.

It is the job of the model’s valuation function to specify exactly how these truth

values get determined:

WhereM (= ⟨W ,R ,I ⟩) is any MPL-model, the valuation forM ,

VM , is de�ned as the two-place function that assigns either 0 or 1to each wff relative to each member ofW , subject to the following

constraints, where α is any sentence letter, φ and ψ are any wffs,

and w is any member ofW :

a) VM (α, w) =I (α, w)b) VM (∼φ, w) = 1 iff VM (φ, w) = 0

c) VM (φ→ψ, w) = 1 iff either VM (φ, w) = 0 or VM (ψ, w) = 1

d) VM (2φ, w) = 1 iff for each v ∈ W , ifRwv, then VM (φ, v) =1

What about the truth values for complex formulas that contain ∧,∨,↔, and

3? Given the de�nition of these de�ned connectives in terms of the primitive

connectives, it is easy to prove that the following derived conditions hold:

i) VM (φ∧ψ, w) = 1 iff VM (φ, w) = 1 and VM (ψ, w) = 1

ii) VM (φ∨ψ, w) = 1 iff VM (φ, w) = 1 or VM (ψ, w) = 1

iii) VM (φ↔ψ, w) = 1 iff VM (φ, w) =VM (ψ, w)iv) VM (3φ, w) = 1 iff for some v ∈W ,Rwv and VM (φ, v) = 1

So far, we have introduced a general notion of an MPL model, and have

de�ned the notion of a wff’s being true at a world in an MPL model. Next, let

us consider how to de�ne validity.

Remember that our overall strategy is C. I. Lewis’s: we want to construct

different modal systems, since it isn’t obvious which formulas ought to count as


logical truths. The systems will be named: K, D, T, B, S4, S5. Each system will

come with its own de�nition of a model. As a result, different formulas will

come out valid in the different systems. For example, as we’ll see, the formula

2P→22P is going to come out valid in S4 and S5, but not in the other systems.

Here are the de�nitions:

• A K-model is de�ned as any MPL-model

• A D-model is any MPL-model whose accessibility relation is

serial (i.e., any model ⟨W ,R ,I ⟩ in whichR is serial inW )

• A T-model is any MPL-model whose accessibility relation is

re�exive (inW )

• A B-model is any MPL-model whose accessibility relation is

re�exive (inW ) and symmetric

• An S4-model is any MPL model whose accessibility relation is

re�exive (inW ) and transitive

• An S5-model is any MPL model whose accessibility relation is

re�exive (inW ), symmetric, and transitive

A formula φ is valid in modelM (= ⟨W ,R ,I ⟩ iff for every w ∈W ,

VM (φ, w) = 1

A formula is valid in system S (where S is either K, D, T, B, S4, or

S5) iff it is valid in every S-model

Notice that for each system, the valid formulas are de�ned as the formulas

that are valid in every model in which the accessibility relation has a certain

formal feature. The systems differ from from one another by what that formal

feature is. For T it is re�exivity: a formula is T-valid iff it is valid in every

model in which the accessibility relation is re�exive. For S4 the formal feature

is re�exivity + transitivity. Other systems correspond to other formal features.

As before, we’ll use the � notation for validity. But since we have many

modal systems, if we claim that a formula is valid, we’ll need to indicate which

system we’re talking about. Let’s do that by subscripting � with the name of

the system; thus, “�Tφ” means that φ is T-valid.

It’s important to get clear on the status of possible-worlds lingo here. Where

⟨W ,R ,I ⟩ is a model, we call the members ofW “worlds”, and we callR the

“accessibility” relation. Now, there is no question that “possible worlds” is a

vivid way to think about necessity and possibility. But of�cially,W is nothing


but a nonempty set, any old nonempty set. Its members needn’t be the kinds

of things metaphysicians call possible worlds: they can be numbers, people,

bananas—whatever you like. Similarly,R is just de�ned to be any old binary

relation onR ; it needn’t have anything to do with the metaphysics of modality.

Of�cially, then, the possible-worlds talk we use to describe our models is just

talk, not heavy-duty metaphysics. Still, models are usually intended to modelsomething—to depict some aspect of the dependence of truth on the world.

So if modal sentences of English containing ‘necessarily’ and ‘possibly’ aren’t

made true by anything like possible worlds, it’s hard to see why possible worlds

models would shed any light on their meaning, or why truth-in-all-possible-

worlds-models would be a good way of modeling (genuine) validity for modal

statements. At any rate, this philosophical issue should be kept in mind. Back,

now, to the formalism.

6.3.3 Semantic validity proofsGiven our de�nition of validity, one can now show that a certain formula is

valid in a given system. First, a very simple example. Let us prove that the

formula 2(P∨∼P ) is K-valid. That is, we must prove that this formula is valid

in every MPL-model, since that is the de�nition of K-validity. Being valid

in a model means being true at every world in the model. So, consider any

MPL-model ⟨W ,R ,I ⟩, and let w be any world in W . We must prove that

VM (2(P∨∼P ), w) = 1. (As before, I’ll start to omit the subscriptM on VMwhen it’s clear which model we’re talking about.)

i) Suppose for reductio that V(2(P∨∼P ), w) = 0

ii) So, by the truth condition for the 2 in the de�nition of the

valuation function, there is some world, v, such that Rwvand V(P∨∼P, v) = 0

iii) Given the truth condition for the∨, V(P, v) = 0 and V(∼P, v) =0

iv) Since V(∼P, v) = 0, given the truth condition for the ∼,

V(P, v) = 1. But that’s impossible; V(P, v) can’t be both 0and 1.

Thus, �K

2(P∨∼P ). Note that similar reasoning would establish �Kφ, for

any propositional-logic tautology φ. The reason is this: within any world, the


truth values of complex statements of propositional logic are determined by

the truth values of their constituents in that world by the usual truth tables. So

if φ is a PL-tautology, it will be true in any world in any model; hence 2φ will

turn out true in any world in any model.

Another example: let’s show that �T(32(P→Q)∧2P )→3Q. We must

show that V((32(P→Q)∧2P )→3Q, w) = 1 for the valuation V for an arbi-

trarily chosen model and world w in that model.

i) Assume for reductio that V((32(P→Q)∧2P )→3Q, w) = 0

ii) So V(32(P→Q)∧2P, w) = 1 and …

iii) …V(3Q, w) = 0

iv) From ii), 32(P→Q) is true at w, and so V(2(P→Q), v) = 1,

for some world, call it v, such thatRwv

v) From ii), V(2P, w) = 1. So, by the truth condition for the

2, P is true in every world accessible from w; sinceRwv, it

follows that V(P, v) = 1.

vi) From iv), P→Q is true in every world accessible from v ; since

our model is a T-model, R is re�exive. So Rvv; and so

V(P→Q, v) = 1

vii) From v) and vi), by the truth condition for the→, V(Q, v) = 1

viii) Given iii), Q is false at every world accessible from w; this

contradicts vii)

OK, we’ve shown that the formula (32(P→Q)∧2P )→3Q is valid in T.

Suppose we were interested in showing that this formula is also valid in S4.

What more would we have to do? Nothing! To be S4-valid is to be valid in

every S4-model; but a quick look at the de�nitions shows that every S4-model

is a T-model. So, since we already know that the the formula is valid in all

T-models, we already know that it must be valid in all S4-models (and hence,

S4-valid), without doing a separate proof.

Think of it another way. To do a proof that the formula is S4-valid, we need

to do a proof in which we are allowed to assume that the accessibility relation is

both transitive and re�exive. And the proof above did just that. We didn’t ever

use the fact that the accessibility relation is transitive—we only used the fact

that it is re�exive (in line 9). But we don’t need to use everything we’re allowed

to assume.


In contrast, the proof above doesn’t establish that this formula is, say, K-valid.

To be K-valid, the formula would need to be valid in all models. But some

models don’t have re�exive accessibility relations, whereas the proof we gave

assumed that the accessibility relation was re�exive. And in fact the formula

isn’t in fact K-valid, as we’ll show how to demonstrate in the next section.

Consider the following diagram of systems:

S5

S4

==||||||B

``@@@@@@

T

>>~~~~~~

aaBBBBBB

D

OO

K

OO

An arrow from one system to another indicates that validity in the �rst system

implies validity in the second system. For example, if a formula is D-valid, then

it’s also T-valid. The reason is that if something is valid in all D-models, then,

since every T-model is also a D-model (since re�exivity implies seriality), it

must be valid in all T-models as well.

S5 is the strongest system, since it has the most valid formulas. (That’s

because it has the fewest models—it’s easier to be S5-valid because there are

fewer potentially falsifying models.)

Notice that the diagram isn’t linear. That’s because of the following. Both B

and S4 are stronger than T; each contains all the T-valid formulas. But neither

B nor S4 is stronger than the other—each contains valid formulas that the

other doesn’t. (They of course overlap, because each contains all the T-valid

formulas.) S5 is stronger than each; S5 contains all the valid formulas of each.

These relationships between the systems will be exhibited below.

Suppose you are given a formula, and for each system in which it is valid,

you want to give a semantic proof of its validity. This needn’t require multiple

semantic proofs—as we have seen, one semantic proof can do the job. To prove

that a certain formula is valid in a number of systems, it suf�ces to prove that it

is valid in the weakest possible system. Then, that very proof will automatically

be a proof that it is valid in all stronger systems. For example, a proof that a


formula is valid in K would itself be a proof that the formula is D, T, B, S4,

and S5-valid. Why? Because every model of any kind is a K-model, so K-valid

formulas are always valid in all other systems.

In general, then, to show what systems a formula is valid in, it suf�ces to

give a single semantic proof of it, namely, a semantic proof in the weakest

system in which it is valid. There is an exception, however, since neither B

nor S4 is stronger than the other. Suppose a formula is not valid in T, but one

has given a semantic proof its validity in B. This proof also establishes that the

formula is also valid in S5, since every S5 model is a B-model. But one still

doesn’t yet know whether the formula is S4-valid, since not every S4-model is a

B-model. Another semantic proof may be needed: of the formula’s S4-validity.

(Of course, the formula may not be S4-valid.)

So: when a wff is valid in both B and S4, but not in T, two semantic proofs

of its validity are needed.

I’ll present some more sample validity proofs below, but it’s often easier to

do proofs of validity when one has failed to construct a counter-model for a

formula. So let’s look �rst at counter-modeling.

6.3.4 CountermodelsWe have a de�nition of validity for the various systems, and we’ve shown how

to establish validity of particular formulas. Now we’ll investigate establishing

invalidity.

Let’s show that the formula 3P→2P is not K-valid. A formula is K-valid if

it is valid in all K-models, so all we must do is �nd one K-model in which it

isn’t valid. What follows is a procedure for doing this:4

Place the formula in a box

The goal is to �nd some model, and some world in the model, where the

formula is false. Let’s start by drawing a box, which represents some chosen

world in the model we’ll construct. The goal is to make the formula false in

this world. In these examples I’ll always call this �rst world “r”:

3P→2Pr

4This procedure is from Cresswell and Hughes (1996).


Now, since the box represents a world, we should have some way of representing

the accessibility relation. What worlds are possible, relative to r; what worlds

does r “see”? Well, to represent one world (box) seeing another, we’ll draw

an arrow from the �rst to the second. However in the case of this particular

model, we don’t need to make this world r see anything. After all, we’re trying

to construct a K-model, and the accessibility relation of a K-model doesn’t

even need to be serial—no world needs to see any worlds at all. So, we’ll forget

about arrows for the time being.

Make the formula false in the world

We will indicate a formula’s truth value (1 or 0) by writing it above the formula’s

major connective. So to indicate that 3P→2P is to be false in this model, we’ll

put a 0 above its arrow:

0

3P→2Pr

Enter in forced truth values

If we want to make the 3P→2P false in this world, the de�nition of a valuation

function requires us to assign certain other truth values. Whenever a conditional

is false at a world, its antecedent is true at that world and its consequent is false

at that world. So, we’ve got to enter in more truth values; a 1 over the major

connective of the antecedent (3P ), and a 0 over the major connective of the

consequent (2P ):

1 0 0

3P→2Pr

Enter asterisks

When we assign a truth value to a modal formula, we thereby commit ourselves

to assigning certain other truth values to various formulas at various worlds.

For example, when we make 3P true at r, we commit ourselves to making Ptrue at some world that r sees. To remind ourselves of this commitment, we’ll

put an asterisk (*) below 3P . An asterisk below indicates a commitment to there

being some world of a certain sort. Similarly, since 2P is false at r, this means


that P must be false in some world P sees (if it were true in all such worlds,

then by the semantic clause for the 2, 2P would be true at r). We again have a

commitment to there being some world of a certain sort, so we enter an asterisk

below 2P as well:

1 0 0

3P→2P∗ ∗

r

Discharge bottom asterisks

The next step is to ful�ll the commitments we incurred by adding the bottom

asterisks. For each, we need to add a world to the diagram. The �rst asterisk

requires us to add a world in which P is true; the second requires us to add a

world in which P is false. We do this as follows:

1 0 0

3P→2P∗ ∗

r

��

��

��

��???

????

???

1Pa

0Pb

What I’ve done is added two more worlds to the diagram: a and b. P is true in

a, but false in b. I have thereby satis�ed my obligations to the asterisks on my

diagram, for r does indeed see a world in which P is true, and another in which

P is false.

The of�cial model

We now have a diagram of a K-model containing a world in which 3P→2Pis false. But we need to produce an of�cial model, according to the of�cial

de�nition of a model. A model is an ordered triple ⟨W ,R ,I ⟩, so we must

specify the model’s three members.

The set of worlds We �rst must specify the set of worlds,W . W is simply

the set of worlds I invoked:

W = {r, a,b}


But what are r, a, and b? Let’s just take them to be the letters ‘r’, ‘a’, and ‘b’. No

reason not to—the members ofW , recall, can be any things whatsoever.

The accessibility relation Next, for the accessibility relation. This is

represented on the diagram by the arrows. In our model, there is an arrow

from r to a, an arrow from r to b, and no other arrows. Thus, the diagram

represents that r sees a, that r sees b, and that there are no further cases of

seeing. Now, remember that the accessibility relation, like all relations, is a set

of ordered pairs. So, we simply write out this set:

R = {⟨r, a⟩, ⟨r,b⟩}

That is, we write out the set of all ordered pairs ⟨w1, w2⟩ such that w1 “sees”

w2.

The interpretation function Finally, we need to specify the interpreta-

tion function, I , which assigns truth values to sentence letters at worlds. In

our model, I must assign 1 to P at world a, and 0 to P at world b. Now, our

of�cial de�nition requires an interpretation to assign a truth value to each of

the in�nitely many sentence letters at each world; but so long as P is true at

world a and false at world b, it doesn’t matter what other truth values I assigns.

So let’s just (arbitrarily) choose to make all other sentence letters false at all

worlds in the model. We have, then:

I (P, a) = 1

I (P, b) = 0

for all other sentence letters α and worlds w, I (α, w) = 0

That’s it—we’re done. We have produced a model in which 3P→2P is false

at some world; hence this formula is not valid in all models; and hence it’s not

K-valid: 2K

3P→2P .

Check the model

At the end of this process, it’s a good idea to check to make sure that your

model is correct. This involves various things. First, make sure that you’ve

succeeded in producing the correct kind of model. For example, if you’re trying

to produce a T-model, make sure that the accessibility relation you’ve written


down is re�exive. (In our case, we were only trying to construct a K-model, and

so for us this step is trivial.) Secondly, make sure that the formula in question

really does come out false at one of the worlds in your model.

Simplifying models

Sometimes a model can be simpli�ed. Consider the diagram of the �nal version

of the model above:

1 0 0

3P→2P∗ ∗

r

��

��

��

��???

????

???

1Pa

0Pb

We needn’t have used three worlds in the model. When we discharged the �rst

asterisk, we needed to put in a world that r sees, in which P is true. But we

needn’t have made that a new world—we could have simply have made P true in

r. Of course we couldn’t haven’t done that for both asterisks, because that would

have made P both true and false at r. So, we could make one simpli�cation:

1 1 0 0

3P→2P∗ ∗

r

��

00

0Pb

The of�cial model would then look as follows:

W : {r,b}R : {⟨r, r⟩, ⟨r,b⟩}I (P, r) = 1; otherwise, everything false

Adapting models to different systems

We have showed that 3P→2P is not K-valid. Now, let’s show that this formula

isn’t D-valid, i.e. that it is false in some world of some model with a serial


accessibility relation (i.e., some “D-model”). Well, we haven’t quite done this,

since the model above does not have a serial accessibility relation. But we can

easily change this, as follows:

1 1 0 0

3P→2P∗ ∗

r

��

00

0Pb

00

Of�cial model:

W : {r,b}R : {⟨r, r⟩, ⟨r,b⟩, ⟨b,b⟩}I (P, r) = 1; otherwise, everything false

That was easy—adding the fact that b sees itself didn’t require changing any-

thing else in the model.

Suppose we want now to show that 3P→2P isn’t T-valid. Well, we’ve

already done so! Why? Because we’ve already produced a T-model in which

this formula is false. Look back at the most recent model. Its accessibility

relation is re�exive. So it’s a T-model already. In fact, that accessibility relation

is also already transitive, so it’s already an S4-model.

So far we have established that 2K,D,T,S4

3P→2P . What about B and S5?

It’s easy to revise our model to make the accessibility relation symmetric:

1 1 0 0

3P→2P∗ ∗

r

OO

��

00

0Pb

00

Of�cial model:

W : {r,b}

R : {⟨r,r⟩,⟨r,b⟩,⟨b,b⟩,⟨b,r⟩}


I (P, r) = 1; otherwise, everything false

Now, we’ve got a B-model, too. What’s more, we’ve also got an S5-model:

notice that the accessibility relation is an equivalence relation. (In fact, it’s also

a total relation.)

So, we’ve succeeded in establishing that 3P→2P is not valid in any of our

systems. Notice that we could have done this more quickly, if we had given the

�nal model in the �rst place. After all, this model is an S5, S4, B, T, D, and

K-model. So one model establishes that the formula isn’t valid in any of the

systems.

In general, in order to establish that a formula is invalid in a number of

systems, try to produce a model for the strongest system (i.e., the system with

the most requirements on models). If you do, then you’ll automatically have a

model for the weaker systems. Keep in mind the diagram of systems:

S5

S4

==||||||B

``@@@@@@

T

>>~~~~~~

aaBBBBBB

D

OO

K

OO

An arrow from one system to another, recall, indicates that validity in the �rst

system implies validity in the second. The arrows also indicate facts about

invalidity, but in reverse: when an arrow points from one system to another,

then invalidity in the second system implies invalidity in the �rst. For example,

if a wff is invalid in T, then it is invalid in D. (That’s because every T-model is

a D-model; a countermodel in T is therefore a countermodel in D.)

When our task is to discover which systems a given formula is invalid in,

usually only one countermodel will be needed—a countermodel in the strongest

system in which the formula is invalid. But there is an exception involving B

and S4. Suppose a given formula is valid in S5, but we discover a model showing

that it isn’t valid in B. That model is automatically a T, D, and K-model, so we

know that the formula isn’t T, D, or K-valid. But we don’t yet know about that

formula’s S4-validity. If it is S4-invalid, then we will need to produce a second


countermodel, an S4 countermodel. (Notice that the B-model couldn’t alreadybe an S4-model. If it were, then its accessibility relation would be re�exive,

symmetric, and transitive, and so it would be an S5 model, contradicting the

fact that the formula was S5-valid.)

Additional steps in countermodelling

I gave a list of steps in constructing countermodels:

1. Place the formula in a box

2. Make the formula false in the world

3. Enter in forced truth values

4. Enter asterisks

5. Discharge bottom asterisks

6. The of�cial model

We’ll need to adapt this list.

Above asterisks Let’s try to get a countermodel for 32P→23P in all the

systems in which it is invalid, and a semantic validity proof in all the systems in

which it is valid. We always start with countermodelling before doing semantic

validity proofs, and when doing countermodelling, we start by trying for a

K-model. After the �rst few steps, we have:

1 0 0

32P→23P∗ ∗

r

}}{{{{

{{{{

{{{

!!CCCC

CCCC

CCC

1

2Pa

0

3Pb

At this point, we’ve got a true 2, and a false 3. Take the �rst: a true 2P . This

doesn’t commit us to adding a world in which P is true; rather, it commits us

to making P true in every world that a sees. Similarly, a zero over a 3, over

3P in world b in this case, commits us to making P false in every world that


b sees. We indicate such commitments, commitments in every world seen, by

putting asterisks above the relevant modal operators:

1 0 0

32P→23P∗ ∗

r

��~~~~

~~~~

~~~~

��@@@

@@@@

@@@@

@

∗1

2Pa

∗0

3Pb

Now, how can we discharge these asterisks? In this case, when trying to

construct a K-model, we don’t need to do anything. Since a, for example,

doesn’t see any world, then automatically P is true in every world it sees; the

statement “for every world, w, if Raw then V(P, w) = 1” is vacuously true.

Same goes for b—P is automatically false in all worlds it sees. So, we’ve got a

K-model in which 32P→23P is true.

Now let’s turn the model into a D-model. Every world must now see at

least one world. Let’s try:

1 0 0

32P→23P∗ ∗

r

��~~~~

~~~~

~~~~

��@@@

@@@@

@@@@

@

∗1

2Pa

��

∗0

3Pb

��1

Pc

00

0

Pd

00

I added worlds c and d, so that a and b would each see at least one world.

(Further, worlds c and d each had to see a world, to keep the relation serial.

I could have added still more worlds that c and d saw, but then they would


themselves need to see some worlds…So I just let c and d see themselves.) But

once c and d were added, discharging the upper asterisks in worlds a and b

required making P true in c and false in d (since a sees c and b sees d).

Let’s now try for a T-model. This will involve, among other things, letting

a and b see themselves. But this gets rid of the need for worlds c and d, since

they were added just to make the relation serial. I’ll try:

1 0 0

32P→23P∗ ∗

r

��~~~~

~~~~

~~~~

��@@@

@@@@

@@@@

@00

∗1 1

2Pa

00

∗0 0

3Pb

00

When I added arrows, I needed to make sure that I correctly discharged the

asterisks. This required nothing of world r, since there were no top asterisks

there. There were top asterisks in worlds a and b; but it turned out to be easy

to discharge these asterisks—I just needed to let P be true in a, but false in b.

Notice that I could have moved straight to this T-model—which is itself a

D-model—rather than �rst going through the earlier mere-D-model. However,

this won’t always be possible—sometimes you’ll be able to get a D-model, but

no T-model.

At this point let’s verify that our model does indeed assign the value 0 to

our formula 32P→23P . First notice that 2P is true in a (since a only sees

one world—itself—and P is true there). But r sees a. So 32P is true at r. Now,

consider b. b only sees one world, itself, and P is false there. So 3P must also

be false there. But r sees b. So 23P is false at r. But now, the antecedent of

32P→23P is true, while its consequent is false, at r. So that conditional is

false at r. Which is what we wanted.

Onward. Our model is not a B-model, since a, for example, doesn’t see r,

despite the fact that r sees a. So let’s try to make this into a B-model. This

involves making the relation symmetric. Here’s how it looks before I try to


discharge the top asterisks in a and b:

1 0 0

32P→23P∗ ∗

r

??

��~~~~

~~~~

~~~~

__

��@@@

@@@@

@@@@

@00

∗1 1

2Pa

00

∗0 0

3Pb

00

Now I need to make sure that all top asterisks are discharged. For example,

since a now sees r, I’ll need to make sure that P is true at r. However, since

b sees r too, P needs to be false at r. But P can’t be both true and false at r.

So we’re stuck, in trying to get a B-model in which this formula is false. This

suggests that maybe it is impossible—that is, perhaps this formula is true in all

worlds in all B-models—that is, perhaps the formula is B-valid. So, the thing

to do is try to prove this: by supplying a semantic validity proof.

So, let ⟨W ,R ,I ⟩ be any model in whichR is re�exive and symmetric, let

V be its valuation function, and let w be any member ofW ; we must show that

V(32P→23P, w) = 1.

i) Suppose for reductio that V(32P→23P, w) = 0

ii) Then V(32, w) = 1 and …

iii) …V(23P, w) = 0

iv) By i), for some v,Rwv and V(2P, v) = 1.

v) By symmetry,Rvw.

vi) From iv), via the truth condition for 2, we know that P is true

at every world accessible from v; and so, by v), V(P, w) = 1.

vii) By iii), there is some world, call it u, such that Rw u and

V(3P, u) = 0.

viii) By symmetry, w is accessible from u.

ix) By vi), P is false in every world accessible from u; and so by

viii), V(P, w) = 0, contradicting vi)


Just as we suspected: the formula is indeed B-valid. So we know that it is

S5-valid (the proof we just gave was itself a proof of its S5-validity). But what

about S4-validity? Remember the diagram—we don’t have the answer yet. The

thing to do here is to try to come up with an S4-model, or an S4 semantic

validity proof. Usually, the best thing to do is to try for a model. In fact, in the

present case this is quite easy: our T-model is already an S4-model. So, we’re

done. Our answer to what systems the formula is valid and invalid in comes in

two parts:

Invalidity: we have an S4-model, which we put down of�cially as

follows:

W = {r,a,b}R = {⟨r, r⟩, ⟨a, a⟩, ⟨b,b⟩, ⟨r, a⟩, ⟨r,b⟩}I (P, a) = 1, all others false

This model is itself also a T, D, and K-model (since its accessibility

relation is re�exive and serial), so: 2K,D,T,S4

32P→23P .

Validity: we gave a semantic proof of B-validity above. This was

itself a proof of S5-validity, as I noted. So �B,S5

32P→23P

I’ll do one more example: 32P→3232P . We can get a T-model as

follows:


∗1 0 0 0

32P→3232P∗ ∗

r

00

��

��

I discharged the second

bottom asterisk in

r by letting r see b

∗1 1 0

2P 232P∗

a

00

��

Notice how commitments

to speci�c truth values for

different formulas are recorded

by placing the formulas

side by side in the box

∗0 0 1

32P P∗

b

00

��0

Pc

00

Of�cial model:

W = {r, a,b, c}R = {⟨r, r⟩, ⟨a, a⟩, ⟨b,b⟩, ⟨c, c⟩, ⟨r, a⟩, ⟨r,b⟩, ⟨a,b⟩, ⟨b,c⟩}I (P,b) = 1, all else 0

Now consider what happens when we try to turn this model into a B-model.

World b must see back to world a. But then the false 32P in b con�icts with the

true 2P in a. So it’s time for a validity proof. In constructed this validity proof,

we can be guided by failed attempt to construct a countermodel (assuming all

of our choices in constructing that countermodel were forced). In the following

proof that the formula is B-valid, I chose variables for worlds that match up

with the countermodel above:

i) Suppose for reductio that V(32P→3232P, r ) = 0, in some

world r in some B-model ⟨W ,R ,I ⟩


ii) So V(32P, r ) = 1 and . . .

iii) V(3232P, r ) = 0

iv) From ii), there’s some world, call it a, such that V(2P,a) = 1andR ra

v) From iii), sinceR ra, V(232P,a) = 0

vi) And so, there’s some world, call it b , such that V(32P, b ) =0 andRab

vii) By symmetry, Rba. And so, given vi), V(2P,a) = 0. This

contradicts iv)

We now have a T-model for the formula, and a proof that it is B-valid. The

B-validity proof shows the formula to be S5-valid; the T-model shows it to

be K- and D-invalid. We still don’t yet know about S4. So let’s return to the

T-model above, and see what happens when we try to make its accessibility

relation transitive. World a must then see world c, which is impossible since

2P is true in a and P is false in c. So we’re ready for a S4-validity proof (the

proof looks like the B-validity proof at �rst, but then diverges):

i) Suppose for reductio that V(32P→3232P, r ) = 0, for some

world r in some S4-model ⟨W ,R ,V ⟩ii) So V(32P, r ) = 1 and . . .

iii) V(3232P, r ) = 0

iv) From ii), there’s some world, call it a, such that V(2P,a) = 1andR ra

v) From iii), sinceR ra, V(232P,a) = 0

vi) And so, there’s some world, call it b , such that V(32P, b ) =0 andRab

vii) By re�exivity,Rb b , so given vi), V(2P, b ) = 0

viii) And so, there’s some world, call it c , such that V(P, c) = 0 and

Rb c .

ix) From vi) and viii), given transitivity, we have Rac . And so,

given iv), V(P, c) = 1, contradicting viii)


Daggers

There’s another kind of step in constructing models. When we make a condi-

tional false, we’re forced to enter certain truth values for its components: 1 for

the antecedent, 0 for the consequent. But consider making a disjunction true. A

disjunction can be true in more than one way. The �rst disjunct might be true,

or the second might be true, or both could be true. So we have a choice for

how to go about making a disjunction true. Similarly for making a conditional

true, a conjunction false, or a biconditional either true or false.

When one has a choice about which truth values to give the constituents

of a propositional compound, it’s best to delay making the choice as long as

possible. After all, some other part of the model might force you to make one

choice rather than the other. If you investigate the rest of the countermodel,

and nothing has forced your hand, you may need then to make a guess: try one

of the truth value combinations open to you, and see whether you can �nish

the countermodel. If not, go back and try another combination.

To remind ourselves of these choice points, we will place a dagger (†) un-

derneath the major connective of the formula in question. Consider, as an

example, constructing a countermodel for the formula 3(3P∨2Q)→(3P∨Q).Throwing caution to the wind and going straight for a T-model, we have after

a few steps:

∗1 0 0 0 0 0

3(3P∨2Q)→(3P ∨Q)∗

r

00

��1 0

3P∨2Q P†

a

00

We still have to decide how to make 3P∨2Q true in world a: which disjunct

to make true? Well, making 2P true won’t require adding another world to

the model, so let’s do that. We have, then, a T-model:


∗1 0 0 0 0 0

3(3P∨2Q)→(3P ∨Q)∗

r

00

��∗

1 1 1 0

3P∨2Q P†

a

00

W : {r,a}R : {⟨r, r⟩, ⟨a,a⟩, ⟨r,a⟩}I (Q,a) = 1, all else 0

OK, let’s try now to upgrade this to a B-model. We can’t simply leave

everything as-is while letting world a see back to world r, since 2Q is true

in a and Q is false in r. But there’s another possibility. We weren’t forced to

discharge the dagger in world a by making 2Q true. So let’s explore the other

possibility; let’s make 3P true:

∗1 0 0 0 0 0

3(3P∨2Q)→(3P ∨Q)∗

r

00 OO

��1 1 0

3P∨2Q P∗ †

a

00 OO

��1

Pb

00


W : {r,a,b}R : {⟨r, r⟩, ⟨a,a⟩, ⟨b,b⟩, ⟨r,a⟩, ⟨a, r⟩, ⟨a,b⟩, ⟨b,a⟩}I (P,b) = 1, all else 0

What about an S4-model? We can’t just add the arrows demanded by

transitivity to our B-model, since 3P is false in world r and P is true in world

b. What we can do instead is revisit the choice of which disjunct of 3P∨2Q to

make true. Instead of making 3P true, we can make 2Q true, as we did when

we constructed our T-model. In fact, that T-model is already an S4-model.

So, we have countermodels in both S4 and B. The �rst resulted from

one choice for discharging the dagger in world a, the second from the other

choice. An S5-model, though, looks impossible. When we made the �rst

choice—making the right disjunct of 3P∨2Q true—we were able to make the

accessibility relation symmetric, and when we made the second choice—making

the left disjunct of 3P∨2Q true—we were able to make the accessibility rela-

tion transitive. It would seem to be impossible, then, to make the accessibility

both transitive and symmetric. Here is an S5-validity proof, based on this

reasoning. Note the “separation of cases” reasoning:

i) Suppose for reductio that in some world r in some S5-model,

V(3(3P∨2Q)→(3P∨Q), r ) = 0. Then V(3(3P∨2Q), r ) =1 and …

ii) …V(3P∨Q, r ) = 0

iii) Given i), for some world a,R ra and V(3P∨2Q,a) = 1. So,

either V(3P,a) = 1 or V(2Q,a) = 1

iv) The �rst possibility leads to a contradiction:

a) Suppose V(3P,a) = 1. Then for some world b ,Rab and

V(P, b ) = 1b) R is transitive, so given a) and iii),R r b .

c) given ii), V(3P, r ) = 0, and so, given b), V(P, b ) = 0,

which contradicts a).

v) So does the second:

a) Suppose V(2Q,a) = 1.

b) R is symmetric. So, given iii), Ra r ; and so, given a),

V(Q, r ) = 1


c) But given ii), V(Q, r ) = 0—contradiction.

vi) Either way we have a contradiction.

So we have demonstrated that �S5

3(3P∨2Q)→(3P∨Q).

Summary of steps

Here, then, is a �nal list of the steps for constructing countermodels:

1. Place the formula in a box

2. Make the formula false in the world

3. Enter in forced truth values

4. Enter in daggers, and after all forced moves over

5. Enter asterisks

6. Discharge asterisks (hint: do bottom asterisks �rst)

7. Back to step 3.

8. The of�cial model

6.3.5 Schemas, validity, and invalidityLet’s digress, for a moment, to clarify the notions of validity and invalidity, as

applied to formulas and schemas.

Formulas—of�cial wffs, that is—are the strings of symbols that are sanc-

tioned by the of�cial rules of grammar of our object language. Formulas of

MPL include P∨(Q→R), 2P→3Q, and so on. Schemas are devices of our

metalanguage that are used to talk about in�nitely many formulas at once. We

used schemas, for instance, in section 2.5 to state the axioms of propositional

logic; we said: “each instance of the schema φ→(ψ→φ) is an axiom of PL”.

And we have used schemas throughout the book to de�ne the notion of a wff,

and to de�ne valuation functions. For example, earlier in this chapter we said

that a valuation function must assign the value 1 to any instance of the schema

∼φ relative to any world iff it assigns 0 to the corresponding instance of φrelative to that world. Schemas are not formulas, since they contain schematic

variables (“φ” and “ψ” here) that are not part of the primitive vocabulary of

the object language.


When we de�ned the notion of validity, what we de�ned was the notion of

a valid formula. (That’s because validity is de�ned in terms of truth in a model,

which itself was de�ned only for formulas.) So it’s not, strictly speaking, correct

to apply the notions of validity or invalidity to schemas.

However, it’s often interesting to show that every instance of a given schema

is valid. (Instances are schemas are formulas, and so the notion of validity can

be properly applied to them.) It’s easy, for example, to show that every instance

of the schema 2(φ→φ) is valid in each of our modal systems. (Let φ be any

MPL-wff, and take any world w in any model. Since the rules for evaluating

propositional compounds within possible worlds are the classical ones, φ→φmust be true at w, no matter what truth value φ has at w. Hence 2(φ→φ) is

true in any world in any model, and so is valid in each system.)

There is, therefore, a kind of indirect notion of schema-validity: validity

of all instances. How about the invalidity of schemas? Here we must take

great care. In particular, the notion of a schema, all of whose instances are

invalid, is not a particularly interesting notion. Take, for instance, the schema

3φ→2φ. We showed earlier that a certain instance of this schema, namely

3P→2P is invalid in each of our systems. However, the schema 2φ→3φ also

has plenty of instances that are valid in various systems. The following formula,

for example, is an instance of 3φ→2ψ, and can easily be shown to be valid in

each of our systems:

3(P→P )→2(P→P )

Thus, even intuitively terrible schemas like 3φ→2φ have some valid instances.

(For an extreme example of this, consider the schema φ. Even this has some

valid instances: P→P , for one.) So it’s not interesting to inquire into whether

each instance of a schema is invalid. What is interesting is to inquire into

whether a given schema has some instances that are invalid. We can show, for

example, that the schema 3φ→2φ has some invalid instances (3P→2P , for

one), and hence is in this way unlike the schema 2(φ→φ).So when dealing with schemas, it will often be of interest to ascertain

whether each instance of the schema is valid; it will rarely (if ever) be of interest

to ascertain whether each instance of the schema is invalid.


6.4 Axiomatic systems of MPLWe turn next to provability in modal logic. We’ll approach this axiomatically:

we’re going to write down axioms, which are sentences of propositional modal

logic that seem clearly to be logical truths, and we’re going to write down rules

of inference, which say which sentences can be logically inferred from which

other sentences.

We’re going to continue to follow C. I. Lewis in constructing multiple

modal systems, since it’s so unclear which sentences of MPL are logical truths.

Hence, we’ll need to formulate multiple axiomatic systems. These systems will

contain different axioms from one another. As a result, different theorems will

be provable in the different systems.

We will, in fact, give these systems the same names as the systems we

investigated semantically: K, D, T, B, S4, and S5. (Thus we will subscript

the symbol for theoremhood with the names of systems; `Kφ, for example,

will mean that φ is a theorem of system K.) Our re-use of the system names

will be justi�ed in sections ?? and ??, where we will establish soundness and

completeness for each system. Given soundness and completeness, for each

system, exactly the same formulas are provable as are valid.

6.4.1 System KOur �rst system, K, is the weakest system—i.e., the system with the fewest

theorems.

Rules: MP NEC (“necessitation”)

φφ→ψ φ

ψ 2φ


Axioms: PL axioms: for any MPL-wffs φ,ψ, and χ , the fol-

lowing are axioms:

(A1) φ→(ψ→φ)(A2) (φ→(ψ→χ ))→((φ→ψ)→(φ→χ ))(A3) (∼ψ→∼φ)→((∼ψ→φ)→ψ)

K-schema: for any MPL wffsφ andψ, the following

is an axiom:

2(φ→ψ)→(2φ→2ψ)

As before, a proof is de�ned as a series of wffs, each of which is either an

axiom or follows from earlier lines in proof by a rule, and a theorem is de�ned

as the last line of any proof.

This axiomatic system (like all the modal systems we will study) is an exten-sion of propositional logic, in the sense that it includes all of the theorems of

propositional logic, but then adds more theorems. It includes all of proposi-

tional logic because one of its rules is the propositional logic rule MP, and each

propositional logic axiom is one of its axioms. It adds theorems by adding a

new rule of inference (NEC), and a new axiom schema (the K-schema) (as well

as adding new wffs—wffs containing the 2—to the stock of wffs that can occur

in the PL axioms.)

The rule of inference, NEC (for “necessitation”), says that if you have a

formula φ on a line, then you may infer the formula 2φ. This may seem

unintuitive. After all, can’t a sentence be true without being necessarily true?

Yes; but the rule of necessitation doesn’t contradict this. Remember that every

line in every axiomatic proof is a theorem. So whenever one uses necessitation in

a proof, one is applying it to a theorem. And necessitation does seem appropriate

when applied to theorems: ifφ is a theorem, then 2φ ought also to be a theorem.

Think of it this way. The worry about the rule of necessitation is that it isn’t

a truth-preserving rule: intuitively speaking, its premise can be true when

its conclusion is false. The answer to the worry is that while necessitation

doesn’t preserve truth, it does preserve logical truth, which is all that matters

in the present context. For in the present context, we’re only using NEC in a

de�nition of theoremhood. We want our theorems to be, intuitively, logical

truths; and provided that our axioms are all logical truths and our rules preserve

logical truth, the de�nition will yield only logical truths as theorems. We will

return to this issue.


Let’s investigate what one can prove in K. The simplest sort of distinctively

modal proof consists of �rst proving something from the PL axioms, and then

necessitating it, as in the following proof of 2((P→Q)→(P→P ))

1. P→(Q→P ) (A1)

2. P→(Q→P ))→((P→Q)→(P→P )) (A2)

3. (P→Q)→(P→P ) 1,2 MP

4. 2((P→Q)→(P→P )) 3, NEC

Using this technique, we can prove anything of the form 2φ, where φ is

provable in PL. And, since the PL axioms are complete (I mentioned but did

not prove this fact in chapter ??), that means that we can prove 2φ whenever

φ is a tautology—i.e., a valid wff of PL. But constructing proofs from the PL

axioms is a pain in the neck!—and anyway not what we want to focus on in

this chapter. So let’s introduce the following time-saving shortcut. Instead of

writing out proofs of tautologies, let’s instead allow ourselves to write any PL

tautology at any point in a proof, annotating simply “PL”.5

Thus, the previous

proof could be shortened to:

1. (P→Q)→(P→P ) PL

2. 2((P→Q)→(P→P )) 1, NEC

Furthermore, consider the wff 2P→2P . Clearly, we can construct a proof

of this wff from the PL axioms: begin with any proof of the tautology Q→Qfrom the PL axioms, and then construct a new proof by replacing each occur-

rence of Q in the �rst proof with 2P . (This is a legitimate proof, even though

2P isn’t a wff of propositional logic, because when we stated the system K, the

schematic letters φ,ψ, and χ in the PL axioms are allowed to be �lled in with

any wffs of MPL, not just wffs of PL.) So let us also include lines like this in

our modal proofs:

2P→2P PL

Why am I making such a fuss about this? Didn’t I just say in the previous

paragraph that we can write down any tautology at any time, with the annotation

“PL”? Well, strictly speaking, 2P→2P isn’t a tautology. A tautology is a valid

5How do you know whether something is a tautology? Figure it out any way you like: do a

truth table, or a natural deduction derivation—whatever.


wff of PL, and 2P→2P isn’t even a wff of PL (since it contains a 2). But it

is the result of beginning with some PL-tautology (Q→Q, in this case) and

uniformly changing sentence letters to chosen modal wffs (in this case, Qs to

2P s); hence any proof of the PL tautology may be converted into a proof of

it; hence the “PL” annotation is just as justi�ed here as it is in the case of a

genuine tautology. So in general, MPL wffs that result from PL tautologies in

this way may be written down and annotated “PL”.

Back to investigating what we can prove in K. As we’ve seen, we can prove

that tautologies are necessary—we can prove 2φ whenever φ is a tautology.

One can also prove in K that contradictions are impossible. For instance,

∼3(P∧∼P ) is a theorem of K:

1. ∼(P∧∼P ) PL

2. 2∼(P∧∼P ) 1, NEC

3. 2∼(P∧∼P )→∼∼2∼(P∧∼P ) PL

4. ∼∼2∼(P∧∼P ) 2, 3, MP

But line 4 is a de�nitional abbreviation of ∼3(P∧∼P ).Let’s introduce another time-saving shortcut. Note that the move from 2 to

4 in the previous proof is just a move from a formula to a propositional logical

consequence of that formula. Let’s allow ourselves to move directly from any

lines in a proof, φ1 . . .φn, to any propositional logical consequence ψ of those

lines, by “PL”. Thus, the previous proof could be shorted to:

1. ∼(P∧∼P ) PL

2. 2∼(P∧∼P ) 1, NEC

3. ∼∼2∼(P∧∼P ) 2, PL

Why is this legitimate? Suppose that ψ is a propositional logical semantic

consequence of φ1 . . .φn. Then the conditional φ1→(φ2→·· · (φn→ψ) . . . ) is

a PL-valid formula, and so, given the completeness of the PL axioms, is a

theorem of K. That means that if we have φ1, . . . ,φn in an axiomatic K-proof,

then we can always prove the conditional φ1→(φ2→·· · (φn→ψ) . . . ) using the

PL-axioms, and then use MP repeatedly to infer ψ. So inferring ψ directly,

and annotating “PL”, is justi�ed. (As with the earlier “PL” shortcut, let’s use

this shortcut when the conditionalφ1→(φ2→·· · (φn→ψ) . . .) results from some

tautology by uniform substitution, even if it contains modal operators and so

isn’t strictly a tautology.)


So far our modal proofs have only used necessitation and the PL axioms.

What about the K-axioms? The point of the K-schema is to enable “distribution

of the 2 over the→”. That is, if you ever have the formula 2(φ→ψ), then you

can always move to 2φ→2ψ as follows:

2(φ→ψ)2(φ→ψ)→(2φ→2ψ) K axiom

2φ→2ψ MP

Distribution of the 2 over the→, plus the rule of necessitation, combine

to give us a powerful proof strategy. Whenever one can prove the conditional

φ→ψ, then one can prove the modal conditional 2φ→2ψ as well, as follows.

First prove φ→ψ, then necessitate it to get 2(φ→ψ), then distribute the 2

over the arrow to get 2φ→2ψ. This procedure is one of the core K-strategies,

and is featured in the following proof of 2(P∧Q)→(2P∧2Q):

1. (P∧Q)→P PL

2. 2[(P∧Q)→P] NEC

3. 2[(P∧Q)→P]→[2(P∧Q)→2P] K axiom

4. 2(P∧Q)→2P 3,4 MP

5. 2(P∧Q)→2Q Insert steps similar to 1-4

6. 2(P∧Q)→(2P∧2Q) 4,5, PL

Notice that the preceding proof, like all of our proofs since we introduced

the time-saving shortcuts, is not a K-proof in the of�cial de�ned sense. Lines 1,

5, and 6 are not axioms, nor do they follow from earlier lines by MP or NEC;

similarly for line 6.6

So what kind of “proof” is it? It’s a metalanguage proof:

an attempt to convince the reader, by acceptable standards of rigor, that some

real K-proof exists. A reader could use this metalanguage proof as a blueprint

for constructing a real proof. She would begin by replacing line 1 with a proof

from the PL axioms of the conditional (P∧Q)→P . (As we know from chapter

??, this could be a real pain in the neck!—but the completeness of PL assures us

that it is possible.) She would then replace line 5 with lines parallel to lines 1-4,

but which begin with a proof of (P∧Q)→Q rather than (P∧Q)→P . Finally,

in place of line 6, she would insert a proof from the PL axioms of the sen-

tence (2(P∧Q)→2P )→[(2(P∧Q)→2Q)→(2(P∧Q)→(2P∧2Q))], and then

use modus ponens twice to infer 2(P∧Q)→(2P∧2Q).Another example: (2P∨2Q)→2(P∨Q):

6A further (even pickier) reason: the symbol ∧ isn’t allowed in wffs; the sentences in the

proof are mere abbreviations for of�cial MPL-wffs.


1. P→(P∨Q) PL

2. 2(P→(P∨Q)) 1, NEC

3. 2(Q→(P∨Q)) PL, NEC

4. 2P→2(P∨Q) 2, K

5. 2Q→2(P∨Q) 3, K

6. (2P∨2Q)→2(P∨Q) 4,5 PL

Here I’ve introduced another time-saving shortcut that I’ll use more and more

as we progress: doing two (or more) steps at once. Line 3 is really short for:

3a. Q→(P∨Q) PL

3b. 2(Q→(P∨Q)) 3a, NEC

And line 4 is short for:

4a. 2(P→(P∨Q))→(2P→2(P∨Q)) K axiom

4b. 2P→2(P∨Q) 2, 4a, MP

One further comment about this last proof: it illustrates a strategy that is

common in modal proofs. We were trying to prove a conditional formula

whose antecedent is a disjunction of two modal formulas. But the modal

techniques we had developed didn’t deliver formulas of this form. They only

showed us how to put 2s in front of PL-tautologies, and how to distribute 2s

over→s. They only yield formulas of the form 2φ and 2φ→2ψ, whereas the

formula we were trying to prove looks different. To overcome this problem,

what we did was to use the modal techniques to prove two conditionals, namely

2P→2(P∨Q) and 2Q→2(P∨Q), from which the desired formula, namely

(2P∨2Q)→2(P∨Q), follows by propositional logic. The trick, in general, is

this: remember that you have PL at your disposal. Simply look for one or

more modal formulas you know how to prove which, by PL, imply the formula

you want. Assemble the desired formulas, and then write down your desired

formula, annotating “PL”. In doing so, it may be helpful to recall PL inferences

like the following:


φ→ψψ→φφ↔ψ

φ→(ψ→χ )(φ∧ψ)→χ

φ→ψφ→χφ→(ψ∧χ )

φ→χψ→χ(φ∨ψ)→χ

φ→ψ∼φ∨ψ

φ→∼ψ∼(φ∧ψ)

The next example illustrates our next major modal proof technique: com-

bining two 2 statements to get a single 2 statement. Let us construct a K-proof

of (2P∧2Q)→2(P∧Q):

1. P→(Q→(P∧Q)) PL

2. 2[P→(Q→(P∧Q))] NEC

3. 2P→2(Q→(P∧Q)) 2, K

4. 2(Q→(P∧Q))→[2Q→2(P∧Q)] K axiom

5. 2P→[2Q→2(P∧Q)] 3,4 PL

6. (2P∧2Q)→2(P∧Q) 5, PL

(If you wanted to, you could skip step 5, and just go straight to 6 by propositional

logic, since 6 is a propositional logical consequence of 3 and 4; I put it in for

perspicuity.)

The general technique illustrated by the last problem applies anytime you

want to move from several 2 statements to a further 2 statement, where the in-

side parts of the �rst 2 statements imply the inside part of the �nal 2 statement.

More carefully: it applies whenever you want to prove a formula of the form

2φ1→(2φ2→·· · (2φn→2ψ) . . . ), provided you are able to prove the formula

φ1→(φ2→·· · (φn→ψ) . . . ). (The previous proof was an instance of this because

it involved moving from 2P and 2Q to 2(P∧Q); and this is a case where one

can move from the inside parts of the �rst two formulas (namely, P and Q), to

the inside part of the third formula (namely, P∧Q)—by PL.) To do this, one

begins by proving the conditionalφ1→(φ2→·· · (φn→ψ) . . . ), necessitating it to

get 2[φ1→(φ2→·· · (φn→ψ) . . . )], and then distributing the 2 over the arrows

repeatedly using K-axioms and PL to get 2φ1→(2φ2→·· · (2φn→2ψ) . . . ).One cautionary note in connection with this last proof. One might think to

make it more intuitive by using conditional proof:


1. 2P∧2Q assume for conditional proof

2. 2P 1, PL

3. 2Q 1, PL

4. P→(Q→(P∧Q)) PL

5. 2[P→(Q→(P∧Q))] NEC

6. 2P→2(Q→(P∧Q)) 5, K

7. 2(Q→(P∧Q)) 6,2, MP

8. 2Q→2(P∧Q) 7,K

9. 2(P∧Q) 3,8 MP

10. (2P∧2Q)→2(P∧Q) 1-9, conditional proof

But this is not a legal proof, since our axiomatic system allows neither assump-

tions nor conditional proof.

In fact, our decision to omit conditional proof was not at all arbitrary. Given

our rule of necessitation, we couldn’t add conditional proof to our system. If we

did, proofs like the following would become legal:

1. P assume for conditional proof

2. 2P 1, NEC

3. P→2P 1,2, conditional proof

Thus, P→2P would turn out to be a K-theorem. But we don’t want that: after

all, a statement P might be true without being necessarily true.

Once we have a soundness proof (section 6.5), we’ll be able to show that

P→2P isn’t a K-theorem. But as we just saw, one can construct a K-proof

from {P} of 2P (recall the notion of a proof from a set Γ, from section 2.5.)

It follows that the deduction theorem (section 2.5.2), which says that if there

exists a proof of ψ from {φ}, then there exists a proof of φ→ψ, fails for K (it

likewise fails for all the modal systems we will consider.) So there will be no

conditional proof in our axiomatic modal systems. (Of course, to convince

yourself that a given formula is really a tautology of propositional logic, you

may sketch a proof of it to yourself using conditional proof in some standard

natural deduction system for nonmodal propositional logic; and then you may

write that formula down in one of our axiomatic MPL proofs, annotating “PL”.)

Back to techniques for constructing proofs in K. The following proof of

22(P∧Q)→22P illustrates a technique for proving formulas with “nested”

modal operators:


1. (P∧Q)→P PL

2. 2(P∧Q)→2P 1, NEC, K

3. 2[2(P∧Q)→2P] 2, NEC

4. 22(P∧Q)→22P 3, K

Notice in line 3 that we necessitated something that was not a PL theorem.

That’s ok; we’re allowed to necessitate any K-theorems, even those whose proofs

were distinctly modal. Notice also how this proof contains two instances of our

basic K-strategy. This strategy involves obtaining a conditional, necessitating

it, then distributing the 2 over the→. We did this �rst using the conditional

(P∧Q)→P ; that led us to a conditional, 2(P∧Q)→2P . Then we started the

strategy over again, using this as our initial conditional.

So far we have no techniques dealing with the 3, other than eliminating it

by de�nition. It will be convenient to derive some shortcuts. For one, there

are the following theorem schemas, which may collectively be called “modal

negation”, or “MN” for short:

`K∼2φ→3∼φ `

K3∼φ→∼2φ

`K∼3φ→2∼φ `

K2∼φ→∼3φ

I’ll do one of these; the rest can be an exercise:

1. ∼∼φ→φ PL

2. 2∼∼φ→2φ 1, NEC, K

3. ∼2φ→∼2∼∼φ 2, PL

The �nal line, 3, is the de�nitional equivalent of ∼2φ→3∼φ.

It will also be worthwhile to know that an analog of the K axiom for the 3

is a K-theorem:

“K3”: 2(φ→ψ)→(3φ→3ψ)

K3 is, by de�nition of the 3, the same formula as:

2(φ→ψ)→(∼2∼φ→∼2∼ψ):

How are we going to construct a K-proof of this theorem? In a natural de-

duction system we would use conditional proof and reductio ad absurdum,

but these strategies are not available to us here. What we must do instead is

look for a formula we know how to prove in K, which is PL-equivalent to the

formula we want to prove. Here is such a formula:


2(φ→ψ)→(2∼ψ→2∼φ)

This is equivalent, given PL, to what we want to show; and it looks like the

result of necessitating a tautology and then distributing the 2 over the→ a

couple times—just the kind of thing we know how to do in K. Here, then, is

the desired proof of K3:

1. (φ→ψ)→ (∼ψ→∼φ) PL

2. 2(φ→ψ)→2(∼ψ→∼φ) 1, NEC, K

3. 2(∼ψ→∼φ)→(2∼ψ→2∼φ) K

4. 2(φ→ψ)→(2∼ψ→2∼φ) 2,3 PL

5. 2(φ→ψ)→(∼2∼φ→∼2∼ψ) 4,PL

In doing proofs, let’s also allow ourselves to refer to earlier theorems proved,

rather than repeating their proofs. The importance of K3 may be illustrated

by the following proof of 2P→(3Q→3(P∧Q)):

1. P→[Q→(P∧Q)] PL

2. 2P→2[Q→(P∧Q)] 1, NEC, K

3. 2[Q→(P∧Q)]→[3Q→3(P∧Q)] K3

4. 2P→[3Q→3(P∧Q)] 2,3, PL

In general, the K3 rule allows us to complete proofs of the following sort.

Suppose we wish to prove a formula of the form:

O1φ1→(O2φ2→(. . .→(Onφn→3ψ) . . .)

where the Oi s are modal operators, all but one of which are 2s. (Thus, the

remaining Oi is the 3.) This can be done, provided that ψ is provable in K from

the φi s. The basic strategy is to prove a nested conditional, the antecedents

of which are the φi s, and the consequent of which is ψ; necessitate it; then

repeatedly distribute the 2 over the→s, once using K3, the rest of the times

using K. But there is one catch. We need to make the application of K3 last,after all the applications of K. This in turn requires the conditional we use to

have theφi that is underneath the 3 as the last of the antecedents. For instance,

suppose that φ3 is the one underneath the 3. Thus, what we are trying to

prove is:

2φ1→(2φ2→(3φ3→(2φ4→(…→(2φn→3ψ)…)

In this case, the conditional to use would be:


φ1→(φ2→(φn→(φ4→(…→(φn−1→(φ3→ψ)…)

In other words, one must swap one of the other φi s (I arbitrarily chose φn)

with φ3. What one obtains at the end will therefore have the modal statements

out of order:

2φ1→(2φ2→(2φn→(2φ4→(…→(2φn−1→(3φ3→3ψ)…)

But that problem is easily solved; this is equivalent in PL to what we’re trying

to get. (Recall that φ→(ψ→χ ) is logically equivalent in PL to ψ→(φ→χ ).)

Why do we need to save K3 for last? The strategy of successively distribut-

ing the box over all the nested conditionals comes to a halt as soon as the K3

theorem is used. Let me illustrate with an example. Suppose we wish to prove

`K 3P→(2Q→3(P∧Q)). We might think to begin as follows:

1. P→(Q→(P∧Q)) PL

2. 2[P→(Q→(P∧Q))] 1, Nec

3. 3P→3(Q→(P∧Q)) K3

4. ?

But now what? What we need to �nish the proof is:

3(Q→(P∧Q))→(2Q→3(P∧Q)).

But neither K nor K3 gets us this. The remedy is to begin the proof with a

different conditional:

1. Q→(P→(P∧Q)) PL

2. 2(Q→(P→(P∧Q))) 1, Nec

3. 2Q→2(P→(P∧Q)) 2, K, MP

4. 2(P→(P∧Q))→(3P→3(P∧Q)) K3

5. 2Q→(3P→3(P∧Q)) 3, 4, PL

6. 3P→(2Q→3(P∧Q)) 5, PL

One can, then, prove a number of theorems in K. Nevertheless, K is a

very weak system. You can’t prove the formula 2P→3P in K. (We’ll be able

to demonstrate this after section 6.5.) Relatedly, one can’t prove in K that

tautologies are possible or that contradictions aren’t necessary.


6.4.2 System DSystem D results from adding a new axiom schema to system K:

System D

Rules: MP, NEC

Axioms: PL axioms

K-schema: 2(φ→ψ)→(2φ→2ψ)D-schema: 2φ→3φ

Notice that since D includes all the K axioms, we retain all the K-theorems.

The addition of the D-schema just adds more theorems. In fact, all of our

systems will build on K in this way, by adding new axioms to K. (K is so weak

that it isn’t of much independent interest; we study it because it’s the common

basis of all of the other systems we’re studying.)

An example of what we can do now is prove that tautologies are possible:

1. P∨∼P PL

2. 2(P∨∼P ) 1, NEC

3. 2(P∨∼P )→3(P∨∼P ) D

4. 3(P∨∼P ) 2,3 MP

Let’s do one more theorem, 22P→23P :

1. 2P→3P D

2. 2(2P→3P ) 1, NEC

3. 22P→23P 2, K

Like K, system D is very weak. As we will see later, we can’t prove 2φ→φin D. Therefore, D doesn’t seem to be a correct logic for metaphysical, or

nomic, or technological necessity, for surely, if something is metaphysically,

nomically, or technologically necessary, then it must be true. (If something is

true in all metaphysically possible worlds, or all nomically possible worlds, or

all technologically possible worlds, then surely it must be true in the actual

world, and so must be plain old true.) But perhaps there is some interest in

D anyway; perhaps D is a correct logic for moral necessity. Suppose we read

2φ as “One ought to make φ be the case”, and, correspondingly, read 3φ as

“One is permitted to make φ be the case”. Then the fact that 2φ→φ cannot

be proved in D would be a virtue, for from the fact that something ought be


done, it certainly doesn’t follow that it is done. The D-axiom, on the other

hand, would correspond to the principle that if something ought to be done

then it is permitted to be done, which does seem like a logical truth. But I won’t

go any further into the question of whether D in fact does give a correct logic

for moral necessity. That’s philosophy, not logic.

6.4.3 System TT is the �rst system we have considered that has any plausibility of being a

correct logic for a wide range of concepts of necessity (metaphysical necessity,

for example):

Rules: MP, NEC

Axioms: PL axioms

K-schema: 2(φ→ψ)→(2φ→2ψ)T-schema: 2φ→φ

Recall that in the case of K, we proved a theorem schema, K3, which was

the analog for the 3 of the K-axiom schema. Let’s do the same thing here; let’s

prove a theorem schema T3, which is the analog for the 3 of the T axiom

schema:

T3: φ→3φ

1. 2∼φ→∼φ T

2. φ→∼2∼φ 1, PL

2 is just the de�nition of φ→3φ. Thus, we have established that for every

wff φ,`Tφ→3φ. So let’s allow ourselves to write down formulas of the form

φ→3φ, annotating simply “T3”.

Notice that instances of the D-axioms are now theorems: 2φ→φ is a T

axiom, we just proved that φ→3φ is a theorem; and from these two by PL we

can prove 2φ→3φ. Thus, T is an extension of D: every theorem of D remains

a theorem of T. (Since D was an extension of K, T too is an extension of K.)

6.4.4 System BOur systems so far don’t allow us to prove anything interesting about iteratedmodalities, i.e., sentences with consecutive boxes or diamonds. Which such


sentences should be theorems? The B axiom schema decides some of these

questions for us; here is system B:

Rules: MP, NEC

Axioms: PL axioms

K-schema: 2(φ→ψ)→(2φ→2ψ)T-schema: 2φ→φB-schema: 32φ→φ

Note that we retain the T axiom schema in B. Thus, B is an extension of T

(and hence of K and D as well.)

As with K and T, we can establish a theorem schema that is the analog for

the 3 of B’s characteristic axiom schema. (The proof illustrates techniques for

“moving” ∼s through strings of modal operators.)

B3: φ→23φ:

1. 32∼φ→∼φ B

2. φ→∼32∼φ 1, PL

3. ∼32∼φ↔2∼2∼φ MN

4. ∼2∼φ→3φ PL (since 3 abbreviates ∼2∼)

5. 2∼2∼φ→23φ 4, NEC, K, MP

6. φ→23φ 2, 3, 5, PL

As an example, let’s show that `B[2P∧232(P→Q)]→2Q:

7

1. 32(P→Q)→(P→Q) B

2. 232(P→Q)→2(P→Q) 1, Nec, K, MP

3. 2(P→Q)→(2P→2Q) K

4. 232(P→Q)→(2P→2Q) 2, 3 PL

5. [2P∧232(P→Q)]→2Q 4, PL

6.4.5 System S4The characteristic axiom of our next system, S4, is a different principle gov-

erning iterated modalities:

7This is # 41 from the handout “MPL theorems”.


Rules: MP, NEC

Axioms: PL axioms

K-schema: 2(φ→ψ)→(2φ→2ψ)T-schema: 2φ→φS4-schema: 2φ→22φ

S4 contains the S4-schema but does not contain the B-schema. Symmetri-

cally, B lacks the S4-schema, but of course contains the B-schema. As a result,

some instances of the B-schema are not provable in S4, and some instances of

the S4-schema are not provable in B (we’ll be able to show this after section

6.5). Hence, although S4 and B are each extensions of T, neither B nor S4 is

an extension of the other.

As before, we have a theorem schema that is the analog for the 3 of the S4

axiom schema:

S43: 33φ→3φ:

1. 2∼φ→22∼φ S4

2. 2∼φ→∼3φ MN

3. 22∼φ→2∼3φ 2, NEC, K, MP

4. 2∼3φ→∼33φ MN

5. ∼3φ→2∼φ MN

6. 33φ→3φ 5,1,3,4, PL

Let’s do one fairly dif�cult example: (3P∧2Q)→3(P∧2Q).8 How should

we approach this problem?

My thinking is as follows. We saw in the K section above that the following

sort of thing may always be proved: 2φ→(3ψ→3χ ), whenever the conditional

φ→(ψ→χ ) can be proved. So we need to try to work the problem into this form.

As-is, the problem doesn’t quite have this form. But something very related

does have this form, namely: 22Q→(3P→3(P∧2Q)) (since the conditional

2Q→(P→(P∧2Q)) is a tautology). This thought inspires the following proof:

1. 2Q→(P→(P∧2Q)) PL

2. 22Q→2(P→(P∧2Q)) 1, Nec, K, MP

3. 2(P→(P∧2Q))→(3P→3(P∧2Q)) K3

4. 22Q→(3P→3(P∧2Q)) 2, 3 PL

5. 2Q→22Q S4

6. (3P∧2Q)→3(P∧2Q) 4, 5 PL

8This is half of #56 from the “MPL Theorems” handout.


6.4.6 System S5Here, instead of the B or S4 schemas, we add the S5 schema to T:

Rules: MP, NEC

Axioms: PL axioms

K-schema: 2(φ→ψ)→(2φ→2ψ)T-schema: 2φ→φS5-schema: 32φ→2φ

First let’s prove the analog of the S5-schema for the 3:

S53: 3φ→23φ

1. 32∼φ→2∼φ S5

2. ∼3φ→2∼φ MN

3. 3∼3φ→32∼φ 2, NEC, K3, MP

4. ∼23φ→3∼3φ MN

5. 2∼φ→∼3φ MN

6. 3φ→23φ 4,3,1,5, PL

Next, note that the B and S4 axioms are now derivable as theorems. The B

axiom, 32φ→φ, is trivial:

1. 32φ→2φ S5

2. 2φ→φ T

3. 32φ→φ 1,2 PL

And now the S4 axiom, 2φ→22φ. This is a little harder. I used the B3

theorem, which we can now appeal to since the theoremhood of the B-schema

has been established.

1. 2φ→232φ B3

2. 32φ→2φ S5

3. 2(32φ→2φ) 2, Nec

4. 232φ→22φ 3, K, MP

5. 2φ→22φ 4, 1, PL


6.4.7 Substitution of equivalents and modal reductionLet’s conclude our discussion of provability in modal logic by proving two

simple meta-theorems. First, let’s prove the following result about system K:

Substitution of equivalents: Where S is any of our modal sys-

tems, and wff χβ results from wff χ by changing occurrences

of wff α to occurrences of wff β:

if `Sα↔β then `

Sχ↔χβ

Proof of substitution of equivalents: Suppose (*) `Sα↔β. We proceed

by induction.

Base case: here χ is a sentence letter. Then either i) changing αs to

βs has no effect, in which case χβ is just χ , in which case obviously

`S

(χ↔χβ); or ii) χ is α, in which case χβ is β, and we know `S

(χ↔χβ) since we are given (*).

Induction case: We now assume the result holds for some formulas χ1

and χ2—that is, we assume that `Sχ1↔χβ1 and `

Sχ2↔χβ2 —and

we show the result holds for ∼χ1, 2χ1, and χ1→χ2.

Take the �rst case. We must show that the result holds for ∼χ1—

i.e., we must show that `S∼χ1↔(∼χ1)

β. (∼χ1)

βis just∼χβ1 , so we

must show `S∼χ1↔∼χ

β1 . But ∼χ1↔∼χ

β1 follows by PL from

χ1↔χβ1 , and the inductive hypothesis tells us that: `Sχ1↔χβ1 .

Take the second case. We must show `S(χ1→χ2)

β. The inductive

hypothesis tells us that `Sχ1↔χβ1 , and so (since S includes PL):

`S(χ1→χ2)↔ (χ1

β→χ2)

The inductive hypothesis also tells us that `S(χ2↔χ2

β), from

which, using propositional logic in S, we obtain:

`S

(χ1β→χ2)↔ (χ1

β→χ2β)

Now from the two indented equivalences, again using propositional

logic in S, we have:

`S (χ1→χ2)↔ (χ1β→χ2

β)


But note that (χ1β→χ2

β) is just the same formula as (χ1→χ2)β

. So

we’ve shown what we wanted to show.

Finally, take the third case. We must show that `S

2χ1↔2χβ1 .

This follows from the inductive hypothesis `Sχ1↔χβ1 . For the

inductive hypothesis implies `Sχ1→χ

β1 , by PL; and then, using

NEC and a K-axiom, we have `S2χ1→2χβ1 . A parallel argument

establishes `S2χβ1 →2χ1; and then the desired conclusion follows

by using PL. That completes the inductive proof.

The following examples illustrate the power of substitution of equivalents.

In our discussion of K we proved the following two theorems:

2(P∧Q)→(2P∧2Q)

(2P∧2Q)→2(P∧Q)

Hence (by PL), 2(P∧Q)↔(2P∧2Q) is a K-theorem. Given substitution

of equivalents, whenever we prove a theorem in which the formula 2(P∧Q)occurs as a subformula, we can infer that the result of changing 2(P∧Q) to

2P∧2Q is also a K-theorem—without having to do a separate proof.

Similarly, given the modal negation theorems, we know that all instances

of the following schemas are theorem of K (and hence of every other system):

2∼φ↔∼3φ

3∼φ↔∼2φ

Call these “the duals equivalences”.9

Given the duals equivalences, we can swap

∼3φ and 2∼φ, or ∼2φ and 3∼φ, within any theorem of any system, and

the result will also be a theorem of that system. So we can “move” ∼s through

series of modal operators at will. For example, it’s easy to show that each of the

following is a theorem of each system S:

9Given the duals equivalences, the 2 is related to the 3 the way the ∀ is related to the

∃ (since ∀x∼φ↔∼∃xφ, and ∃x∼φ↔∼∀xφ are logical truths). This shared relationship,

which holds between the 2 and the 3, and between the ∀ and the ∃, is called “duality”; 2 and

3 are said to be duals, as are ∀ and ∃. This logical analogy would be neatly explained by a

metaphysics according to which necessity just is truth in all worlds and possibility just is truth

in some worlds!


1. 332∼φ↔332∼φ this is a theorem of S, since it has the

form ψ→ψ

2. 33∼3φ↔332∼φ this is the result of changing 2∼φ on

the left of 1 to ∼3φ. Since 1 is a theo-

rem of S, this too is a theorem of S, by

substitution of equivalents via a duals

equivalence

3. 3∼23φ↔332∼φ changed 3∼3φ in 2 to ∼23φ, so by

substitution of equivalents via a duals

equivalence this too is a theorem of S

4. ∼223φ↔332∼φ this follows from 3 and a MN theorem

by PL, so it too is a theorem of S

(Note how this greatly simpli�es the process of establishing the existence of

theorems such as K3, T3, B3, S43, and S53!)

Our second meta-theorem concerns only system S5:

Modal reduction theorem for S5 Where O1 . . .On are modal op-

erators and φ is a wff:

`S5O1 . . .Onφ↔Onφ

That is, whenever a formula has a string of modal operators in front, it is always

equivalent to the result of deleting all the modal operators except the innermost

one. For example, 223232232323φ and 3φ are provably equivalent in

S5; i.e., 223232232323φ↔3φ is a theorem of S5). This follows from

the fact that the following equivalences are all theorems of S5:

a) 32φ↔2φ

b) 22φ↔2φ

c) 23φ↔3φ

d) 33φ↔3φ

The left-to-right direction of a) is just S5; the right-to-left is T3; b) is T and

S4; c) is T and S53; and d) is S43 and T3. Thus, by repeated applications of

these equivalences, using substitution of equivalents, we can reduce strings of


modal operators to the innermost operator. (It is straightforward to convert

this argument into a more rigorous inductive proof.)10

6.5 Soundness in MPL11

At this point, we have de�ned twelve logical systems: six semantic systems and

six axiomatic systems. But each semantic system was paired with an axiomatic

system to which we gave the same name. The time has come to justify this

pairing. In this section and the next, we show that for each semantic system,

exactly the same wffs are counted valid in that system as are counted theorems

by the axiomatic system of the same name. That is, for each of our systems, S

(for S = K, D, T, B, S4, and S5), we will prove soundness and completeness:

S-soundness every S-theorem is S-valid

S-completeness every S-valid formula is a S-theorem

Our study of modal logic has reversed that of history. We began with

semantics, because that is the more intuitive approach. Historically (as we

noted earlier), the axiomatic systems came �rst, in the work of C. I. Lewis.

Given the uncertainty over what formulas ought to be counted as axioms, modal

logic was in disarray. The discovery by the teenaged Saul Kripke in the late

1950s of the possible-worlds semantics we studied in section ??, and of the

correspondence between simple constraints (re�exivity, transitivity, etc.) on

the accessibility relation in his models and Lewis’s axiomatic systems, was a

major advance in the history of modal logic.

The soundness and completeness theorems have practical as well as the-

oretical value. First, once we’ve proved soundness, we will for the �rst time

have a method for establishing that a given formula is not a theorem: construct

a countermodel for that formula, thus establishing that the formula is not valid,

and then conclude via soundness that the formula is not a theorem. Second,

given completeness, if we want to know that a given formula is a theorem, it

10The modal reduction formula, the duals equivalences, and substitution of equivalents

together let us “reduce” strings of operators that include ∼s as well as modal operators. Simply

use the duals equivalents to drive any ∼s in the string to the far right hand side, then use the

modal reduction theorem to eliminate all but the innermost modal operator.

11The proofs of soundness and completeness in this and the next section are from Cresswell

and Hughes (1996).


suf�ces to show that it is valid. Since semantic validity proofs are comparatively

easy to construct, it’s nice to be able to use them rather than axiomatic proofs.

Let’s begin with soundness. We’re going to prove a general theorem, which

we’ll use in several soundness proofs. First we’ll need a piece of terminology.

Where Γ is any set of modal wffs, let’s call “K+Γ” the axiomatic system that

consists of the same rules of inference as K (MP and NEC), and which has

as axioms the axioms of K (instances of the K- and PL- schemas), plus the

members of Γ. Here, then, is the theorem:

Theorem 6.1 If Γ is any set of modal wffs and M is an MPL-

model in which each wff in Γ is valid, then every theorem of

K+Γ is valid inM

Modal systems of the form K+Γ are commonly called normal. Normal

modal systems contain all the K-theorems, plus possibly more. What Theorem

6.1 gives us is a method for constructing a soundness proof for any normal

system. Since all the systems we have studied here (K, D, etc.) are normal, this

method is suf�ciently general for us. Here’s how the method works for system

T. System T has the same rules of inference as K, and its axioms are all the

axioms of K, plus the instances of the T-schema. In the “K+Γ” notation, T = K

+ {2φ→φ :φ is an MPL wff}. To establish soundness for T, all we need to do

is show that every instance of the T-schema is valid in all re�exive models; for

we may then conclude by Theorem 6.1 that every theorem of T is valid in all

re�exive models. This method can be applied to each of our systems: for any

system, S, to establish S’s soundness it will suf�ce to show that the S’s “extra-K”

axioms are valid in all of the S-models.

To prove Theorem 6.1, we will �rst prove two lemmas:

Lemma 6.2 All PL and K-axioms are valid in all MPL-models

Lemma 6.3 For every MPL-model,M , MP and Necessitation

preserve validity inM

Theorem 6.1 follows from the lemmas: assume that every wff in Γ is valid in a

given MPL-modelM , and consider any theorem φ of K+Γ. That theorem is

a last line in a proof in which each line is either an axiom K+Γ, or follows from

earlier lines in the proof by MP or NEC. But axioms of K+Γ are either PL

axioms, K axioms, or members of Γ. The �rst two classes of axioms are valid in

all MPL-models, by Lemma 6.2, and so are valid inM ; and the �nal class of

axioms are valid inM by hypothesis. Thus, all axioms in the proof are valid


inM . Moreover, by Lemma 6.3, the rules of inference in the proof preserve

validity inM . Therefore, by induction, every line in the proof is valid inM .

Hence the last line in the proof, φ, is valid inM .

We now need to prove the lemmas.

Proof of Lemma 6.2

From our proof of soundness for PL (section 2.6), we know that the PL truth

tables generate the value 1 for each PL axiom, no matter what truth value its

immediate constituents have. But here in MPL, the truth values of conditionals

and negations are determined at a given world by the truth values at that world

of its immediate constituents via the PL truth tables. So any PL axiom must

have truth value 1 at any world, regardless of what truth values its immediate

constituents have. PL-axioms, therefore, are true at every world in every model,

and so are valid in every model. We need now to show that any K axiom—i.e.,

any formula of the form 2(φ→ψ)→ (2φ→2ψ)—is valid in any model:

i) Suppose for reductio that V(2(φ→ψ)→(2φ→2ψ), w) = 0,

for some model ⟨W ,R ,I ⟩, whose valuation is V, and some

w ∈Wii) So V(2(φ→ψ), w) = 1 and…

iii) …V((2φ→2ψ), w) = 0

iv) Given iii), V(2φ, w) = 1 and …

v) …V(2ψ, w) = 0

vi) Given v), for some v,Rwv and V(ψ, v) = 0

vii) Given iv), sinceRwv, V(φ, v) = 1

viii) Given ii), sinceRwv, V(φ→ψ, v) = 1

ix) Lines vi), vii), and viii) contradict, given the truth condition

for the→

Proof of Lemma 6.3

We must show that the rules MP and NEC preserve validity in any given model.

That is, we must show that if the inputs to one of these rules is valid in some

model, then that rule’s output must also be valid in that model.


First MP. Let φ and φ→ψ be valid in model ⟨W ,R ,I ⟩; we must show that

ψ is also valid in that model. That is, where V is this model’s valuation, and

w is any member of W , we must show that V(ψ, w) = 1. Since φ and φ→ψare valid in this model, V(φ→ψ, w) = 1, and V(φ, w) = 1; but by the truth

condition for→, V(ψ, w) must also be 1.

Next NEC. Suppose φ is valid in modelM . We must show that 2φ is

valid inM , i.e., that 2φ is true at each world inM , i.e., that for each world,

w, φ is true at every world accessible from w. But since φ is valid inM , φ is

true in every world inM , and hence is true at every world accessible from w.

6.5.1 Soundness of KWe can now construct soundness proofs for the individual systems. I’ll do this

for some of the systems, and leave the veri�cation of soundness for the other

systems as exercises.

First K. In the “K+Γ” notation, K is just K+∅, and so it follows immediately

from Theorem 6.1 that every theorem of K is valid in every MPL-model. So

K is sound.

6.5.2 Soundness of TT is K+Γ, where Γ is the set of all instances of the T-schema. So, given Theorem

6.1, to show that every theorem of T is valid in all T-models, it suf�ces to show

that all instances of the T-schema are valid in all T-models:

i) Assume for reductio that V(2φ→φ, w) = 0 for some world win some T-model (i.e., some model with a re�exive accessibility

relation)

ii) So V(2φ, w) = 1 and…

iii) …V(φ, w) = 0

iv) Rww, by re�exivity. So, from ii), V(φ, w) = 1, contradicting

iii)

6.5.3 Soundness of BB is K+ Γ, where Γ is the set of all instances of the T- and B- schemas. Given

Theorem 6.1, it suf�ces to show that every instance of the B-schema and every


instance of the T-schema is valid in every B-model. So, choose an arbitrary

model with a re�exive and symmetric accessibility relation, whose valuation is

V, and let w be any world in that model. We must show that V counts each

instance of the T-schema and the B-schema as being true at w. The proof

of the previous section shows that the T-axioms are true at w. Now for the

B-axioms:

i) Assume for reductio that V(32φ→φ, w) = 1.

ii) So V(32φ, w) = 1 and…

iii) …V(φ, w) = 0

iv) By ii), V(2φ, v) = 1, for some v such thatRwv.

v) By symmetry,Rvw. So, given iv), V(φ, w) = 1, contradicting

iii)

6.6 Completeness of MPLNext, completeness: for each system, we’ll show that every valid formula is a

theorem. As with soundness, most of the work will go into developing some

general-purpose machinery. At the end we’ll use the machinery to construct

completeness proofs for each system.

We’ll be constructing a kind of completeness proof known as a “Henkin-

proof”, after Leon Henkin, who used similar methods to demonstrate com-

pleteness for (nonmodal) predicate logic.

6.6.1 Canonical modelsFor each of our systems, we’re going to show how to construct a certain special

model, the canonical model for that system. The canonical model for a system,

S, will be shown to have the following feature:

If a formula is valid in the canonical model for S, then it is a theorem of S

This suf�cient condition for theoremhood can then be used to give complete-

ness proofs, as the following example brings out. Suppose we can demonstrate

that the accessibility relation in the canonical model for T is re�exive. Then,

since T-valid formulas are by de�nition true in every world in every model

with a re�exive accessibility relation, we know that every T-valid formula is


valid in the canonical model for T. But then the italicized statement tells us

that every T-valid formula is a theorem of T. So we would have established

completeness for T.

The trick for constructing canonical models will be to let the worlds in these

models be sets of formulas (remember, worlds are allowed to be anything we

like). And we’re going to construct the interpretation function of the canonical

model in such a way that a formula will be true at a world iff the formula is a

member of the set that is the world. Working out this idea will occupy us for

awhile.

6.6.2 Maximal consistent sets of wffsTo carry out this idea of constructing worlds as sets of formulas that are true at

those worlds, we’ll need to put some constraints on the nature of these sets of

wffs. It’s part of the de�nition of a valuation function that for any wff φ and

any world w, either φ or ∼φ is true at w. That means that any set of wffs that

we’re going to call a world had better contain either φ or ∼φ. Moreover, we’d

better not let such a set contain both φ and ∼φ, since a formula can’t be both

true and false at a world. Other constraints must be introduced as well.

Where S is any of our axiomatic systems, let’s de�ne the following notions:

A set of MPL-wffs, Γ, is S-inconsistent iff for some φ1 . . .φn ∈ Γ,

`S∼(φ1∧· · ·∧φn). Γ is S-consistent iff it is not S-inconsistent

A set of MPL-wffs, Γ, is maximal iff for every MPL-wff φ, either

φ or ∼φ is a member of ΓA set is maximal S-consistent iff it is both maximal and S-consistent

A set is S-inconsistent if it contains some wffs (�nite in number) that are

provably (in S) contradictory; it is S-consistent if it contains no such wffs. This

notion of consistency is proof-theoretic: it has to do with what can be proved in

axiomatic systems, not with truth in models. Furthermore, S-consistency has to

do with provability in system S. It therefore requires more than the mere absence

of contradictions. Thus consider a set of wffs that contains no contradictions

(i.e., for no φ does the set contain both φ and ∼φ), but which contains both

2P and ∼P . This set would be T-inconsistent, since ∼(2P∧∼P ) is a theorem

of T.

A maximal S-consistent set of wffs contains, for each formula, either that

formula or its negation; and it contains nothing that is disprovable in S. Maximal

consistent sets are �t sets to be worlds in our canonical models.


6.6.3 De�nition of canonical modelsWe’re now ready to de�ne canonical models. It may not be fully clear at this

point why the de�nition is phrased as it is; you’ll need to take it on faith that

the de�nition will get us where we want to go.

The canonical model for S is the MPL-model ⟨W ,R ,I ⟩ where:

• W is the set of all maximal S-consistent sets of wffs

• Rww ′ iff 2−(w)⊆ w ′

• I (α, w) = 1 iff α ∈ w, for each sentence letter α and each

w ∈W

(where 2−(∆) is de�ned as the set of wffs φ such that 2φ is a

member of ∆)

Let’s think for a bit about this de�nition. As promised, we have de�ned

the members ofW to be maximal S-consistent sets of wffs. And note that allmaximal S-consistent sets of wffs are included.

Accessibility is de�ned using the “2−” notation. Think of this operation

as “stripping off the boxes”: to arrive at 2−(∆) (“the box-strip of set ∆”),

begin with set ∆, discard any formula that doesn’t begin with a 2, line up

the remaining formulas, and then strip one 2 off of the front of each. The

de�nition of accessibility, therefore, says Rww ′ iff for each wff 2φ that is a

member of w, the wff φ is a member of w ′.The de�nition of accessibility in the canonical model says nothing about

formal properties like transitivity, re�exivity, and so on. As a result, it is not

true by de�nition that the canonical model for S is an S-model. T-models,

for example, must have re�exive accessibility relations, whereas the de�nition

of the accessibility relation in the canonical model for T says nothing about

re�exivity. As we will eventually see, the canonical model for each system S

turns out to be an S-model, but this fact must be proven; it’s not built into the

de�nition of a canonical model.

An atomic wff (sentence letter) is de�ned to be true at a world iff it is a

member of that world. Thus, for atomic wffs, truth and membership coincide.

What we really need to know, however, is that truth and membership coincide

for all wffs, including complex wffs. Proving this turns out to be a big task,

which will occupy us for several sections. We’ll need �rst to assemble some


�repower: a number of preliminary lemmas and theorems which will eventually

be used to prove that membership and truth coincide for all wffs in canonical

models. We’ll then �nally be able to give completeness proofs.

6.6.4 Features of maximal consistent setsWe’ll begin with some lemmas governing the behavior of maximal consistent

sets:

Lemma 6.1 Where Γ is any maximal S-consistent set of wffs:

6.1a for any wff φ, exactly one of φ, ∼φ is a member of Γ

6.1b φ→ψ ∈ Γ iff either φ /∈ Γ or ψ ∈ Γ6.1c if φ and φ→ψ are both members of Γ then so is ψ

Lemma 6.2 Where Γ is any maximal S-consistent set of wffs,

6.2a if `Sφ then φ ∈ Γ

6.2b if φ ∈ Γ and `Sφ→ψ then ψ ∈ Γ

I’ll prove Lemma 6.1, and leave 6.2 to the reader.

6.1a: we know from the de�nition of maximality that at least one ofφ or∼φis in Γ. But it cannot be that both are in Γ, for then Γwould be S-inconsistent (it

would contain the �nite subset {φ,∼φ}; but since all modal systems incorporate

propositional logic, it is a theorem of S that ∼(φ∧∼φ).)6.1b: Suppose �rst that φ→ψ is in Γ, and suppose for reductio that φ is

in Γ but ψ is not. Then, by 6.1a, ∼ψ is in Γ; but now Γ is S-inconsistent by

containing the subset {φ,φ→ψ,∼ψ}. Suppose for the other direction that either

φ is not in Γ or ψ is in Γ, and suppose for reductio that φ→ψ isn’t in Γ. By

6.1a, ∼(φ→ψ) is in Γ. Now, if φ /∈ Γ then ∼φ ∈ Γ, but then Γ would contain

the S-inconsistent subset {∼(φ→ψ),∼φ}. And if on the other hand, ψ ∈ Γthen Γ again contains an S-inconsistent subset: {∼(φ→ψ),ψ}. Either possibility

contradicts Γ’s S-consistency.

6.1c: Direct consequence of 6.1b.


6.6.5 Maximal consistent extensionsNext let’s show that if we begin with an S-consistent set ∆, we can “expand” it

into a maximal S-consistent set Γ. (We prove this because we’ll need to know

that there exist enough maximal consistent sets in a canonical model’s W in

order for the model to do its thing.)

Theorem 6.3 If ∆ is an S-consistent set of wffs, then there exists

some maximal S-consistent set of wffs, Γ, such that ∆⊆ Γ

Proof of Theorem 6.3: In outline, we’re going to build up Γ as follows. We’re

going to start by dumping all the formulas in∆ into Γ. Then we will go through

all the wffs in the language of MPL, φ1, φ2,…, one at a time. For each of these

wffs, we’re going to dump either it or its negation into Γ, depending on which

choice would be S-consistent. After we’re done, our set Γ will obviously be

maximal; it will obviously contain ∆ as a subset; and, we’ll show, it will also be

S-consistent.

So, let φ1, φ2,…be a list—an in�nite list, of course—of all the wffs of

MPL.12

Our strategy, recall, is to construct Γ by starting with ∆, and then

12We need to be sure that there is some way of arranging all the wffs of MPL into such a

list. Here is one method. Consider the following list of the primitive expressions of MPL:

( ) ∼ → 2 P1 P2 . . .1 2 3 4 5 6 7 . . .

Since we’ll need to refer to what position an expression has in this list, the positions of the

expressions are listed underneath those expressions. (E.g., the position of the 2 is 5.) Now,

where φ is any wff, call the rating of φ the sum of the positions of the occurrences of its

primitive expressions. (The rating for the wff (P1→P1), for example, is 1+ 6+ 4+ 6+ 2= 19.)

We can now construct the listing of all the wffs of MPL by an in�nite series of stages: stage 1,

stage 2, etc. In stage n, we append to our growing list all the wffs of rating n, in alphabeticalorder. The notion of alphabetical order here is the usual one, given the ordering of the primitive

expressions laid out above. (E.g., just as ‘and’ comes before ‘nad’ in alphabetical order, since

‘a’ precedes ‘n’ in the usual ordering of the English alphabet, ∼2P2 comes before 2∼P2 in

alphabetical order since ∼ comes before the 2 in the ordering of the alphabet of MPL. Note

that each of these wffs are inserted into the list in stage 15, since each has rating 15.) In stages

1-5 no wffs are added at all, since every wff must have at least one sentence letter and P1 is

the sentence letter with the smallest position. In stage 6 there is one wff: P1. Thus, the �rst

member of our list of wffs is P1. In stage 7 there is one wff: P2, so P2 is the second member of

the list. In every subsequent stage there are only �nitely many wffs; so each stage adds �nitely

many wffs to the list; each wff gets added at some stage; so each wff eventually gets added after

some �nite amount of time to this list.


going through this list one-by-one, at each point adding either φi or ∼φi .

Here’s how we do this more carefully. Let’s begin by de�ning an in�nite

sequence of sets, Γ0,Γ1, . . . :

i) Γ0 is ∆

ii) Γn+1 is Γn ∪ {φn+1} if that is S-consistent; otherwise Γn+1 is

Γn ∪{∼φn+1}

Note the recursive nature of the de�nition: the next member of the sequence

of sets, Γn+1 is de�ned as a function of the previous member of the sequence,

Γn.

Next let’s prove that each member in this sequence—that is, each Γi —is an

S-consistent set. We do this inductively, by �rst showing that Γ0 is S-consistent,

and then showing that if Γn is S-consistent, then so will be Γn+1.

Obviously, Γ0 is S-consistent, since ∆ was stipulated to be S-consistent.

Next, suppose thatΓn is S-consistent; we must show thatΓn+1 is S-consistent.

Look at the de�nition of Γn+1. What Γn+1 gets de�ned as depends on whether

Γn∪{φn+1} is S-consistent. If Γn∪{φn+1} is S-consistent, then Γn+1 gets de�ned

as that very set Γn ∪ {φn+1}, and so of course is S-consistent. So we’re ok in

that case.

The remaining possibility is that Γn ∪{φn+1} is S-inconsistent. In that case,

Γn+1 gets de�ned as Γn∪{∼φn+1}. So must show that in this case, Γn∪{∼φn+1}is S-consistent. Suppose for reductio that it isn’t. The conjunction of some

�nite subset of its members must therefore be provably false in S. Since Γn was

S-consistent, the �nite subset must contain∼φn+1, and so there existψ1 . . .ψm ∈Γn such that `

S∼(ψ1∧· · ·∧ψm∧∼φn+1). Furthermore, since Γn ∪ {φn+1} is S-

inconsistent, it too contains a �nite subset that is provably false in S. Since Γnis S-consistent, the �nite subset must contain φn+1, so there exist χ1 . . .χp ∈ Γn

such that `S ∼(χ1∧· · ·∧χp∧φn+1). But notice that ∼(ψ1∧· · ·∧ψm∧χ1∧· · ·∧χp)is a PL-semantic-consequence of the formulas ∼(ψ1∧· · ·∧ψm∧∼φn+1) and

∼(χ1∧· · ·∧χp∧φn+1). It follows that `S∼(ψ1∧· · ·∧ψm∧χ1∧· · ·∧χp) (each of

our modal systems “contains PL”: given the completeness of the PL axioms,

one can always move within an MPL axiomatic proof from some formulas to a

PL-semantic-consequence of those formulas.) Since ψ1 . . .ψm and χ1 . . .χp are

all members of Γn, this contradicts the fact that Γn is S-consistent.

We have shown that all the sets in our sequence Γi are S-consistent. Let

us now de�ne Γ to be the union of all the sets in the in�nite sequence—i.e.,


{φ :φ ∈ Γi for some i}. We must now show that Γ is the set we’re after: that i)

∆⊆ Γ, ii) Γ is maximal, and iii) Γ is S-consistent.

Any member of ∆ is a member of Γ0 (since Γ0 was de�ned as ∆), hence is a

member of one of the Γi s, and hence is a member of Γ. So ∆⊆ Γ.

Any wff of MPL is in the list somewhere—i.e., it is φi for some i. But by

de�nition of Γi , either φi or ∼φi is a member of Γi ; and so one of these is a

member of Γ. Γ is therefore maximal.

Suppose for reductio that Γ is S-inconsistent; there must then existψ1 . . .ψm∈ Γ such that `

S∼(ψ1∧· · ·∧ψm). By de�nition of Γ, each of these ψi ’s are

members of Γ j , for some j . Let k be the largest such j . Note next that, given

the way the Γi ’s are constructed, each Γi is a subset of all subsequent ones.

Thus, all of the ψi ’s are members of Γk , and thus Γk is S-inconsistent. But that

can’t be—we showed that all the Γi ’s are S-consistent. QED.

6.6.6 “Mesh”Our ultimate goal is to show that in canonical models, a wff is true at a world

iff it is a member of that world. If we’re going to be able to show this, we’d

better be able to show things like this:

(2) If 2φ is a member of world w, then φ is a member of every

world accessible from w

(3) If 3φ is a member of world w, then φ is a member of some

world accessible from w

We’ll need to be able to show (2) and (3) because it’s part of the de�nition of

truth in any MPL-model (whether canonical or not) that 2φ is true at w iff φis true at each world accessible from w, and that 3φ is true at w iff φ is true at

some world accessible from w. Think of it this way: (2) and (3) say that the

modal statements that are members of a world w in a canonical model “mesh”

with the members of the other worlds in that canonical model. This sort of

mesh had better hold if truth and membership are going to coincide.

(2) we know to be true straightaway, since it follows from the de�nition of

the accessibility relation in canonical models. The de�nition of the canonical

model for S, recall, stipulated that w ′ is accessible from w iff for each wff 2φin w, the wff φ is a member of w ′. (3), on the other hand, doesn’t follow

immediately from our de�nitions; we’ll need to prove it. Actually, it will be

convenient to prove something slightly different which involves only the 2:


Lemma 6.4 If ∆ is a maximal S-consistent set of wffs containing

∼2φ, then there exists a maximal S-consistent set of wffs Γsuch that 2−(∆)⊆ Γ and ∼φ ∈ Γ

(Given the de�nition of accessibility in the canonical model and the de�nition

of the 3 in terms of the 2, Lemma 6.4 basically amounts to (3).)

Proof of Lemma 6.4: Let ∆ be as described. The �rst (and biggest) step is to

establish:

(*) 2−(∆)∪{∼φ} is S-consistent.

Suppose for reductio that (*) is false. By the de�nition of S-inconsistency, the

following is a theorem of S:

∼(χ1∧· · ·∧χm)

for someχ1 . . .χm ∈2−(∆)∪{∼φ}. The following is a PL-semantic-consequence

of this formula, and hence, since S includes PL, is also a theorem of S:

∼(χ1∧· · ·∧χm∧∼φ)Now go through the list χ1 . . .χm, and if it contains any wffs that are not

members of 2−(∆), drop them from the list. Call the resulting list ψ1 . . .ψn.

Each of the ψi s, note, is a member of 2−(∆).13The only wff that could have

been dropped, in moving from the χi s to the ψi s, is ∼φ (since each χi was

a member of 2−(∆)∪ {∼φ}); the following wff is therefore a PL-semantic-

consequence of the previous wff, and so is itself a theorem of S:

∼(ψ1∧· · ·∧ψn∧∼φ)Next, begin a proof in S with a proof of∼(ψ1∧· · ·∧ψn∧∼φ), and then continue

as follows:

.

.

∼(ψ1∧· · ·∧ψn∧∼φ)ψ1→(ψ2→·· · (ψn→φ) . . . ) PL

2(ψ1→(ψ2→·· · (ψn→φ) . . . ) NEC

.

.

2ψ1→(2ψ2→·· · (2ψn→2φ) . . . ) K, PL (×n)

∼(2ψ1∧· · ·∧2ψn∧∼2φ) PL

13If 2−(∆) is empty then there will be no ψi s. In that case, let’s regard “ψ1∧· · ·∧ψn∧∼φ”

as standing for ∼φ.


This proof establishes that `S∼(2ψ1∧· · ·∧2ψn∧∼2φ). But since 2ψ1…2ψn,

and ∼2φ are all in∆, this contradicts∆’s S-consistency (2ψ1…2ψn are mem-

bers of ∆ because ψ1…ψn are members of 2−(∆).)We’ve established (*): 2−(∆) ∪ {∼φ} is S-consistent. It therefore has a

maximal S-consistent extension, Γ, by Theorem 6.3. Since 2−(∆)∪{∼φ} ⊆ Γ,

we know that 2−(∆)⊆ Γ and that ∼φ ∈ Γ. Γ is therefore our desired set; the

proof of Lemma 6.4 is complete.

6.6.7 The coincidence of truth and membership in canon-ical models

We’re now in a position to put all of our lemmas to work, and prove that

canonical models have the desired property that the wffs true at a world are

exactly the members of that world:

Theorem 6.5 WhereM (= ⟨W ,R ,I ⟩) is the canonical model

for any normal modal system, S, for any wffφ and any w ∈W ,

VM (φ, w) = 1 iff φ ∈ w

Proof of Theorem 6.5: We’ll use induction. The base case is when φ has zero

connectives—i.e., φ is a sentence letter. In that case, the result is immediate:

by the de�nition of the canonical model, I (φ, w) = 1 iff φ ∈ w; but by the

de�nition of the valuation function, VM (φ, w) = 1 iff I (φ, w) = 1.

Now the inductive step. We suppose (ih) that the result holds for φ, ψ, and

show that it holds for ∼φ, φ→ψ, and 2φ as well:

∼ : We must show that ∼φ is true at w iff ∼φ ∈ w.

i) ∼φ ∈ w iff φ /∈ w (6.1a)

ii) φ /∈ w iff φ is not true at w (ih)

iii) φ is not true at w iff ∼φ is true at w (truth cond. for ∼)iv) so, ∼φ is true at w iff ∼φ ∈ w (i, ii, iii)

→ : We must show that φ→ψ is true at w iff φ→ψ ∈ w:

i) φ→ψ is true at w iff either φ is not true at w or ψ is true at

w (truth cond for→)


ii) So, φ→ψ is true at w iff either φ /∈ w or ψ ∈ w (ih)

iii) So, φ→ψ is true at w iff φ→ψ ∈ w (6.1b)

2 : We must show that 2φ is true at w iff 2φ ∈ w.

First the forwards direction. Assume 2φ is true at w; then φ is true at every

world w ′ such thatRww ′. By the ih, we have (+) φ is a member of every such

w ′. Now suppose for reductio that 2φ /∈ w; by 6.1a, ∼2φ ∈ w. Since w is

maximal S-consistent, by Lemma 6.4, we know that there exists some maximal

S-consistent set Γ such that 2−(w) ⊆ Γ and ∼φ ∈ Γ. By de�nition of W , Γis a world; by de�nition of R , RwΓ; and so by (+) Γ contains φ. But Γ also

contains ∼φ, which contradicts its S-consistency.

Now the backwards direction. Assume 2φ ∈ w. Then by de�nition ofR ,

for every w ′ such thatRww ′, φ ∈ w ′. By the ih, φ is true at every such world;

hence by the truth condition for 2, 2φ is true at w. QED.

What was the point of proving theorem 6.5? The whole idea of a canonical

model was to be that a formula is valid in the canonical model for S iff it is a

theorem of S. This fact follows fairly immediately from Theorem 6.5:

Corollary 6.6: φ is valid in the canonical model for S iff `Sφ

Proof of Corollary 6.6: Let ⟨W ,R ,I ⟩ be the canonical model for S. Suppose

`Sφ. Then, by 6.2a, φ is a member of every maximal S-consistent set, and

hence φ ∈ w, for every w ∈ W . By 6.5, φ is true in every w ∈ W , and so is

valid in this model. Now for the other direction: suppose 0Sφ. Then {∼φ} is

S-consistent, and so by theorem 6.3, has a maximal consistent extension; thus,

∼φ ∈ w for some w ∈W ; by theorem 6.5, ∼φ is therefore true at w, and so φis not true at w, and hence φ is not valid in this model.

So, we’ve gotten where we wanted to go: we’ve shown that every system

has a canonical model, and that a wff is valid in the canonical model iff it is a

theorem of the system. We now use this fact to prove completeness for our

various systems:

6.6.8 Completeness of systems of MPLCompleteness of K

K’s completeness follows immediately. Any K-valid wff is valid in all MPL-

models, and so is valid in the canonical model for K, and so, by corollary 6.6, is

a theorem of K.


For any other system, S, all we need to do to prove S-completeness is to

show that the canonical model for S is an S-model. That is, we must show

that the accessibility relation in the canonical model for S satis�es the formal

constraint for system S (seriality for D, re�exivity for T and so on). This will

be made clear in the proof of completeness for D:

Completeness of D

Let us show that in the canonical model for D, the accessibility relation,R , is

serial. Let w be any world in that model. We showed above that 3(P→P ) is

a theorem of D, and so is a member of w by 6.2a, and so is true at w by 6.5.

Thus, by the truth condition for 3, there must be some world accessible to win which P→P is true; and hence there must be some world accessible to w.

Now for D’s completeness. Let φ be D-valid. It is then valid in all D-

models, i.e., all models with a serial accessibility relation. But we just showed

that the canonical model for D has a serial accessibility relation. φ is therefore

valid in that model, and hence by 6.6, `D φ.

Completeness of T

All we need to do is to prove that the accessibility relation in the canonical model

for T is re�exive; given that, every T-valid formula is valid in the canonical

model for T, and hence by 6.6, every T-valid formula is a T-theorem.

Let w be any world in the canonical model for T. For any φ, `T

2φ→φ;

thus, by 6.2b, it follows that for any φ, if 2φ ∈ w then φ ∈ w. But this is the

de�nition ofRww.

Completeness of B

We must show that the accessibility relation in the canonical model for B is

re�exive and symmetric. Re�exivity can be demonstrated in the same way as it

was for T, since every T-theorem is a B-theorem.

Now for symmetry: in the canonical model for B, suppose thatRwv. We

must show thatRvw—that is, that for any 2ψ in v, ψ ∈ w. So, suppose that

2ψ ∈ v. By 6.5, 2ψ is true at v; sinceRwv, by the de�nition of 3 it follows

that 32ψ is true at w, and hence is a member of w by 6.5. Since `B

32ψ→ψ,

by 6.2b, ψ ∈ w.


Completeness of S4

We must show that the accessibility relation in the canonical model for S4 is

re�exive and transitive. Again, re�exivity can be demonstrated as it was for

T; transitivity remains. SupposeRwv andRv u. We must showRw u—that

is, for any 2ψ ∈ w,ψ ∈u. If 2ψ ∈ w, since `S4 2ψ→22ψ, by 6.2b we have

22ψ ∈ w. By 6.5, 22ψ is true at w; hence by the truth condition for 2, 2ψ is

true at v; again by the truth condition for 2, ψ is true at u; by 6.5, ψ ∈ u.

Completeness of S5

We must show that the accessibility relation in the canonical model for S5 is

re�exive, symmetric, and transitive. But since each T, B, and S4 theorem is

an S5 theorem, the proofs of re�exivity, symmetry, and transitivity from the

previous three sections apply here.

Chapter 7

Variations on Propositional ModalLogic

As we have seen, possible worlds are useful for giving a semantics for propo-

sitional modal logic. Possible worlds are useful in other areas of logic as

well. In this chapter we will brie�y examine two other uses for possible worlds:

semantics for tense logic, and semantics for intuitionist propositional logic.

7.1 Propositional tense logic1

7.1.1 The metaphysics of timePropositional modal logic concerned the logic of the non-truth-functional

sentential operators “it is necessary that” and “it is possible that”. Another

set of sentential operators that can be similarly treated are propositional tenseoperators, such as “it will be the case that”, “it has always been the case that”,

etc.

A full logical treatment of natural language obviously requires that we

pay attention to temporal notions. Some philosophers, however, think that it

requires nothing beyond standard predicate logic. This was the view of many

early logicians, most notably Quine.2

Here are some examples of how Quine

would regiment temporal sentences in predicate logic:

1See Gamut (1991b, section 2.4); Cresswell and Hughes (1996, pp. 127-134).

2See, for example, Quine (1953b).

172

CHAPTER 7. VARIATIONS ON MPL 173

Everyone who is now an adult was once a child

∀x(Axn→∃t[E t n∧C x t])

A dinosaur once trampled a mammal

∃x∃y∃t (E t n∧D x ∧M y ∧T xy t )

Comments:

• ‘n’ is to be a name for the present time

• The predicate “E” is to be a predicate for the earlier-than re-

lation over moments of time. Thus, “E t n” means that t is a

time that is before the present moment; and so, “∃t (E t n∧ . . .”means that there exists some time, t , before the present mo-

ment, such that …”.

• we add in a new place for all predicates, for the time at which

the object satis�es the predicate. Thus, instead of saying

“C x”—“x is a child”—we say “C x t”: “x is a child at t”

• the quanti�er ∃x is atemporal, ranging over all objects at all

times. That’s how we can say that there is a thing, x, that

is a dinosaur, and which, at some previous time, trampled a

mammal.

So: we can use Quine’s strategy to represent temporal notions using standard

predicate logic. But some philosophers reject the conception of time that is

presupposed by Quine’s strategy. First, Quine presupposes that the past, present,

and future are equally real. After all, his symbolization of “A dinosaur once

trampled a mammal” says that there is such a thing as a dinosaur. Quine’s view is

that time is “space-like”. Other times are as real as the present, just temporally

distant, just as other places are equally real but spatially distant. Second, Quine

presupposes a distinctive metaphysics of change. Quine accounts for change

by adding argument places to temporary predicates like ‘is a child’ and ‘is an

adult’. For him, the statement ‘Ted is an adult’ is incomplete in something

like the way ‘Philadelphia is north of’ is complete: its predicate has an un�lled

argument place. When all of a sentence’s argument places are �lled, it can no

longer change its truth value; as a result (according to some), Quine’s approach

leaves no room for genuine change.

Arthur Prior (1967; 1968) and others reject Quine’s picture of time. Ac-

cording to Prior, rather than reducing notions of past, present, and future to

notions about what is true at times, we must instead include certain special


temporal expressions—sentential tense operators—in our most basic languages,

and develop an account of their logic. Thus he initiated the study of tense logic.One of Prior’s tense operators was P, symbolizing “it was the case that”.

Grammatically, P behaves like the ∼ and the 2: it attaches to a complete

sentence and forms another complete sentence. Thus, if R symbolizes “it is

raining”, then P R symbolizes “it was raining”. If a sentence letter occurs by

itself, outside of the scope of all temporal operators, then for Prior it is to

be read as present-tensed. Thus, it was appropriate to let R symbolize “It is

raining”—i.e., it is now raining.

Suppose we symbolize “there exists a dinosaur” as ∃xD x. Prior would then

symbolize “There once existed a dinosaur” as:

P∃xD x

And according to Prior, P∃xD x is not to be analyzed as saying that there exist

dinosaurs located in the past. For him, there is no further analysis of P∃xD x.

Prior’s attitude toward P is like everyone else’s attitude toward the ∼: no one

thinks that ∼∃xU x, “there are no unicorns”, is to be analyzed as saying that

there exist unreal unicorns. Further, Prior can represent the fact that I am

now, but have not always been, an adult, without adding argument places for

times to predicates. Symbolizing ‘is an adult’ with ‘A’, and ‘Ted’ with ‘t ’, Prior

would write: At ∧P∼At (“Ted is an adult, but it was the case that: Ted isn’t an

adult”). For Prior, the sentence At (“Ted is an adult”) is a complete statement,

but nevertheless can alter its truth value.

7.1.2 Tense operatorsOne can study various tense operators. Here is one group:

Gφ: it is, and is always going to be the case that φ

Hφ: it is, and always has been the case that φ

Fφ: it either is, or will at some point in the future be the case that,

φ

Pφ: it either is, or was at some point in the past the case that φ

Notice how these tense operators come in interde�nable pairs (related to

each other as the 2 and the 3):


Gφ iff ∼F∼φHφ iff ∼P∼φ

Thus one could start with just two of them, G and H say, and de�ne the others.

And one could use these two to de�ne further tense operators, for example Aand S, for “always” and “sometimes”

Aφ iff Hφ∧Gφ

Sφ iff ∼H∼φ∨∼G∼φ (i.e., iff Pφ∨Fφ)

There are further tense operators that are not de�nable in terms of those

we’ve been considering so far. There are, for example, metrical tense operators,

which concern what happened or will happen at speci�c temporal distances in

the past or future:

Pxφ: it was the case x minutes ago that φ

Fxφ: it will be the case in x minutes that φ

We will not consider metrical tense operators further.

The (nonmetrical) tense operators, as interpreted above, “include the

present moment”. For example, if Gφ is now true, then φ must now be true.

One could specify an alternate interpretation on which they do not include the

present moment:

Gφ: it is always going to be the case that φ

Hφ: it always has been the case that φ

Fφ: it will at some point in the future be the case that φ

Pφ: it was at some point in the past the case that φ

Whether we take the tense operators as including the present moment will

affect what kind of logic we develop for them. For example, the “2-like”

operators G and H will obey the T-principle (Gφ and Hφ will imply φ) if they

are interpreted as including the present moment, but not otherwise.


7.1.3 Syntax of tense logicIn this chapter we will study only propositional tense logic. Its syntax is straight-

forward: each tense operator has the grammar of the∼ and the 2. For example,

if we take the tense operators G and H as basic, then we could begin with the

de�nition of a wff from propositional logic (section 2.1) and add the following

clause:

If φ is a wff then so are:

GφHφ

7.1.4 Possible worlds semantics for tense logicLet’s turn now to semantics. The most natural semantics for tense logic is

a possible worlds-style semantics, in which we think of the members of Was times rather than possible worlds, we think of the accessibility relation as

the temporal ordering relation, and we think of the interpretation function as

assigning truth values to sentence letters at times.

(A Priorean faces hard philosophical questions about the use of such a

semantics, since according to him, the semantics doesn’t accurately model the

metaphysics of time. The questions are like those questions that confront

someone who uses possible worlds semantics for modal logic, but doesn’t think

that possible worlds are part of the metaphysics of modality.)

This change in how we think about possible-worlds models doesn’t require

any change to the de�nition of a model from section ??. A model,M , is still an

ordered triple, whose �rst member is a nonempty set, whose second member

is a binary relation over that set, and whose third member is a function that

assigns to each sentence letter a truth value relative to each member of the

�rst member ofM . To signify the change in how we’re thinking about these

models, however, let’s change our notation. Let’s call a model’s �rst member T ,

rather thanW , and let’s use variables like t , t ′, etc., for its members. And since

we’re thinking of a model’s second member as a relation of temporal ordering—

the at-least-as-early-as relation over times—let’s rename it too: “≤”. (If we were

interpreting the tense operators as not including the present moment, then

we would think of the temporal ordering relation as the strictly-earlier-than

relation, and would write it “<”.) Thus, instead of writing “Rww ′”, we write:

t ≤ t ′.


We’ll need to update the de�nition of the valuation function. The clauses

for the propositional connectives remain the same; what we need to add is

clauses for the tense operators. Let’s take just G and H as primitive; here are

the clauses:

VM (Gφ, t ) = 1 iff for every t ′ such that t ≤ t ′,VM (φ, t ′) = 1

VM (Hφ, t ) = 1 iff for every t ′ such that t ′ ≤ t ,VM (φ, t ′) = 1

If we de�ne F and P as ∼G∼ and ∼H∼, respectively, then we get the following

derived clauses:

VM (Fφ, t ) = 1 iff for some t ′ such that t ≤ t ′,VM (φ, t ′) = 1

VM (Pφ, t ) = 1 iff for some t ′ such that t ′ ≤ t ,VM (φ, t ′) = 1

Call an MPL-model, thought of in this way, a “PTL-model” (for “Priorean

Tense Logic”). And say that a wff is PTL-valid iff it is true in every time in every

PTL-model. Given our discussion of system K from chapter 6, we already

know a lot about PTL-validity. The truth condition for the G is the same as

the truth condition for the 2 in MPL. Thus, for each K-valid formula φ of

MPL, there is a PTL-valid formula of tense logic: simply replace each 2 in

φ with G. Replacing 2s with Gs in the K-valid formula 2(P∧Q)→2P results

in the PTL-valid formula G(P∧Q)→GP , for example. Similarly, the result of

replacing 2s with Hs in a K-valid formula also results in a PTL-valid formula.

But there are further cases of PTL-validity that depend on the interaction

between different tense operators, and hence have no direct analog in MPL.

For example, we can demonstrate that �PTL

φ→GPφ:

i) Suppose for reductio that in some PTL-modelM (= ⟨T ,≤,I ⟩) and some t ∈ T , VM (φ→GPφ, t ) = 0. (I henceforth

drop the subscriptM .)

ii) So V(φ, t ) = 1 and …

iii) …V(GPφ, t ) = 0.

iv) Given iii), by the truth condition for G: for some t ′ ∈ T , t ≤ t ′

and V(Pφ, t ) = 0v) Given iv), by the (derived) truth condition for P: for every

t ′′ ∈ T , if t ′′ ≤ t ′ then V(φ, t ′′) = 0vi) letting t ′′ in v) be t , given that t ≤ t ′ (iv), we have: V(φ, t ) = 0,

contradicting ii).

Similarly, one can show that �PTL

φ→HFφ.


7.1.5 Formal constraints on ≤PTL-validity is not a good model for logical truth in tense logic. We have

so far placed no constraints on the formal properties of the relation ≤ in a

PTL-model. That means that there are PTL models in which the ≤ looks

nothing like a temporal ordering. We don’t normally think that time could

consist of a number of wholly temporally disconnected points, for example,

or of many points each of which is at-least-as-early-as all of the rest, and so

on, but there are PTL-models answering to these strange descriptions. As we

have de�ned them, PTL-valid formulas must be true at every world in everyPTL-model, even these strange models. This means that many tense-logical

statements that ought, intuitively, to count as logical truths, are in fact not

PTL-valid.

The formula GP→GGP is an example. It is PTL-invalid, for consider a

model with three times, t1, t2, and t3, where t1 ≤ t2, t2 ≤ t3, and t1 6≤ t3, and in

which P is true at t1 and t2, but not at t3:

•P

t1(( •t2

P

(( •t3

∼PIn this model, GP→GGP is false at time t1. But GP→GGP is, intuitively, a

logical truth. If it is and will always be raining, then surely it must also be

true that: it is and always will be the case that: it is and always will be raining.

The problem, of course, is that the ≤ relation in the model we considered is

intransitive, whereas, one normally assumes, the at-least-as-early-as relation

must be transitive.

So: a more interesting notion of validity for PTL formulas results from

considering only PTL-models with transitive ≤ relations. Doing this validates

every instance of the “S4” schemas:

Gφ→GGφ

Hφ→HHφ

There are other interesting constraints on ≤ that one might impose. One

might impose re�exivity, for example. This is natural to impose if we are

construing the tense operators as including the present moment; not otherwise.

Imposing re�exivity validates the “T-schemas” Gφ→φ and Hφ→φ.

One might also impose “connectivity” of some sort. Connective principles

disallow, in various ways, “incomparable” pairs of times—pairs of times neither


of which bears the ≤ relation to the other. The strongest sort disallows allincomparable pairs:

“Strong connectivity”: for every t ′, t ′′, either t ′ ≤ t ′′ or t ′′ ≤ t ′

A weaker sort merely disallows incomparable pairs when each member of the

pair is after or before some one time:

“Weak connectivity”: for every t , t ′, t ′′, IF: either t ≤ t ′ and t ≤ t ′′,or t ′ ≤ t and t ′′ ≤ t , THEN: either t ′ ≤ t ′′ or t ′′ ≤ t ′

Weak connectivity disallows “branches” in the temporal order but allows dis-

tinct timelines wholly disconnected from one another; strong connectivity

insures that all times are part of a single non-branching structure. Each sort

validates every instance of the following schemas:

G(Gφ→ψ)∨G(Gψ→φ)H(Hφ→ψ)∨H(Hψ→φ)

Here’s a sketch of a validity proof for the �rst schema:

i) Assume for reductio that V(G(Gφ→ψ)∨G(Gψ→φ), t ) = 0in some PTL-model whose accessibility relation is weakly

connected.

ii) So V(G(Gφ→ψ), t ) = 0 and V(G(Gψ→φ), t ) = 0

iii) so there exist times, t ′ and t ′′, such that t ≤ t ′ and t ≤ t ′′, and

V(Gφ→ψ, t ′) = 0 and V(Gψ→φ, t ′′) = 0

iv) thus, Gφ is true at t ′, ψ is false at t ′, Gψ is true at t ′′, and φis false at t ′′

v) but by weak connectivity, t ′ ≤ t ′′ or t ′′ ≤ t ′. Either way iv)

leads to a contradiction.

There are other constraints one might impose, for example anti-symmetry(no distinct times bear ≤ to each other), density (between any two times there is

another time), or eternality (there exists neither a �rst nor a last time). In some

cases, imposing a constraint validates an interesting schema being validated.

Further, some constraints are more philosophically controversial than others.


Notice that one should not impose symmetry on ≤. Obviously if one time

is at least as early as another, then the second time needn’t be at least as early

as the �rst. Moreover, imposing symmetry would validate the “B” schemas

FGφ→φ and PHφ→φ; but these clearly ought not to be validated. Take the

�rst, for example: it doesn’t follow from it will be the case that it is always going tobe the case that I’m dead that I’m (now) dead.

So far we have been interpreting the tense operators as including the present

moment. That led us to call the temporal ordering relation in our models “≤”,

and require that it be re�exive. What if we instead interpreted the tense

operators as not including the present moment? We would then call the

temporal ordering relation “<”, and think of it as the earlier-than relation; and

we would no longer require that it be re�exive. Indeed, it would be natural to

require that it be irre�exive: that it never be the case that t < t .

We have considered only the semantic approach to tense logic. What of a

proof-theoretic approach? Given the similarity between tense logic and modal

logic, it should be no surprise that axiom systems similar to those of section

6.4 can be developed for tense logic. Moreover, the techniques developed in

sections 6.5-6.6 can be used to give soundness and completeness proofs for

tense-logical axiom systems, relative to the possible-worlds semantics that we

have developed in this section.

7.2 Intuitionist propositional logic

7.2.1 Kripke semantics for intuitionist propositional logic3

As we saw in section 3.4, intuitionists think of meaning in proof-theoretic terms,

rather than truth-theoretic terms, and as a result reject classical propositional

logic in favor of intuitionist propositional logic. We have already developed a

proof-theory for intuitionist logic: we began with the original sequent calculus

and then dropped double-negation elimination while adding ex falso. But we

still need a semantics. What should such a semantics look like?

In this book we have been thinking of logical truth, on the semantic

conception—i.e., validity—as “truth no matter what”. It is natural for in-

tuitionists to think, rather, in terms of “provability no matter what”. We will

lay out a semantics for intuitionist propositional logic—due to Saul Kripke—

that is based on this idea. The semantics will be like that of possible-worlds

3See (Priest, 2001, chapter 6)


semantics for propositional modal logic, and so it will include valuation func-

tions that assign the values 1 and 0 to formulas relative to the members of a

set W . But the idea is to now think of the members of W as stages in the

construction of proofs, rather than as possible worlds, and to think of 1 and 0 as

“proof statuses”, rather than truth values. That is, we are to think of V(φ, w) = 1as meaning that formula φ has been proved at stage w.

Let us treat the ∧ and the ∨ as primitive connectives. Here is Kripke’s

semantics for intuitionist propositional logic. (To emphasize the different way

we are regarding the “worlds”, we renameW “S ”, for stages in the construction

of proofs, and we will use the variables s , s ′, etc., for its members.)

A intuitionism-model is a triple ⟨S ,R ,I ⟩, such that:

i) S is a non-empty set (“proof stages”)

ii) R is a binary relation over S (“accessibility”) that is re�exive,

transitive, and obeys the heredity condition: for any sentenceletter α, if I (α, s) = 1 andR s s ′ then I (α, s ′) = 1

iii) I is a function from sentence letters and stages to truth values

(“interpretation function”).

Given any such model, we can de�ne the corresponding valuation function

thus:

a) VM (α, s) =I (α, s) for any sentence letter α

b) VM (φ∧ψ, s) = 1 iff VM (φ, s) = 1 and VM (ψ, s) = 1

c) VM (φ∨ψ, s) = 1 iff VM (φ, s) = 1 or VM (ψ, s) = 1

d) VM (∼φ, s) = 1 iff for every s ′ such thatR s s ′, VM (φ, s ′) = 0

e) VM (φ→ψ, s) = 1 iff for every s ′ such thatR s s ′, either VM (φ, s ′) =0 or VM (ψ, s ′) = 1

Note that the truth conditions for the → and the ∼ at stage s no longer

depend exclusively on what s is like; they are sensitive to what happens at

stages accessible from s . Unlike the ∧ and the ∨, → and ∼ are not “truth

functional” (relative to a stage); they behave like modal operators.

Let us think intuitively about these models. We are to think of each member

of S as a stage in the construction of mathematical proofs. At any stage, one


has come up with proofs of some things but not others. When V assigns 1 to a

formula at a stage, that means intuitively that as of that state of information,

the formula has been proven. The assignment of 0 means that the formula has

not been proven thus far (though it might nevertheless in the future.)

The holding of the accessibility relationR represents which future stages

are possible, given one’s current stage. If s ′ is accessible from s , that means that

s ′ contains all the proofs in s , plus perhaps more. Given this understanding,

re�exivity and transitivity are obviously correct to impose, as is the heredity

condition, since (on the somewhat idealized conception of proof we are oper-

ating with) one does not lose proved information when constructing further

proofs. But the accessibility relation will not in general be symmetric: for

sometimes one will come across a new proof that one did not formerly have.

Let’s also think through why the truth conditions for →,∧,∨ and ∼ are

intuitively correct. Intuitionists, recall, associate with each propositional con-

nective, a conception of what proofs of formulas built using that connective

must be like:

a proof of ∼φ is a proof that φ leads to a contradiction

a proof of φ∧ψ is a proof of φ and a proof of ψ

a proof of φ∨ψ is a proof of φ or a proof of ψ

a proof of φ→ψ is a construction that can be used to turn any proof

of φ into a proof of ψ

This is what inspires the de�nition of a valuation function. As of a time, one

has proved φ∧ψ iff one has proved both φ and ψ then. As of a time, one has

proved φ∨ψ iff one has proved one of the disjuncts. As for ∼, a proof of ∼φ,

according to an intuitionist, is a proof that φ leads to a contradiction. But i) if

one has proved that φleads to a contradiction, then in no future stage could

one prove φ (at least if one’s methods of proof are consistent); and ii) if one

has not proved that φ leads to a contradiction, this leaves open the possibility

of a future stage at which one proves φ. Thus the valuation condition for ∼ is

justi�ed.4

As for→: if one has a method of converting proofs of φ into proofs

4I’m fudging here a bit. Are the stages idealized so that one has already proven everyone

one can in principle prove? Clearly not, for then any formula assigned 1 at any accessible

stage should already be assigned 1 at that stage. But if stages are not idealized in this way,

then why suppose that the assignment of 0 at a stage to ∼φ (failure to prove that φ leads to a

contradiction) insures that there is some future stage at which φ is proved? A similar worry

confronts the valuation condition for→.


of ψ, then there could never be a possible future in which one has a proof of

φ but not one of ψ. Conversely, if one lacks such a method, then it should be

possible one day to have a proof of φ without being able to convert it into a

proof of ψ, and thus without then having a proof of ψ.

We can now de�ne intuitionist semantic consequence and validity in the

obvious way:

�Iφ iff VM (φ, s) = 1 for each stage s in each intuitionist modelM

Γ �Iφ iff for every intuitionist modelM and every stage s inM ,

if VM (γ , s) = 1 for each γ ∈ Γ, then VM (φ, s) = 1

7.2.2 Examples and proofsFirst some examples. Let’s establish that:

Q � P→Q

Take any model and any stage s ; assume that V(Q, s) = 1 and V(P→Q, s) = 0.

Thus, for some s ′, R s s ′ and V(P, s ′) = 1 and V(Q, s ′) = 0. But this violates

heredity.

Next:

P→Q �∼Q→∼P (contraposition):

Suppose V(P→Q, s) = 1 and V(∼Q→∼P, s) = 0. Given the latter, there’s

some stage s ′ such thatR s s ′ and V(∼Q, s ′) = 1 and V(∼P, s ′) = 0. Given the

latter, for some s ′′, R s ′ s ′′ and V(P, s ′′) = 1. Given the former, V(Q, s ′′) = 0.

Given transitivity,R s s ′′. Given the truth of P→Q at s , either V(P, s ′′) = 0 or

V(Q, s ′′) = 1. Contradiction.

In section 3.4 we asserted (but did not prove) that ∅ ` P∨∼P is an unprov-

able sequent. Here we’ll establish:

2 P∨∼P :

Here’s a model in which P∨∼P is valuated as 0 in stage r:


0 0 0

P∨∼P∗

r

��

00

∗1

Pa

00

As in section ??, we use asterisks to remind ourselves of commitments that

concern other worlds/stages. The asterisk is under ∼P in stage r because a

negation with value 0 carries a commitment to including some stage at which

the negated formula is 1. The asterisk is over the P in stage a because of the

heredity condition: a sentence letter valuated 1 carries a commitment to make

that letter 1 in every accessible stage. (Likewise, negations and conditionals

valuated as 1 generate top-asterisks, and conditionals valuated as 0 generate

bottom-asterisks). The of�cial model:

S : {r,a}R : {⟨r, r⟩, ⟨a,a⟩, ⟨r,a⟩}I (P,a) = 1, all other atomics 0 everywhere

(I’ll skip the of�cial models from now on.) So, 2 P∨∼P . Given the soundness

of the intuitionist proof system (proved below), it follows that ∅ ` P∨∼P is

indeed an unprovable sequent.

Next example:

∼∼P 2 P :


∗1 0 0

∼∼P P∗

r

00

��∗1 0

P ∼P∗

a

00

Note: since ∼∼P is 1 at r, that means that ∼P must be 0 at every stage at

which r sees. Now, Rrr, so ∼P must be 0 at r. So r must see some stage in

which P is 1. World a takes care of that.

Next:

∼(P∧Q) 2 (∼P∨∼Q):

∗1 0 0 0 0 0

∼(P∧Q) ∼P∨∼Q∗ ∗

r

zzttttttttttt

%%JJJJJJJJJJJ00

∗1 0 0

P P∧Qa

00

∗1 0 0

Q P∧Qb

00

Next example:

∼P∨∼Q �∼(P∧Q):

Suppose V(∼P∨∼Q, s) = 1 and V(∼(P∧Q), s) = 0. Given the latter, for some

s ′, R s s ′ and V(P∧Q, s ′) = 1. So, V(P, s ′) = 1 and V(Q, s ′) = 1. Given the


former, either V(∼P, s) = 1 or V(∼Q, s) = 1. If the former then V(P, s ′) = 0; if

the latter V(Q, s ′) = 0. Contradiction either way.

Final example:

P→(Q∨R) 2 (P→Q)∨(P→R):

We begin thus:

∗1 0 0 0

P→(Q∨R) (P→Q)∨(P→R)∗ ∗

r

00

We now discharge the bottom-asterisks:

∗1 0 0 0

P→(Q∨R) (P→Q)∨(P→R)∗ ∗

r

xxrrrrrrrrrrrrrrr

&&LLLLLLLLLLLLLLLLL00

∗1 0

P Qa

00

P Rb

00

7.2.3 Soundness and other facts about intuitionist validityIn this section we establish a few facts about intuitionist validity.

Semantic consequence and conditionals

Let’s establish two facts connecting the relation of semantic consequence to

conditionals.

Deduction theorem: If φ �Iψ then �

Iφ→ψ

Converse deduction theorem: if �Iφ→ψ then φ �

Iψ


Proof: for the �rst, suppose φ �ψ, and suppose for reductio that V(φ→ψ, s) =0. Then for some s ′ (that s sees), V(φ, s ′) = 1 and V(ψ, s ′) = 0—contradicts

φ �ψ.

As for the second, suppose�φ→ψ, and suppose for reductio that V(φ, s) = 1while V(ψ, s) = 0. By �φ→ψ, V(φ→ψ, s) = 1. By re�exivity, either V(φ, s) = 0or V(ψ, s) = 1. Contradiction.

Intuitionistic consequence implies classical consequence

That is: if Γ �Iφ then Γ �

PLφ, where �

PLis the relation of classical semantic

consequence (from section 2.3)).

Proof: Suppose Γ �Iφ, and let I be a PL-interpretation in which every

member of Γ is true; we must show that VI (φ) = 1. (VI , recall, is the classical

valuation forI .) Consider the intuitionist model with just one stage, r, in which

formulas have the same valuations as they have in the classical interpretation—

i.e., ⟨{r},{⟨r, r⟩},I *⟩, where I *(α, r) = I (α) for each sentence letter α. It’s

easy to check that since the intuitionist model has only one stage, the classical

and intuitionist truth conditions collapse in this case, so that for every wff φ,

VI *(φ, r) =VI (φ). So, since every member of Γ is true in I , every member of

Γ is true at r in the intuitionist model. Since Γ �Iφ, it follows that φ is 1 at r

in the intuitionist model; and so, φ is true in the classical interpretation—i.e.,

VI (φ) = 1.

The heredity condition holds for all formulas

(i.e., for any wff φ, whether atomic or no, and any stage, s , in any intuitionist

model, if V(φ, s) = 1 and R s s ′ then V(φ, s ′) = 1). Call this claim “General

heredity”.

Proof: by induction. The base case is just the of�cial heredity condition.

Next we make the inductive hypothesis: heredity is true for formulas φ and ψ;

we must now show that heredity also holds for ∼φ, φ→ψ, φ∧ψ, and φ∨ψ:

∼ : Suppose for reductio that V(∼φ, s) = 1,R s s ′, and V(∼φ, s ′) = 0. Given

the latter, for some s ′′, R s ′ s ′′ and V(φ, s ′′) = 1. By transitivity, R s s ′′. This

contradicts V(∼φ, s) = 1.

→ : Suppose for reductio that V(φ→ψ, s) = 1, R s s ′, and V(φ→ψ, s ′) = 0.

Given the latter, for some s ′′,R s ′ s ′′ and V(φ, s ′′) = 1 and V(ψ, s ′′) = 0; but by

transitivity,R s s ′′—contradicts the fact that V(φ→ψ, s) = 1.


∧: Suppose for reductio that V(φ∧ψ, s) = 1, R s s ′, and V(φ∧ψ, s ′) = 0.

Given the former, V(φ, s) = 1 and V(ψ, s) = 1. By the inductive hypothesis,

V(φ, s ′) = 1 and V(ψ, s ′) = 1—contradiction.

∨ : Suppose for reductio that V(φ∨ψ, s) = 1, R s s ′, and V(φ∨ψ, s ′) = 0.

Given the former, either V(φ, s) = 1 or V(ψ, s) = 1; and so, given the inductive

hypothesis, either φ or ψ is 1 in s ′. That violates V(φ∨ψ, s ′) = 0.

Soundness for intuitionism

Say that a sequent Γ `φ is intuitionistically valid (“I-valid”) iff in for every stage

in every intuitionist model, IF every member of Γ is 1 at that stage (“V(Γ, s) = 1”,

for short), THEN V(φ, s) = 1. What we wish to prove is that the system of

section 3.4 is sound—that is, that every sequent that is provable in that system

is I-valid.

Proof: Since a provable sequent is the last sequent in any proof, all we need

to show is that every sequent in any proof is I-valid. And to do that, all we need

to show is that i) the rule of assumptions generates I-valid sequents, and ii) all

the other rules preserve I-validity.

The rule of assumptions generates sequents of the form φ `φ, which are

clearly I-valid.

As for the other rules:

∧I: Here we assume that the inputs to ∧I are I-valid, and show that its

output is I-valid. That is, we assume that Γ `φ and ∆ `ψ are I-valid sequents,

and we must show that it follows that Γ, ∆ `φ∧ψ is also I-valid. So, consider

any model with valuation V and any stage s such that V(Γ∪∆)=1, and suppose

for reductio that V(φ∧ψ, s) = 0. Since Γ `φ is I-valid, V(φ, s) = 1; since∆ `ψis I-valid, V(ψ, s) = 1; contradiction.

∧E: Assume that Γ `φ∧ψ is I-valid, and suppose for reductio that V(Γ, s) =1 and V(φ, s) = 0, for some stage s in some model. By the I-validity of Γ `φ∧ψ,

V(φ∧ψ, s) = 1, so V(φ, s) = 1. Contradiction. The case of ψ is parallel.

∨I: Assume that Γ `φ is I-valid and suppose for reductio that V(Γ, s) = 1but V(φ∨ψ, s) = 0. Thus V(φ, s) = 0—contradiction.

∨E: Assume that Γ ` φ∨ψ, ∆1,φ ` Π, and ∆2,ψ ` Π are all I-valid, and

suppose for reductio that V(Γ ∪∆1 ∪∆2, s) = 1 but V(Π, s) = 0. The �rst

assumption tells us that V(φ∨ψ, s) = 1, so either φ or ψ is 1 at s . If the former,

then the second assumption tells us that V(Π, s) = 1; if the second, then the

third assumption tells us that V(Π, s) = 1. Either way, we have a contradiction.


DN (the remaining version: if Γ ` φ then Γ ` ∼∼φ): assume Γ ` φ is

I-valid, and suppose for reductio that V(Γ, s) = 1 but V(∼∼φ, s) = 1. From

the latter, for some s ′, R s s ′ and V(∼φ, s ′) = 1. So V(φ, s ′) = 0 (since R is

re�exive). From the former and the fact that Γ `φ, V(φ, s) = 1. This violates

general heredity.

RAA: Suppose that Γ,φ `ψ∧∼ψ is I-valid, and suppose for reductio that

V(Γ, s) = 1 but V(∼φ, s) = 0. Then for some s ′, R s s ′ and V(φ, s ′) = 1. Since

V(Γ, s) = 1 (i.e., all members of Γ are 1 at s), by general heredity V(Γ, s ′) = 1 (all

members of Γ are 1 at s’). Thus, since Γ,φ `ψ∧∼ψ is I-valid, V(ψ∧∼ψ, s ′) = 1.

But that is impossible. (If V(ψ∧∼ψ, s ′) = 1, then V(ψ, s ′) = 1 and V(∼ψ, s ′) = 1;

but from the latter and the re�exivity ofR it follows that V(ψ, s ′) = 0.)

→I: Suppose thatΓ,φ `ψ is I-valid, and suppose for reductio that V(Γ, s) = 1but V(φ→ψ, s) = 0. Given the latter, for some s ′, R s s ′ and V(φ, s ′) = 1 and

V(ψ, s ′) = 0. Given general heredity, V(Γ, s ′) = 1. And so, given that Γ,φ `ψ is

I-valid, V(ψ, s ′) = 1—contradiction.

→E: Suppose that Γ `φ and ∆ `φ→ψ are both I-valid, and suppose for

reductio that V(Γ ∪∆, s) = 1 but V(ψ, s) = 0. Since Γ ` φ and ∆ ` φ→ψ,

V(φ, s) = 1 and V(φ→ψ, s) = 1. Given the latter, and given thatR is re�exive,

either V(φ, s) = 0 or V(ψ, s) = 1. Contradiction.

EF (ex falso): Suppose Γ `φ∧∼φ is I-valid, and suppose for reductio that

V(Γ, s) = 1 but V(ψ, s) = 0. Given the former and the I-validity of Γ `φ∧∼φ,

V(φ∧∼φ, s) = 1, which is impossible.

Note that the rule we dropped, double-negation elimination, does notpreserve I-validity. For suppose it did. The sequent∼∼P `∼∼P is obviously I-

valid, and thus the sequent∼∼P ` P , which follows from it by double-negation

elimination, would also be I-valid. But it isn’t: above we gave a model and a

stage r in which ∼∼P is 1 and P is 0.

Chapter 8

Counterfactuals1

There are certain conditionals in natural language that are not well-

represented either by propositional logic’s material conditional or by

modal logic’s strict conditional. In this chapter we consider “counterfactual”

conditionals—conditionals that (loosely speaking) have the form:

If it had been that φ, then it would have been that ψ

For instance:

If I had struck this match, it would have lit

The counterfactuals that we typically utter have false antecedents (hence

the name), and are phrased in the subjunctive mood. They must therefore

be distinguished from English conditionals phrased in the indicative mood.

Counterfactuals are generally thought to semantically differ from indicative

conditionals. A famous example: the counterfactual conditional ‘If Oswald

hadn’t shot Kennedy, someone else would have’ is false (assuming that certain

conspiracy theories are false and Oswald was acting alone); but the indicative

conditional ‘If Oswald didn’t shoot Kennedy then someone else did’ is true

(we know that someone shot Kennedy, so if it wasn’t Oswald, it must have been

someone else.) The semantics of indicative conditionals is an important topic

in its own right, but we won’t take up that topic here.

We represent the counterfactual with antecedent φ and consequent ψ thus:

φ2→ψ1This section is adapted from my notes from Ed Gettier’s fall 1988 modal logic class.

190

CHAPTER 8. COUNTERFACTUALS 191

What should the logic of this new connective be, if it is to accurately represent

natural language counterfactuals?

8.1 Natural language counterfactualsWell, let’s have a look at how natural language counterfactuals behave. Our

survey will provide guidance for our main task: developing a semantics for 2→.

As we’ll see, counterfactuals behave very differently from both material and

strict conditionals.

8.1.1 Not truth-functionalOur system for counterfactuals should have the following features:

∼P 2 P2→Q

Q 2 P2→Q

For consider: I did not strike the match; but it doesn’t logically follow that

if I had struck the match, it would have turned into a feather. So if 2→ is to

represent ‘if it had been that…, it would have been that…’, ∼P should not

semantically imply P2→Q. Similarly, George W. Bush (somehow) won the last

United States presidential election, but it doesn’t follow that if the newspapers

had discovered beforehand that Bush had an affair with Al Gore, he would

still have won. So our semantics had better not count P2→Q as a semantic

consequence of Q either.

These implications hold for the material conditional, however:

∼P � P→Q

Q � P→Q

We have our �rst difference in logical behavior between counterfactuals and

the material conditional→.

8.1.2 Can be contingentIt’s not true, presumably, that if Oswald hadn’t shot Kennedy, then someone else

would have (assuming that the conspiracy theory is false). But the conspiracy

theory might have been true; in a possible world in which there is a conspiracy,


it would be true that if Oswald hadn’t shot Kennedy, someone else would

have. Thus, our logic should allow counterfactuals to be contingent statements.

Just because a counterfactual is true, it should not follow logically that it is

necessarily true; and just because a counterfactual is false, it should not follow

logically that it is necessarily false. Our semantics for 2→, that is, should have

the following features:

P2→Q 22(P2→Q)

∼(P2→Q) 2→2∼(P2→Q)

One reason this is important is that it shows an obstacle to using the strict

conditional⇒ to represent natural language counterfactuals. For remember

that P⇒Q is de�ned as 2(P→Q). As a result:

P⇒Q �S4,S5

2(P⇒Q)

∼(P⇒Q) �S5

2∼(P⇒Q)

So if, as is commonly supposed, the logic of the 2 is at least as strong as S4, we

have a logical mismatch between counterfactuals and the⇒.

8.1.3 No augmentationThe→ and the⇒ obey the argument form augmentation:

P→Q P⇒Q(P∧R)→Q (P∧R)⇒Q

That is, P→Q �PL(P∧R)→Q and P⇒Q �

K,…(P∧R)⇒Q. However, natural

language counterfactuals famously do not obey augmentation. Consider:

If I were to strike the match, it would light.

Therefore, if I were to strike the match and I was in outer space, it

would light.

So, our next desideratum is that the corresponding argument should not hold

good for 2→ (that is, P2→Q 2 (P∧R)2→Q.)


8.1.4 No contraposition→ and⇒ obey contraposition:

φ→ψ φ⇒ψ∼ψ→∼φ ∼ψ⇒∼φ

But counterfactuals do not. Suppose I’m on the �ring squad, and we shoot

someone dead. My gun was loaded, but so were those of the others. Then the

premise of the following argument is true, while its consequent is false:

If my gun hadn’t been loaded, he would still be dead.

Therefore, if he weren’t dead, my gun would have been loaded.

8.1.5 Some implicationsHere is an argument form that intuitively should hold for the 2→:

P2→Q

P→Q

The counterfactual conditional should imply the material conditional. 2→ will

then obey modus ponens and modus tollens:

P P2→QP2→Q ∼QQ ∼P

The reason is, of course, that modus ponens and modus tollens are valid for

the→. (Note that it’s not inconsistent to say that modus tollens holds for the

2→ and also that contraposition fails.)

Another implication: the strict conditional should imply the counterfactual:

P⇒Q

P2→Q

To see that these implications should hold, consider �rst the argument from

the strict conditional to the counterfactual conditional. Surely, if P entails Q,

then if P were true, Q would be as well. As for the counterfactual implying the


material, suppose that you think that if P were true, Q would also be true. Now

suppose that someone tells you that P is true, but that Q is false. Wouldn’t

you then need to give up your original claim that if P were to be true, then

Q would be true? It seems so. So, the statement P2→Q isn’t consistent with

P∧∼Q—that is, it isn’t consistent with the denial of P→Q.

8.1.6 Context dependenceYears ago, a few of us were at a restaurant in NY—Red Smith, Frank

Graham, Allie Reynolds, Yogi [Berra] and me. At about 11.30 p.m., Ted

[Williams] walked in helped by a cane. Graham asked us what we thought

Ted would hit if he were playing today. Allie said, “due to the better

equipment probably about .350.” Red Smith said. “About .385.” I said,

“due to the lack of really great pitching about .390.” Yogi said, “.220.”

We all jumped up and I said, “You’re nuts, Yogi! Ted’s lifetime average is

.344.” “Yeah” said Yogi “but he is 74 years old.” –Buzzie Bavasi, baseball

executive.

Who was right? If Ted Williams had played at the time the story was told,

would he or wouldn’t he have hit over .300?

Clearly, there’s no one answer. The �rst respondents were imagining

Williams playing as a young man. Understood that way, the answer is no

doubt: yes, he would have hit over .300. But Berra took the question a different

way: he was imagining Williams hitting as he was then: a 74 year old man.

Berra took the others off guard, by deliberately (?—this is Yogi Berra we’re

talking about…) shifting how the question was construed, but he didn’t make a

semantic mistake in so doing. It’s perfectly legitimate, in other circumstances

anyway, to take the question in Berra’s way. (Imagine Williams talking to him-

self �ve years after he retired: “These punks today! If I were playing today,

I’d still hit over .300!”) Counterfactual sentences can mean different things

depending on the conversational context in which they are uttered.

Another example:

If Syracuse were in Louisiana, Syracuse winters would be warm.

True or false? It might seem true: Louisiana is in the south. But wait—perhaps

Louisiana would include Syracuse by extending its borders north to Syracuse’s

actual latitude.

Would Syracuse be warm in the winter? Would Williams hit over .300?

No one answer is correct, once and for all. Which answer is correct depends


on the linguistic context. Whether a counterfactual is true or whether it is false

depends in part on what the speaker means to be saying, and what her audience

takes her to be saying, when she utters the counterfactual. Would Syracuse be

warm?—in some contexts, it would be correct to say yes, and in others, to say

no. When we imagine Syracuse being warm, we imagine reality being different

in certain respects from actuality. In particular, we imagine Syracuse as being

in Louisiana. In other respects, we imagine a situation that is a lot like reality—

we don’t imagine a situation, for example, in which Syracuse and Louisiana

are both located in China. Now, when considering counterfactuals, there is

a question of what parts of reality we hold constant. In the Syracuse-Louisiana

case, we seem to have at least two choices. Do we hold constant the location of

Syracuse, or do we hold constant the borders of Louisiana? The truth value of

the counterfactual depends on which we hold constant.

What determines which things are to be held constant, when we evaluate

the truth value of a counterfactual? It large part: the context of utterance of

the counterfactual. Suppose I am in the middle of the following conversation:

“Syracuse restaurants struggle to survive because the climate there is so bad:

no one wants to go out to eat in the winter. If Syracuse were in Louisiana,

its restaurants would do much better.” In such a context, an utterance of the

counterfactual “If Syracuse were in Louisiana, Syracuse winters would be warm”

would be regarded as true. But if this counterfactual were uttered in the midst of

the following conversation, it would be regarded as false: “You know, Louisiana

is statistically the warmest state in the country. Good thing Syracuse isn’t in

Louisiana, because that would ruin the statistic.”

Does just saying a sentence, intending it to be true, make it true? Well,

sort of! When a certain sentence has a meaning that is partly determined by

context, then when a person utters that sentence with the intention of saying

something true, that tends to create a context in which the sentence is true.

Compare ‘�at’—we’ll say “the table is �at”, and thereby utter a truth. But when

a scientist looks at the same table and says “you know, macroscopic objects are

far from being �at. Take that table, for instance. It isn’t �at at all—when viewed

under a microscope, it can be seen to have a very irregular surface”. The term

‘�at’ has a certain amount of vagueness—how �at does a thing have to be to

count as being “�at”? Well, the amount required is determined by context.2

2See Lewis (1979).


8.2 The Lewis/Stalnaker approachHere is the core idea of David Lewis (1973) and Robert Stalnaker (1968) of how

to interpret counterfactual conditionals. Consider a counterfactual conditional

P2→Q. To determine its truth, Lewis and Stalnaker instruct us to consider

the possible world that is as similar to reality as possible, in which P is true.

Then, the counterfactual is true in the actual world if and only if Q is true in

that possible world. Consider Lewis’s example:

If kangaroos had no tails, they would topple over.

When we consider the possible world that would be actual if kangaroos had

no tails, we do not depart gratuitously from actuality. For example, we do

not consider a world in which kangaroos have wings, or crutches. We do not

consider a world with different laws of nature, in which there is no gravity. We

keep the kangaroos as they actually are, but remove the tails, and we keep the

laws of nature as they actually are. It seems that the kangaroos would then fall

over.

Take the examples of the previous section, in which I got you to give

differing answers to certain sentences. Consider:

If Syracuse were in Louisiana, Syracuse winters would be warm.

How does the contextual dependence of this sentence work, on the Lewis-

Stalnaker view? By supplying different standards of comparison of similarity.

Think about similarity, for a moment: things can be similar in certain respects,

while not being similar in other respect. A blue square is similar to a blue circle

in respect of color, not in respect of shape. Now, when we answer af�rmative

to this counterfactual, according to Lewis and Stalnaker, when we consider

the possible world most similar to the actual world in which Syracuse is in

Louisiana, we are using a kind of similarity that weights heavily Louisiana’s

actual borders. When we count the counterfactual false, we are using a kind of

similarity that weights very heavily Syracuse’s actual location.

8.3 Stalnaker’s system3

I now lay out Stalnaker’s system, SC (for “Stalnaker-Counterfactuals”). The

idea is to add the 2→ to propositional modal logic.

3See Stalnaker (1968). The version of the theory I present here is slightly different from

Stalnaker’s original version; see Lewis (1973, p. 79).


8.3.1 Syntax of SCThe primitive vocabulary of SC is that of propositional modal logic, plus the

connective 2→. Here’s the grammar:

De�nition of SC-wffs:


ii) if φ, ψ are wffs then (φ→ψ), ∼φ, 2φ, and (φ2→ψ) are wffs

iii) nothing else is a wff

8.3.2 Semantics of SCFirst, let’s de�ne two formal properties for relations: strong connectivity and

anti-symmetry. Let R be a binary relation over set A:

strong connectivity (in A): for any x, y ∈A, either Rxy or Ry x

anti-symmetry: for any x, y, if Rxy and Ry x then x = y

More notation: where R is a three-place relation, let’s abbreviate “Rxy z” as

“Rz xy”. And, where u is any object, let “Ru” be the two-place relation that holds

between objects x and y iff Ru xy. (Think of Ru as the result of “plugging” up

one place of R with u.)

We can now de�ne SC-models:

An SC-model is an ordered triple ⟨W ,�, V⟩, where

i) W is a nonempty set (“worlds”)

ii) V is a function that assigns truth values to formulas rela-tive to members of W (“valuation function”)

iii) � is a three-place relation overW (“nearness relation”; read

“x �z y” as “x is at least as near to/similar to z as is y”.)

iv) V and � satisfy the following conditions:

C1: for any w,�w is strongly connected (inW )

C2: for any w, �w is transitive

C3: for any w, �w is anti-symmetric

C4: for any x, y, x �xy (“Base”)


C5: for any SC-wff,φ, providedφ is true in at least one world,

then for every z, there’s some w such that V(φ, w) =1, and such that for any x, if V(φ, x) = 1 then w �z x(“Limit”)

v) For all SC-wffs, φ, ψ and for all w ∈W ,

a) V(∼φ, w) = 1 iff V(φ, w) = 0b) V(φ→ψ, w) = 1 iff either V(φ, w) = 0 or V(ψ, w) = 1c) V(2φ, w) = 1 iff for any v, V(φ, v) = 1d) V(φ2→ψ, w) = 1 iff for any x, IF [V(φ, x) = 1 and for

any y such that V(φ, y) = 1, x �w y] THEN V(ψ, x) = 1

Phew! Let’s look into what this means.

First, notice that much of this is exactly the same as for our MPL models—

we still have the set of worlds, and formulas being given truth values at worlds.

We’ll still say that φ is true “at w” iff V(φ, w) = 1.

What happened to the accessibility relation? It has simply been dropped,

in favor of a simpli�ed truth-clause for the 2—clause c. 2φ is true iff φ is true

at all worlds in the model, not just all accessible worlds. It turns out that this

in effect just gives us an S5 logic for the 2, for you get the same valid formulas

for MPL, whether you make the accessibility relation an equivalence relation, or

a total relation. Clearly, if φ is valid in all equivalence relation models, then it

is valid in all total models, since every total relation is an equivalence relation.

What’s more, the converse is true—if φ is valid in all total models then it’s also

valid in all equivalence relation models. Rough proof: let φ be any formula

that’s valid in all total models, and letM be any equivalence relation model.

We need to show that φ is true in an arbitrary world r ∈W (M ’s set of worlds).

Now, any equivalence relation partitions its domain into non-overlapping

subsets in which each world sees every other world. SoW is divided up into

one or more non-overlapping subsets. One of these, Wr , contains r . Now,

consider a model,M ′, just likeM , but whose set of worlds is justWr .M ′

is a

total model, so φ is valid in it by hypothesis. Thus, in this model, φ is true at

r . But then φ is true at r inM , as well. Why? Roughly: the truth value of φat r inM isn’t affected by what goes on outside r ’s partition, since chains of

modal operators just take us to worlds seen by r , and worlds seen by worlds

seen by r , and…. Such chains will never have us “look at” anything out of r ’s

partition, since these worlds are utterly unconnected to r via the accessibility


relation. So φ’s truth value at r inM is determined by what goes on inWr ,

and so is the same as its truth value at r inM ′.

So, we get the same class of valid formulas whether we require the accessi-

bility relation to be total, or an equivalence relation. Things are easier if we

make it a total relation, because then we can simply drop talk of the accessibility

relation and de�ne necessity as truth at all worlds. The corresponding clause

for possibility is:

e) V(3φ, w) = 1 iff for some v, V(φ, v) = 1

The derived clauses for the other connectives remain the same:

f) V(φ∧ψ, w) = 1 iff V(φ, w) = 1 and V(ψ, w) = 1

g) V(φ∨ψ, w) = 1 iff V(φ, w) = 1 or V(ψ, w) = 1

h) V(φ↔ψ, w) = 1 iff V(φ, w) =V(ψ, w)

Next, what about this nearness relation? Think of this as the similarity

relation we talked about before. In order to evaluate whether x �w y, we place

ourselves in possible world w, and we ask whether x is more similar to our world

than y is. (Recall the bit about counterfactual conditionals being highly context

dependent. In a full treatment of counterfactuals, we would complicate our

semantics by introducing contexts of utterance, and evaluate sentences relative tothese contexts of utterance. The point of this would be to allow different nearness

relations in the different contexts.)

I say “we can think of” � as a similarity relation, but take this with a grain

of salt—just as our de�nitions allow the members ofW to be any old things, so,

� is allowed to be any old relation overW . Just as the members ofW could be

�sh, so the � relation could be any old relation over �sh. (But as before, if the

truth conditions for natural language counterfactuals have nothing in common

with the truth conditions for 2→ statements in our models, the interest in our

semantics is diminished, since our models wouldn’t be modeling the behavior of

natural language counterfactuals.)

The constraints on the formal properties of the nearness relation—certain

of them, at least, seem plausible constraints on � if it is to be thought of as

a similarity relation. C1 simply says that it makes sense to compare any two

worlds in respect of similarity to a given world. C2 has a transparent meaning.

C3 means “no ties”—it says that, relative to a given base world w, it is never

the case that there are two separate worlds x and y such that each is at least as

close to w as the other. C4 is the “base” axiom—it says that every world is at


least as close to itself as every other. Given C3, it has the further implication

that every world is closer to itself than every other. (We de�ne “x is closer to wthan y is” (x ≺w y) to mean x �w y and not: y �w x.) C5 is called the “limit”

assumption: according to it, for any formula φ and any base world w, there

is some world that is a closest world to w in which φ is true (that is, unless φisn’t true at any worlds at all). This rules out the following possibility: there

are no closest φ worlds, only an in�nite chain of φ worlds, each of which is

closer than the previous. Certain of these assumptions have been challenged,

especially C3 and C5. We will consider those issues below.

Note that the valuation function for a model was built directly into the

model. This is in contrast to our earlier de�nitions of models, in which we

equipped models with interpretation functions of different sorts, and then

de�ned the resultant valuation functions accordingly. The valuation function is

built into the de�nition of a SC-model because the limit assumption constrains

SC-models, and the limit assumption is stated in terms of the valuation function.

Given our de�nitions, we can de�ne validity and semantic consequence:

�SCφ iff φ is true at every world in every SC-model

Γ �SCφ iff for every SC-model and every world w in that model,

if every member of Γ is true at w then φ is also true at w

8.4 Validity proofs in SCGiven this semantic system, we can give semantic validity proofs just as we

did for the various modal systems. Consider the formula (P∧Q)→(P2→Q).We want to show that this is SC-valid. Thus, we pick an arbitrary SC-model,

⟨W ,�,V⟩, pick an arbitrary world r ∈W , and show that this formula is true at

r :

i) Suppose for reductio that V((P∧Q)→(P2→Q), r ) = 0

ii) then V(P, r ) = 1,V(Q, r ) = 1, and V(P2→Q, r ) = 0

iii) the truth condition for 2→ says that P2→Q is true at r iff

for every closest P-world (to r ), Q is true as well. So since

P2→Q is false at r , there must be a closest-to-r P-world at

which Q is false—that is, there is some world a such that:

a) V(P,a) = 1


b) for any x, if V(P, x) = 1 then a �r xc) V(Q,a) = 0

iv) from ii), since V(P, r ) = 1,a �r r

v) by “base”, r �r a

vi) so, by anti-symmetry, r = a. But now, Q is both true and false

at r

Here is another validity proof, for the formula {[P2→Q]∧[(P∧Q)2→R]}→[P2→R]. (This formula is worth taking note of, because it is valid despite

its similarity to the invalid formula {[P2→Q]∧[Q2→R]}→[P2→R]):

i) Suppose for reductio that P2→Q and (P∧Q)2→R are true

at r , but P2→R is false there.

ii) Then there’s a world, a, that is a nearest-to-r P world, in

which R is false.

iii) Since P2→Q is true at r , Q is true in all the nearest-to-r Pworlds, and so V(Q,a) = 1.

iv) Note now that a is a nearest-to-r P∧Q world:

• P and Q are both true there, so P∧Q is true there.

• let x be any P∧Q world. x is then a P world. But since a is

a nearest-to-r P world, we know that a �r x. (Remember:

“a is a nearest-to-r P world” means: “V(P,a) = 1, and for

any x, if V(P, x) = 1 then a �r x”)

v) Since (P∧Q)2→R is true at r , it follows that R is true at a.

This contradicts ii).

8.5 Countermodels in SCOur other main concern will be countermodels. As before, we’ll be able to use

failed attempts at constructing countermodels as guides to our validity proofs.

First, consider the truth condition for φ2→ψ:

V(φ2→ψ, w) = 1 iff for every x, if x is a nearest-to-w φ world

then V(ψ, x) = 1


Now, suppose there are no nearest-to-w φ worlds. Then V(φ2→ψ, w) is

automatically 1, since the universally quanti�ed statement is vacuously true.

How could it turn out that there are no nearest-to-w φ worlds? One way

would be if there are no φ worlds whatsoever. That is, if φ is necessarily false.

And in fact, this is the only way there could be no nearest-to-w φ worlds, for

the limit assumption guarantees that if there is at least one φ world, then there

is a nearest-to-w φ world.

Say that a counterfactual φ2→ψ is vacuously true iff φ is false at every world.

What we have seen is that there are two ways a counterfactual φ2→ψ can be

true at a given world w: i) it can be vacuously true, or ii) ψ can be true in all

the nearest-to-w φ worlds (by anti-symmetry, there can be only one). Bear

this in mind when constructing a countermodel—there are two ways to make a

2→ true.

On the other hand, there’s only one way to make a counterfactual false. It

follows from the truth condition for the 2→ that:

V(φ2→ψ, w) = 0 iff there is a nearest-to-w φ world at which ψ is

false.

In constructing models, it is often wise to make forced moves �rst, as we saw

for constructing MPL models. This continues to be a good strategy. It now

means: take care of false 2→s before taking care of true 2→s, for the latter

can occur in two different ways, whereas the former can only occur in one way.

We’ll see how this goes with some concrete examples.

Let’s show that transitivity is invalid for 2→. That is, let’s show that the

following formula is SC-invalid: [(P2→Q)∧(Q2→R)]→(P2→R). I’m going

to do this with diagrams that are a little different from the ones I used for MPL.

The new diagrams again contain boxes (rounded now, to distinguish them from

the old countermodels) in which we put formulas; and we again indicate truth

values of formulas with small numbers above the formulas. Since there is no

accessibility relation, we don’t need the arrows between the boxes. But since we

need to represent the nearness relation, we will arrange the boxes in a vertical

line. At the bottom of the lines goes a box for the world, r , of our model in

which we’re trying to make a given formula false. We place the other worlds

in the diagram above this bottom world r : the further away a world is from

r in the �r ordering, the further above r we place it in the diagram. In this

case we want to make P2→Q and Q2→R true, but P2→R false, so we begin

as follows:


/. -,() *+

1 1 1 0 0

[(P2→Q)∧(Q2→R)]→(P2→R)r

In keeping with the advice I gave a moment ago, let’s “take care of” the false

counterfactual �rst: let’s make P2→R false in r. This means that we need to

have R be false in the nearest-to-r P world. I’ll indicate this as follows:

OO

no P

��

/. -,() *+

1 0 1

P R Qa

/. -,() *+

0 1 1 1 0 0

[(P2→Q)∧(Q2→R)]→(P2→R)r

a is the nearest P world to r. I indicate that in the diagram by making P true

at a, and reminding myself that I can’t put any P-world anywhere between a

and r . Since the counterfactual P2→R was supposed to be false at r, I made

R false in a. Notice that I made Q true in a. This was to take care of the fact

that P2→Q is true in r . This formula says that Q is true in the nearest-to-r Pworld. Well, a is the nearest-to-r P world.

Now for the �nal counterfactual Q2→R. This can be true in two ways—

either there is no Q world at all (the vacuous case), or R is true in the nearest-to-r

Q world. Well, Q is already true in a, so the vacuous case is ruled out. So

we must make R true in the nearest to r Q-world. Where will we put the

nearest-to-r Q world? There are three possibilities. It could be farther away

from, tied with, or closer to r than a. Let’s try the �rst possibility:

/. -,() *+

1 1

Q Rb OO

no Q

��

OO

no P

��

/. -,() *+

1 0 1

P R Qa

/. -,() *+

0 1 0 1 1 0 0

[(P2→Q)∧(Q2→R)]→(P2→R)r


This doesn’t work, because when we make b the nearest-to-r Q world, this

contradicts the fact that we’ve got Q true at a nearer world—namely, a.

Likewise, we can’t make the nearest-to-r Q world be tied with a. Anti-

symmetry would make this world be a. But we need to make R true at the

nearest Q world, whereas R is already false in a.

But the �nal possibility works out just �ne—let the nearest Q world be

closer than a:

OO

no P

��

/. -,() *+

1 0 1

P R Qa

/. -,() *+

1 1 0

Q R Pb OO

no Q

��/. -,() *+

0 1 0 1 1 0 0

[(P2→Q)∧(Q2→R)]→(P2→R)r

Notice that I made P false in b, since I said “no P” of all worlds nearer to r

than a. Here’s the of�cial model:

W = {r,a,b}�

r= {⟨b,a⟩ . . .}

V(P,a) = V(Q,a) = V(Q,b) = V(R,b) = 1; all other atomics false

everywhere else.

A couple comments on the of�cial model. I left out a lot in giving the similarity

relation for this model. First, I left out some of the elements of �r—fully

written out, it would be:

�r

= {⟨r,b⟩,⟨b,a⟩,⟨r,a⟩,⟨r,r⟩,⟨a,a⟩,⟨b,b⟩}

I left out ⟨r,b⟩ because it gets included automatically given the “base” assumption

(C4). Also, the element ⟨r,a⟩ is required to make �r

transitive. The elements

⟨r,r⟩, ⟨a,a⟩, and ⟨b,b⟩ were entered to make that relation re�exive. Why must it

be re�exive? Because re�exivity comes from strong connectivity. (Let w and xbe any members ofW ; we get (x �w x or x �w x) from Strong connectivity of


�w , and hence x �w x.) My plan was to write out enough of �r

so that the rest

could be inferred, given the de�nition of an SC-model.

Secondly, this isn’t a complete writing out of � itself; it is just �r. To be

complete, we’d need to write out �a, and �

b. But in this problem, these latter

two parts of� don’t matter, so it’s permissible to omit them. We’ll do a problem

below where we need to consider more of � than simply �r.

Suppose one has a formula, for example:

(P2→R)→((P∧Q)2→R)

(This is the formula corresponding to the inference pattern of augmentation.)

Suppose we ask: is it SC-valid? The best way to approach this is as follows.

First try to found a countermodel. If we succeed, then the formula is not valid.

If we have trouble �nding a countermodel, then we can try giving a semantic

validity proof (for if the formula is in fact valid, then no countermodel exists).

In this particular case, the formula is invalid, as the following countermodel

shows:

OO

no P∧Q

��

/. -,() *+

1 1 1 0

P∧Q Ra

/. -,() *+

1 1 0

P R Qb OO

no P

��/. -,() *+

0 1 0 0

(P2→R)→[(P∧Q)2→R)r

Ok, I began with the false subjunctive: (P∧Q)2→R. This forced the existence

of a nearest P∧Q world, in which R was false. But since P∧Q was true there,

P was true there; this ruled out the true P2→R in r being vacuously true. So I

was forced to consider the nearest P world. It couldn’t be farther out than a,

since P is true in a. It couldn’t be a, since R was already false there. So I had to

put it nearer than a. Notice that I had to make Q false at b. Why? Well, it was

in the “no P∧Q zone”, and I had made P true in it. Here’s the of�cial model:

W = {r,a,b}�

r= {⟨b,a⟩ . . .}


V(P,a) = V(Q,a) = V(P,b) = V(R,b) = 1; all other atomics 0everywhere else.

Next example: 3P→[(P2→Q)→∼(P2→∼Q)]. An attempt to �nd a coun-

termodel fails at the following point:

OO

no P

��

/. -,

() *+1

1 1 0

P ∼Qa

/. -,() *+

1 0 0 1 0 0 1

3P→[(P2→Q)→∼(P2→∼Q)]r

At world a, we’ve got Q being both true and false. A word about how we got

to that point. I noticed that I had to make two counterfactuals true: P2→Qand P2→∼Q. Now, this isn’t a contradiction all by itself. Remember that

counterfactuals are vacuously true if their antecedents are impossible. So if Pwere impossible, then both of these would indeed be true, without any problem.

But 3P has to be true at r. This rules out those counterfactual’s being vacuously

true. Since P is possible, the limit assumption has the result that there is a closest

P world. This then with the two true counterfactuals created the contradiction.

This reasoning is embodied in the following semantic validity proof:

i) Suppose for reductio that 3P is true at some world r , but

(P2→Q)→∼(P2→∼Q) is false at r .

ii) Then P2→Q and P2→∼Q are both true at r as well.

iii) Since 3P is true at r , P is true at some world. So, by the

limit assumption, we have: there exists a world, a, such that

V(P,a) = 1 and for any x, if V(P,x)=1 then a �r x. For short,

there is a closest-to-r P world.

iv) The truth condition for 2→, applied to P2→Q, gives us that

Q is true at all the closest-to-r P worlds.

v) Similarly, applied to P2→∼Q, we know that ∼Q is true at all

the closest-to-r P worlds.

vi) Thus, both Q and ∼Q would be true at a. Impossible.


Note the use of the limit assumption. It is the limit assumption we must use

when we need to know that there is a nearest φ-world, and we can’t get this

knowledge from other things in the proof.

For the last example of a countermodel, I’ll do a new kind of problem: one

with a counterfactual nested within another counterfactual. Let’s show that the

following formula is SC-invalid: [P2→(Q2→R)]→[(P∧Q)2→R] (this is the

formula corresponding to “importation”). We begin by making the formula

false in r, the actual world of the model. This means making the antecedent true

and the consequent false. Now, since the consequent is a false counterfactual,

we are forced to make there be a nearest P∧Q world in which R is false:

OO

no P∧Q

��

/. -,() *+

1 1 1 0

P∧Q Ra

/. -,() *+

1 0 0

[P2→(Q2→R)]→[(P∧Q)2→R]r

Now we’ve got to make the �rst subjunctive conditional true. We can’t make it

vacuously true, because we’ve already got a P-world in the model: a. So, we’ve

got to put in the nearest-to-r P world. Could it be farther away than a? No,

because a would be a closer P world. Could it be a? No, because we’ve got to

make Q2→R true in the closest P world, and since Q is true but R is false in a,

Q2→R is already false in a. So, we do it as follows:

OO

no P∧Q

��

/. -,() *+

1 1 1 0

P∧Q Ra

/. -,() *+

1 0 1

P Q2→Rb OO

no P

��/. -,() *+

0 1 0 0

[P2→(Q2→R)]→[(P∧Q)2→R]r

Why did I make Q false at b? Well, because b is in the “no P∧Q” zone, and Pis true at b, so Q had to be false there.


Now, the remaining thing to do is to make Q2→R true at b. This requires

some thought. The diagram right now represents “the view from r”—it rep-

resents how near the worlds in the model are to r. That is, it represents the

�r

relation. But the truth value of Q2→R at b depends on “the view from

b”—that is, the �b

relation. So we need to consider a new diagram, in which b

is the bottom, base, world:

OO

no Q

��

/. -,() *+

1 1

Q Rc

/. -,() *+

1 0 1

P Q2→Rb

I made there be a nearest-to-b Q world, and made R true there. Notice that

I kept the old truth values of b from the other diagram. This is because this

new diagram is a diagram of the same worlds as the old diagram; the difference

is that the new diagram represents the �b

nearness relation, whereas the old

one represented a different relation: �r. Now, this diagram isn’t �nished. The

diagram is that of the �b

relation, and that relation relates all the worlds in

the model. So, worlds r and a have to show up somewhere here. But the

safest practice is to put them far away from b, so that there isn’t any possibility

of con�ict with the “no Q” zone that has been established. Thus, the �nal

appearance of this part of the diagram is as follows:

/. -,() *+r

/. -,() *+a

OO

no Q

��

/. -,() *+

1 1

Q Rc

/. -,() *+

1 0 1

P Q2→Rb


The old truth values from r and a still are had (remember that this is another

diagram of the same model, but representing a different nearness relation),

but I left them out because of the fact that they’ve already been written on the

other part of the diagram.

Notice that the order of the worlds in the r-diagram does not in any way

affect the order of the worlds on the b diagram. The nearness relations in

the two diagrams are completely independent, because in our de�nition of ‘SC-

model’, we entered in no conditions constraining the relations between �i and

� j when i 6= j . This sometimes seems unintuitive. For example, we could have

two halves of a model looking as follows:

The view from r The view from ac r

b b

a c

r a

If this seems odd, remember that only some of the features of a diagram are

intended to be genuinely representative. For example, these diagrams are in ink;

this is not intended to convey the idea that the worlds in the model are made

of ink. This feature of the diagram isn’t intended to convey information about

the model. Analogously, the fact that in the left diagram, b is physically closer

to a than c is not intended to convey the information that, in the model, b�ac.

In fact, the diagram of the view from r is only intended to convey information

about �r; it doesn’t carry any information about �

a, �

b, or �

c.

Back to the countermodel. That other part of the diagram, the view from r,

must be updated to include world c. The safest procedure is to put c far away

on the model to minimize possibility of con�ict. Thus, the �nal picture of the

view from r is:


/. -,() *+c

OO

no P∧Q

��

/. -,() *+

1 1 1 0

P∧Q Ra

/. -,() *+

1 0 1

P Q2→Rb OO

no P

��/. -,() *+

0 1 0 0

[P2→(Q2→R)]→[(P∧Q)2→R]r

Again, I haven’t re-written the truth values in world c, because they’re already

in the other diagram, but they are to be understood as carrying over. Now for

the of�cial model:

W = {r,a,b,c}V(P,a) =V(Q,a) =V(P,b) =V(Q,c) =V(R,c) = 1; all else (atom-

ics) 0 elsewhere

�r= {⟨b,a⟩, ⟨a,c⟩ . . .}

�b= {⟨c,a⟩, ⟨a, r⟩ . . .}

Notice that we needed to express two of �’s subrelations—�r

and �b. Remem-

ber that any model has got to contain �i for every world i in the model. For

example, if we were to write out this model completely of�cially, we’d have to

specify �a

and �c. But we don’t bother with those parts of � that don’t matter.

8.6 Logical Features of SCHere we’ll discuss various features of SC, which appear to con�rm that it’s a

good logic for counterfactual conditionals. In part, we will be showing that our

semantics for 2→ matches the logical features of natural language that were

discussed in section 8.1.


8.6.1 Not truth-functionalWe wanted our system for counterfactuals to have the following features:

∼P 2 P2→Q

Q 2 P2→Q

Clearly, it does. The �rst fact is demonstrated by a model in which P is false

at some world, r, and in which there’s a nearest-to-r P world in which Q is

false; the second, by a model in which Q is true at r, P is false at r, and in which

there’s a nearest-to-r P world in which Q is false.

8.6.2 Can be contingentWe wanted it to turn out that:

P2→Q 22(P2→Q)

∼(P2→Q) 2→2∼(P2→Q)

Our semantics does indeed have this result, because the similarity metrics based

on different worlds can be very different. For example: consider a model with

worlds r and a, in which Q is true in the nearest-to-r P world, but in which Qis false at the nearest-to-a P world. P2→Q is true at r and false at a, whence

2(P2→Q) is false at r.

8.6.3 No augmentationEarlier we produced a model containing a world in which this conditional was

false:

[P2→Q]→[(P∧R)2→Q]

Thus, P2→Q 2 (P∧R)2→Q.

8.6.4 No contrapositionLet’s show that P2→Q 2∼Q2→∼P :


OO

no ∼Q

��

/. -,() *+

1 0 0 1

∼Q ∼Pa

/. -,() *+

1 1

P Qb OO

no P

��/. -,() *+

1 0

(P2→Q) (∼Q2→∼P )r

I won’t bother with the of�cial model.

8.6.5 Some implicationsLet’s show that the SC semantics vindicates the inference from the counterfac-

tual conditional to the material conditional, and from the strict conditional to

the counterfactual conditional.

Proof that P2→Q � P→Q:

i) Suppose P2→Q is true at some world r in some SC-model,

and …

ii) …suppose for reductio that P→Q isn’t true there.

iii) Then P is true at r and …

iv) …Q is false at rv) Given “base”, for every world, x, r �r x.

vi) Given iii) and v), r is a closest-to-r P world. So, given i), Qis true at r . Contradicts iv).

Proof that P⇒Q � P2→Q:

i) Suppose P⇒Q is true at r , and suppose for reductio that

P2→Q is false at r .

ii) Then there’s a nearest-to-r P world, a, at which Q is false.

iii) But that can’t be. “P⇒Q” means 2(P→Q). So P→Q is true

at every world. So there can’t be a world like a, in which P is

true and Q is false.


8.6.6 No exportationWe have shown that the SC-semantics reproduces the logical features of natural

language counterfactuals discussed in section 8.1. In the next few sections we

discuss some further logical features of the SC-semantics, and compare them

with the logical features of the→, the⇒, and natural language counterfactuals.

The→ obeys exportation:

(P∧Q)→R

P→(Q→R)

But the ⇒ doesn’t in any system; (P∧Q)⇒R 2S5

P⇒(Q⇒R). Nor does

the 2→; it can be easily shown with a countermodel that (P∧Q)2→R 2SC

P2→(Q2→R).Does the natural language counterfactual obey exportation? Here is an

argument that it does not. The following is true:

(L&H) If Bill had married Laura and Hillary, he would have been a

bigamist.

But one can argue that the following is false:

(L2→H) If Bill had married Laura, then it would have been the case

that if he had married Hillary, he would have been a bigamist.

Suppose Bill had married Laura. Would it then have been true that: if he had

married Hillary, he would have been a bigamist? Well, let’s ask for comparison:

what would the world have been like, had George W. Bush married Hillary

Rodham Clinton? Would Bush have been a bigamist? Here the natural answer

is no. George W. Bush is in fact married to Laura Bush; but when imagining him

married to Hillary Rodham Clinton, we don’t hold constant his actual marriage.

We imagine him being married to Hillary instead. If this is true for Bush, then

one might think it’s also true for Bill in the counterfactual circumstance in

which he’s married to Laura: it would then have been true of him that, if he

had married Hillary, he wouldn’t have still been married to Laura, and hence

would not have been a bigamist.

It’s unclear whether this is a good argument, though, since it assumes that

ordinary standards for evaluating unembedded counterfactuals (“If George had

married Hillary, he would have been a bigamist”) apply to counterfactuals em-

bedded within other counterfactuals (“If Bill had married Hillary, he would have


been a bigamist” as embedded within (L2→H).) Contrary to the assumption, it

seems most natural to evaluate the consequent of an embedded counterfactual

like (L2→H) by holding its antecedent constant. But a defender of the SC

semantics might argue that (L2→H) has a reading on which it is false (recall the

context-dependence of counterfactuals), and hence that we need a semantics

that allows for the failure of exportation.

8.6.7 No importationImportation holds for→, and for⇒ in T and stronger systems:

P→(Q→R) P⇒(Q⇒R)(P∧Q)→R (P∧Q)⇒R

but not for the 2→: above we produced an SC-model with a world in which

the conditional [P2→(Q2→R)]→[(P∧Q)2→R] was false.

The status of importation for natural language counterfactuals is similar

to the status of exportation. If (L2→H) is false (on at least one reading), then

presumably the following could be true (on at least one reading):

If Bill had married Laura, then it would have been the case that if

he had married Hillary, he would have been happy.

without the result of importing being true:

If Bill had married Laura and Hillary, he would have been happy

(if he had married both, his relatives would have hated him, he would have

become a public spectacle, etc.)

8.6.8 No hypothetical syllogism (transitivity)The following inference pattern is valid for→ and⇒ (in all systems):

Q→R Q⇒RP→Q P⇒QP→R P⇒R


but not for 2→, as our proof above of the invalidity of [(P2→Q)∧(Q2→R)]→(P2→R) shows.

Natural language counterfactuals also seem not to obey hypothetical syllo-

gism. I am the oldest child in my family; my brother Mike is the second-oldest.

So the following counterfactuals seem true:4

If I hadn’t been born, Mike would have been my parent’s oldest

child.

If my parents had never met, I wouldn’t have been born.

But the result of applying hypothetical syllogism seems false:

If my parents had never met, Mike would have been their oldest

child.

8.6.9 No transpositionTransposition governs the→:

P→(Q→R)

Q→(P→R)

but not the⇒ (in any of our modal systems); P⇒(Q⇒R) 2S5

Q⇒(P⇒R). Nor

does it govern the 2→; it’s easy to show that P2→(Q2→R) 2SC

Q2→(P2→R).The status of transposition for natural language counterfactuals is sim-

ilar to that of importation and exportation. If we can ignore the effects of

embedding on the evaluation of counterfactuals, then we have the following

counterexample to transposition. It is true that:

4Note that if these two statements are written in the reverse order, it seems far less clear

that they’re both true:

If my parents had never met, I wouldn’t have been born.

If I hadn’t been born, Mike would have been my parent’s oldest child.

It’s natural in this case to interpret the second conditional by holding constant the antecedent

of the �rst conditional. This fact, together with what we observed about embedded counter-

factuals, suggests a systematic dependence of the interpretation of counterfactuals on their

immediate linguistic context. See von Fintel (2001) for a “dynamic” semantics for counterfac-

tuals, which more accurately models this feature of their use.


If Bill Clinton had married Laura Bush, then if he had married

Hillary Rodham, he’d have been married to a Democrat.

But it is not true that:

If Bill Clinton had married Hillary Rodham, then if he had married

Laura Bush, he’d have been married to a Democrat.

8.7 Lewis’s criticisms of Stalnaker’s theoryDavid Lewis has a rival theory of counterfactuals. Like Stalnaker’s theory, it is

based on similarity, but it differs from Stalnaker’s in certain respects. So far, we

have only discussed features of Stalnaker’s system that are shared by Lewis’s.

Let’s turn, now, to the differences.

Lewis challenges Stalnaker’s assumption that �w is always anti-symmetric.

Real similarity relations permit ties. So it seems implausible to rule out the

possibility of two worlds being exactly similar to a given world.

The challenge to Stalnaker here is most straightforward if Stalnaker intends

to be giving truth conditions for natural language counterfactuals, rather than

merely doing model theory. In that case, the setW in an SC-model must be the

set of genuine possible worlds, and � must be a relation of genuine similarity,

in which case it ought to admit ties. But even if Stalnaker is not doing this,

the objection may yet have bite, to the extent that the semantics of natural

language conditionals is like similarity-theoretic semantics.5

The validity of certain formulas depends on the “no ties” assumption; the

following two wffs are SC-valid, but are challenged by Lewis:

(P2→Q)∨ (P2→∼Q) (“Conditional excluded middle”)

[P2→(Q∨R)]→[(P2→Q)∨(P2→R)] (“distribution”)

Take the �rst one, for example. Suppose you gave up anti-symmetry, thereby

allowing ties. Then the following would be a countermodel for the law of

conditional excluded middle:

5For an interesting response to Lewis, see Stalnaker (1981).


OO

no Q

��

/. -,() *+

1 0

P Qa

/. -,() *+

1 0 1

P ∼Qb

/. -,() *+

0 0 0

(P2→Q)∨(P2→∼Q)r

Remember that P2→Q is true only if Q is true in all the nearest P worlds.

In this model, Q is true in one of the nearest P worlds, but not all, so that

counterfactual is false at r. Similarly for P2→∼Q.

A similar model shows that distribution fails if the “no ties” assumption is

given up.

So, should we give up conditional excluded middle? As Lewis concedes,

the principle is initially plausible. An equivalent formulation of conditional

excluded middle is this:

∼(P2→Q)→(P2→∼Q)

But everyone agrees that (P2→∼Q)→∼(P2→Q) is always true, at least, when

P is possibly true. So, in cases where P is possibly true anyway, the question

of whether conditional excluded middle is valid is the question of whether

∼(P2→Q) and P2→∼Q are equivalent to each other. And it does indeed

seem that in ordinary usage, one expresses the negation of a counterfactual

by negating its consequent. To deny the counterfactual “if she had played,

she would have won", one says "no, she wouldn’t have!”, meaning “if she had

played, she would not have won”.

And take the other formula validated by Stalnaker’s theory, distribution. In

reply to: “if the coin had been �ipped, it would have come out either heads or

tails”, one might ask: “which would it have been, heads or tails?”. The thinking

behind the reply is that “if the coin had been �ipped, it would have come up

heads”, or “if the coin had been �ipped, it would have come up tails” must be

true.

So there’s some plausibility to both these formulas. But Lewis says two

things. The �rst is metaphysical: if we’re going to accept the similarity analysis,

we’ve got to give them up, because ties just are possible. The second is purely

semantic: the intuitions aren’t completely compelling. About the coin-�ipping

case, Lewis denies that if the coin had been �ipped, it would have come up


heads, and he also denies that if the coin had been �ipped, it would have come

up tails. Rather, he says, if it had been �ipped, it might have come up heads.

And if it had been �ipped, it might have come up tails. But neither outcome is

such that it would have resulted, had the coin been �ipped.

Concerning excluded middle, Lewis says:

It is not the case that if Bizet and Verdi were compatriots, Bizet

would be Italian; and it is not the case that if Bizet and Verdi were

compatriots, Bizet would not be Italian; nevertheless, if Bizet and

Verdi were compatriots, Bizet either would or would not be Italian.

(Counterfactuals, p. 80.)

Lewis can follow this up by noting that if Bizet and Verdi were compatriots,

Bizet might be Italian, but it’s not the case that if they were compatriots, he

would be Italian.

Here is a related complaint of Lewis’s about Stalnaker’s semantics. In the

last little bit, I’ve used English phrases of the form “if it were the case that φ,

then it might have been the case that ψ”. This conditional Lewis calls “the

‘might’ counterfactual”; he symbolizes it as φ3→ψ, and de�nes it thus:

φ3→ψ=df∼(φ2→∼ψ)

Lewis criticizes Stalnaker’s system for the fact that this de�nition of 3→ doesn’t

work in Stalnaker’s system. Why not? Well, since internal negation is valid in

Stalnaker’s system, φ3→ψ would always imply φ2→ψ—not good, since the

might-conditional in English seems weaker than the would-conditional. So,

Lewis’s de�nition of 3→ doesn’t work in Stalnaker’s system. Moreover, there

doesn’t seem to be any other plausible de�nition. So, Stalnaker can’t de�ne

3→.6

Lewis also objects to Stalnaker’s limit assumption. The following line is

less than one inch long:

Now, consider the counterfactual:

If the line were more than 1 inch long, it would be over 100 miles

long.

6Lewis (1973, p. 80).


Seems false. But if we use Stalnaker’s truth conditions as truth conditions for

natural language counterfactuals, and take our intuitive judgments of similarity

seriously, we seem to get the result that it is true! The reason is that there

doesn’t seem to be a closest world in which the line is more than 1 inch long.

For every world in which the line is, say, 1+k inches long, there’s another world

in which the line has a length closer to its actual length but still more than 1

inch long: say, 1+k2 inches. So there doesn’t seem to be any closest world in

which the line is over 1 inch long.

In light of these criticisms, Lewis proposes a new similarity-based seman-

tics for counterfactuals, which assumes neither anti-symmetry nor the limit

assumption. Let’s look at that system.

8.8 Lewis’s system7

To move from Stalaker’s system to Lewis’s, we can start by just dropping the

anti-symmetry assumption. We also want to drop the limit assumption. But

after dropping the limit assumption, if we made no further adjustments to the

system, we would get unwanted vacuous truths, as we did in the example of the

1 inch line above.8

The truth de�nition for 2→ needs to be changed. Instead

of saying that φ2→ψ is true iff ψ is true in all the nearest φ worlds, we will

instead say that φ2→ψ is true iff either i) φ is true in no worlds (the vacuous

case), or ii) there is some φ world such that for every φ world at least as close,

φ→ψ is true there.

Here is the new system, LC (Lewis-counterfactuals). It is exactly the same

as the Stalnaker system except that limit and anti-symmetry are dropped, and

the parts indicated in boldface are changed:

An LC-model is a three-tuple ⟨W ,�,V⟩, where:

i) W is a nonempty set

ii) V is a function that assigns either 0 or 1 to each wff relative

to each member ofW7See Lewis (1973, pp. 48-49). My formulation does away with the accessibility relation (in

Lewis’s terminology, Si , the set of worlds accessible from world i , is alwaysW , the set of all

worlds in the model), so it is a bit simpler.

8Actually, dropping the limit assumption doesn’t affect the class of valid formulas, which is

the same with or without the limit assumption. (See Lewis (1973, p. 121).


iii) � is a three-place relation overWiv) V and � satisfy the following conditions:

C1: for any w, �w is strongly connected

C2: for any w, �w is transitive

C3: for any x , y, if y �x x then x = y (“base”)v) For all wffs, φ, ψ and for all w ∈W :

a) V(∼φ, w) = 1 iff V(φ, w) = 0b) V(φ→ψ, w) = 1 iff either V(φ, w) = 0 or V(ψ, w) = 1c) V(2φ, w) = 1 iff for any v, V(φ, v) = 1d) V(φ2→ψ, w) = 1 iff EITHER φ is true at no worlds,

OR: there is some world, x , such that V(φ, x) = 1 andfor all y, if y �w x then V(φ→ψ, y) = 1

It may be veri�ed that every LC-valid wff is SC-valid (although not vice

versa).9

Comments on all this: First, notice that the limit and anti-symmetry con-

ditions are simply dropped. Second, the Base condition is modi�ed; now it

says that no world is as close to a world as itself. Before, it said that each world

is at least as close to itself as any other. Stalnaker’s Base condition, plus anti-

symmetry, entails the present Base condition. But Lewis’s system doesn’t have

anti-symmetry, so the Base condition must be stated in the stronger form.10

Third, let’s think about what the truth condition for the 2→ says. First,

there’s the vacuous case: if φ is necessarily false then φ2→ψ comes out true.

But if φ is possibly true, then what the clause says is this: φ2→ψ is true at

w iff there’s some φ world where ψ is true, such that no matter how much

closer to w you go, you’ll never get a φ world where ψ is false. If there is a

nearest-to-w φ world, then this implies that φ2→ψ is true at w iff ψ is true in

all the nearest-to-w φ worlds.

So, thinking of these as truth-conditions for natural-langague counterfac-

tuals for a moment, recall the sentence:

9Here’s a sketch of an argument. Let’s say that an LC model is “Stalnaker-acceptable” iff it

obeys the limit and anti-symmetry assumptions. Suppose that φ is LC-valid. Then it’s true in

all Stalnaker acceptable LC-models. Now, notice that in Stalnaker-acceptable models, Lewis’s

truth-conditions for 2→ are identical to Stalnaker’s. So, φ must be true in all SC-models.

10Why do we want to prohibit worlds being just as close to w as w is to itself? So that P∧Q

semantically implies P2→Q. Otherwise P∧Q could be true at w while P∧∼Q was true at

some world as close to w as w is to itself, in which case P2→Q would turn out false at w.


If the line were over 1 inch long, it would be over 10 inches long.

There’s no nearest world in which the line is over 1 inch long, only an in�nite

series of worlds where the line has lengths getting closer and closer to 1 inch

long. But this doesn’t make the counterfactual true. A counterfactual is true if

its antecedent is impossible, but that’s not true in this case. So the only way the

counterfactual could be true is if the second part of the de�nition is satis�ed—if,

that is, there is some world, x, such that the antecedent is true at x, and the

material conditional (antecedent→ consequent) is true at every world at least

as similar to the actual world as is x. Since the “at least as similar as” relation is

re�exive, this can be rewritten thus:

for some world, x, the antecedent and consequent are both true at

x, and the material conditional (antecedent→ consequent) is true

at every world at least as similar to the actual world as is x

So, is there any such world, x? No. For let x be any world at which the

antecedent and consequent are both true—i.e., any world in which the line is

over 10 inches long. We can always �nd a world that is more similar to the actual

world than x in which the material conditional (antecedent→ consequent) is

false: just choose a world just like x but in which the line is only, say, 2 inches

long.

Let’s see how Lewis’s theory works in the case of a true counterfactual, for

instance:

If I were more than six feet tall, then I would be less than nine feet

tall

(I am, in fact, less than six feet tall.) The situation here is similar to the previous

example in that there is no nearest world in which the antecedent is true. But

now, we can �nd a world x, in which the antecedent and consequent are both

true, and such that the material conditional (antecedent→ consequent) is true

in every world at least as similar to the actual world as is x. Simply take x to

be a world just like the actual world but in which I am, say, six-feet-one. Any

world that is at least as similar to the actual world as this world must be one in

which I’m less than nine feet tall; so in any such world the material conditional

(I’m more than six feet tall→ I’m less than nine feet tall) is true.

Notice that the formulas representing Conditional Excluded Middle and

Distribution come out invalid now, because of the possibility of ties.


Another thing: Lewis gives the following de�nition for the ‘might’-counter-

factual:

φ3→ψ=df∼(φ2→∼ψ)

From this we may obtain a derived clause for the truth conditions of φ3→ψ:

V(φ3→ψ, w) = 1 iff for some x, V(φ, x) = 1, and for any x, if

V(φ, x) = 1 then there’s some y such that y �w x and V(φ∧ψ, y) =1)

That is, φ3→ψ is true at w iff φ is possible, and for any φ world, there’s a

world as close or closer to w in which φ and ψ are both true. In cases where

there is a nearest φ world, this means that ψ must be true in at least one of the

nearest φ worlds.

8.9 The problem of disjunctive antecedentsBefore we leave counterfactual conditionals, I want to talk about one criticism

that has been raised against both Lewis’s and Stalnaker’s systems.11

In neither

system does the formula (P∨Q)2→R semantically imply P2→R. (Take a model

where there is a unique nearest P∨Q world to r, in which Q is true but not P ;

and make there be a unique nearest P world in which R is false.) But shouldn’t

this implication hold? Imagine a conversation between Butch Cassidy and

the Sundance Kid in heaven, after having been surrounded and killed by the

Bolivian army. They say:

If we had surrendered or tried to run away, we would have been

shot.

Intuitively, if this is true, so is this:

If we had surrendered, we would have gotten shot.

In general, one is entitled to conclude from “If P or Q had been the case,

then R would have been the case” that “if P had been the case, R would have

been the case”. If Butch Cassidy and the Sundance Kid could have survived by

11For references, see the bibliography of Lewis (1977).


surrendering, they certainly would not say to each other “If we had surrendered

or tried to run away, we would have been shot”.

Is this a problem for Lewis and Stalnaker? Some have argued this, but

others respond as follows. One must take great care in translating from natural

language into logic. For example,12

no one would want to criticize the law

∼∼P→P on the grounds that “There ain’t no way I’m doing that” doesn’t

imply that I might do that. And there are notorious peculiar things about the

behavior of ‘or’ in similar contexts. Consider:

(1) You are permitted to stay or go.

One can argue that this does not have the form:

(2) Permit(Stay ∨ Go)

After all, suppose that you are permitted to stay, but not to go. If you stay, you

can’t help doing the following act: staying ∨ going. So, surely, you’re permitted

to do that. So, (2) is true. But (1) isn’t; if someone uttered (1) to you when you

were in jail, they’d be lying to you! (1) really means:

(3) You are permitted to stay, AND you are permitted to go.

Similarly, “If either P or Q were true then R would be true” seems usually to

mean “If P were true then R would be true, and if Q were true then R would be

true”. We can’t just expect natural language to translate directly into our logical

language—sometimes the surface structure of natural language is misleading.

12The example is adapted from Loewer (1976).

Chapter 9

Quanti�ed Modal Logic

We’re going to look at Kripke-style semantics for quanti�ed modal logic.

The language is what you get by adding the 2 and 3 to the language

of predicate logic. There are many interesting issues concerning the interaction

of modal operators with quanti�ers.

9.1 Grammar of QMLGrammatically, the language of QML is the same as that of plain old predicate

logic, except that we add the 2. Thus, the one new clause to the de�nition of a

wff is the clause that if φ is a wff, then so is 2φ. The 3 is de�ned as before: as

∼2∼.

9.2 Symbolizations in QMLAdding quanti�ers to our modal language allows us to make new distinctions

when symbolizing sentences of English. In particular, it allows us to make the

famous distinction between de re and de dicto modal statements. A paradigm

instance concerns the ambiguity of the sentence

(1) Some rich person might have been poor.

If we don’t have any quanti�ers around, it seems that we have only one possible

symbolization: as 3P , where P stands for “Some rich person is poor”. But this

symbolization is incorrect, because in no possible world is the statement “Some

224

CHAPTER 9. QUANTIFIED MODAL LOGIC 225

rich person is poor” true! In contrast, (1) seems to have a reading on which it

is true. When we turn to symbolize it in QML, we have better luck. There

seem to be two possibilities:

(1a) 3∃x(Rx∧P x)(1b) ∃x(Rx∧3P x)

(1a) is just the rejected symbolization as 3P , with P becoming ∃x(Rx∧P x).But (1b) is better—translating back into English, it says that there is a person

who is in fact rich, but is such that s/he might have been poor. That’s what the

original sentence meant, on its most natural reading.

The ambiguity of (1) is said to be a “de re/de dicto” ambiguity. (1a) is the “de

dicto” reading—the modal operator 3 attaches to a complete closed sentence.

It’s called “de dicto” because the property of possibility is attributed to the

sentence—a dictum. (1b) is the de re reading—there, the modal operator

attaches to an open sentence. It’s called “de re” (“of the object”), because 3F xcan be thought of as attributing a certain property to an object u when x is

assigned the value u—the modal property of possibly being F.

The following sentence also exhibits a de re/de dicto ambiguity:

Every bachelor is necessarily male.

The two readings are:

2∀x(B x→M x) (De dicto)

∀x(B x→2M x) (De re)

The de dicto reading is true because it is true that in any possible world, anyone

that is in that world a bachelor is, in that world, male. The de re reading is

false because it makes the claim that if any object, u, is a bachelor in the actual

world, that object u is necessarily a bachelor—i.e., the object u is a bachelor in

all possible worlds. That claim is false: many objects are in fact bachelors, but

might have been married.

De�nite descriptions also exhibit a de re/de dicto ambiguity. This may be

illustrated by using Russell’s theory of descriptions (section 5.3.3). Russell’s

method generates an ambiguity when a sentential operator such as “not” is

added to “is G”, for example:

The striped bear is not dangerous


Does this mean:

∼∃x(S x∧B x∧∀y([Sy∧By]→x=y)∧D x)

or does it mean the following?

∃x(S x∧B x∧∀y([Sy∧By]→x=y)∧∼D x)

I.e., is the sentence denying that there is one and only one striped bear that

is dangerous, or is it saying that there is one and only one striped bear, and

that bear is non-dangerous? There’s an ambiguity, which is a matter of the

relative scopes of the de�nite description and the ∼. Sentences with de�nite

descriptions and modal operators have an analogous ambiguity. Consider, for

instance:

The number of the planets is necessarily odd

Letting “N x” mean that x numbers the planets, this could be symbolized in

either of the following ways:

2∃x(N x∧∀y(N y→x=y)∧O x)∃x(N x∧∀y(N y→x=y)∧2O x)

The �rst is de dicto; it says that it’s necessary that: there is one and only one

number of the planets, and that number is odd; that’s false (since there could

have been eight planets.) The second is de re; it says that (in fact) there is one

and only one number of the planets, and that that number is necessarily odd.

That’s true, since the number that is in fact the number of the planets—the

number nine—is indeed necessarily odd.

9.3 A simple semantics for QMLI’m going to simplify our discussion of QML in a couple ways. I won’t consider

axiom systems at all; we’ll go straight to semantics.1

Further, I’ll begin by

considering a very simple semantics, SQML (for “simply QML”). It’s simple in

two ways. First, there will be no accessibility relation. 2φ will be said to be true

iff φ is true in all worlds in the model. In effect, each world is accessible from

every other (and hence the underlying propositional modal logic is S5). Second,

it will be a “constant domain” semantics. (We’ll discuss what this means, and

more complex semantical treatments of QML, below.)

1Cresswell and Hughes (1996, part III) present axiom systems for QML.


An SQML-model is an ordered triple ⟨W ,D,I ⟩ such that:

i) W is a nonempty set (“possible worlds”)

ii) D is a non-empty set (“domain”)

iii) I is a function such that: (“interpretation function”)

a) if α is a constant then I (α) ∈Db) ifΠn

is an n-place predicate thenI (Πn) is a set of ordered

n+1-tuples ⟨u1, . . . , un, w⟩, where u1, . . . , un are members

of D, and w ∈W .

Recall that our semantics for modal propositional logic assigned truth values

to sentence letters relative to possible worlds. We have something similar

here: we relativize the interpretation of predicates to possible worlds. The

interpretation of a two-place predicate, R, for example is a set of ordered

triples, two members of which are in the domain, and one member of which is

a possible world. When ⟨u1, u2, w⟩ is in the interpretation of R, that represents

R’s applying to u1 and u2 in possible world w. In a possible worlds setting, this

relativization makes intuitive sense: a predicate can apply to some objects in

one possible world but fail to apply to those same objects in some other possible

world.

Notice that the interpretations of constants are not relativized in any way to

possible worlds. The interpretation I assigns simply a member of the domain

to a name. This re�ects the common belief that natural language proper

names—which constants are intended to represent—are rigid designators, i.e.,

terms that have the same denotation relative to every possible world (see Kripke

(1972).) We’ll discuss the signi�cance of this feature of our semantics below.

On to the de�nition of the valuation function for an SQML-model. First,

we keep the de�nition of a variable assignment from nonmodal predicate logic

(section 4). Our variable assignments therefore assign members of the domain

to variables absolutely, rather than relative to worlds. (This is an appropriate

choice given our choice to assign constants absolute semantic values.) But the

valuation function will now relativize truth values to possible worlds (as well

as to variable assignments). After all, the sentence ‘F a’, if it represents “Ted is

tall”, should vary in truth value from world to world.

The valuation function VM ,g , for SQML-modelM (= ⟨W ,D,I ⟩)and variable assignment g , is de�ned as the function that assigns


either 0 or 1 to each wff relative to each member ofW , subject to

the following constraints:

i) for any terms α,β, VM ,g (α=β, w) = 1 iff [α]M ,g = [β]M ,g

ii) for any n-place predicate, Π, and any terms α1, . . . ,αn,VM ,g (Πα1 . . .αn, w) = 1 iff ⟨[α1]M ,g , . . . ,[αn]M ,g , w⟩ ∈ I (Π)

iii) for any wffs φ, ψ, and any variable, α,

a) VM ,g (∼φ, w) = 1 iff VM ,g (φ, w) = 0b) VM ,g (φ→ψ, w) = 1 iff either VM ,g (φ, w) = 0 or VM ,g (ψ, w) =

1c) VM ,g (∀αφ, w) = 1 iff for every u ∈D,VM ,gu/α

(φ, w) = 1

d) VM ,g (2φ, w) = 1 iff for every v ∈W ,VM ,g (φ, v) = 1

The derived clauses are what you’d expect, including the following one for

3:

VM ,g (3φ, w) = 1 iff for some v ∈W ,VM ,g (φ, v) = 1

Finally, we have:

φ is valid inM (= ⟨W ,D,I ⟩) iff for every variable assignment, g ,

and every w ∈W ,VM ,g (φ, w) = 1

φ is SQML-valid (“�QML

φ”) iff φ is valid in all SQML models.

Γ SQML-semantically-implies φ (“Γ �SQML

φ”) iff for every world

w in every SQML model, if every member of Γ is true at w, then

so is φ

9.4 Countermodels and validity proofs in SQMLWe’ll again be interested in coming up with countermodels for invalid formulas,

and validity proofs for valid ones. We can use the same pictorial method for

constructing countermodels, asterisks and all, with a few changes. First, there’s

no need for the arrows between worlds, since we’ve dropped the accessibility

relation, thereby making every world accessible to every other. Secondly, we

have predicates and names for atomics instead of sentence letters, so how to


account for this? Let’s look at an example: �nding a countermodel for the

formula (3F a∧3Ga)→3(F a∧Ga).We begin as follows:

∗1 1 1 0 0

(3F a∧3Ga)→3(F a∧Ga)∗ ∗

r

The understars make us create two new worlds:

∗1 1 1 0 0

(3F a∧3Ga)→3(F a∧Ga)∗ ∗

r

1

F aa

1

Gab

We must then discharge the overstar from the false diamond in each world

(since every world is accessible to every other world in our models):

∗1 1 1 0 0 0 0

(3F a∧3Ga)→3(F a∧Ga)∗ ∗ †

r

1 0 0

F a F a∧Gaa

1 0 0

Ga F a∧Gab

(I had to make either F a or Ga false in r—I chose F a arbitrarily.) Now, we’ve

indicated the truth-values that we want the atomics to have. How do we make

the atomics have the TVs we want in the picture?

We do this by introducing a domain for the model, and stipulating what the

names refer to and what objects are in the extensions of the predicates. Let’s

use letters like ‘u’ and ‘v’ as the members of the domain in our models. Now, if

we let the name ‘a’ refer to (the letter) u, and let the extension of F in world r


be {} (the empty set), then the truth value of ‘F a’ in world r will be 0 (false),

since the denotation of a isn’t in the extension of F at world r. Likewise, we

need to put u in the extension of F (but not in the extension of G) in world

a, and put u in the extension of G ((but not in the extension of F ) in world b.

This all may be indicated on the diagram as follows:

a: u

∗1 1 1 0 0 0 0

(3F a∧3Ga)→3(F a∧Ga)∗ ∗ †

F :{}

r

1 0 0

F a F a∧Ga

F : {u} G : {}

a

1 0 0

Ga F a∧Ga

F : {} G : {u}

b

Within each world I’ve included a speci�cation of the extension of each predi-

cate. But the speci�cation of the referent of the name ‘a’ does not go within

any world; it was rather indicated (in boldface) at the top of model. This is

because names, unlike predicates, get assigned semantic values absolutely in a

model, not relative to worlds.

Time for the of�cial model:

W = {r,a,b}D = {u}I (a) = u

I (F ) = {⟨u,a⟩}I (G) = {⟨u,b⟩}

Let’s try one with quanti�ers: 2∃xF x→∃x2F x:

∗ +1 1 0 0

2∃ xF x→∃ x2F x+

r


The overstar above the 2 in the antecedent must be discharged in r itself, since,

remember, every world sees every world in these models. That gives us a true

existential. Now, a true existential is a bit like a true 3—the true ∃xF x means

that there must be some object u from the domain that’s in the extension of Fin r. I’ll put a + under true ∃s and false ∀s, to indicate a commitment to someinstance of some sort or other. Analogously, I’ll indicate a commitment to all

instances of a given type (which would arise from a true ∀ or a false ∃) with a +

above the connective in question.

OK, how do we make ∃xF x true in r? By making “F x” true for some

value of x. Let’s put the letter u in the domain, and make “F x” true when u is

assigned to x. We’ll indicate this by putting a 1 overtop of “F u

x ” in the diagram.

Now, “F u

x ” isn’t a formula of our language—what it indicates is that “F x” is to

be true when u is assigned to x. And to make this come true, we treat it as an

atomic—we put u in the extension of F at r:

∗ +1 1 0 0 1

2∃ xF x→∃ x2F x F u

x+

F : {u}

r

Good. Now we’ve got to attend to the overplus, the + sign overtop the false

∃x2F x. Since it’s a false ∃, we’ve got to make 2F x false for every object in the

domain (otherwise—if there were something in the domain for which 2F x was

true—∃x2F x would be true after all). So far, we’ve got only one object in our

domain, u, so we’ve got to make 2F x false, when u is assigned to the variable

‘x’. We’ll indicate this on the diagram by putting a 0 overtop of “2F u

x ”:

∗ +1 1 0 0 1 0


x 2F u

x+ ∗

F : {u}

r

Ok, now we have an understar, which means we should add a new world to

our model. When doing so, we’ll need to discharge the overstar from the

antecedent. We get:


∗ +1 1 0 0 1 0


x 2F u

x+ ∗

F : {u}

r

0 1 1

F u

x ∃ xF x F v

x+

F : {v}

a

This move requires some explanation. Why the v? Well, I was required to

make F x false, with u assigned to x. Well, that means keeping u out of the

extension of F at a. Easy enough, right? Just make F ’s extension {}? Well,

no—because of the true 2 in r, I’ve got to make ∃xF x true in a. But that means

that something’s got to be in F ’s extension in a! It can’t be u, so I’ll add a new

object, v, to the domain, and put it in F ’s extension in a.

But adding v to the domain of the model adds a complication. We had

an overplus in r—over the false ∃. That meant that, in r, for every member of

the domain, 2F x is false. So, 2F x is false in r when v is assigned to x. That

creates another understar, requiring the creation of a new world. The model

then looks as follows:


∗ +1 1 0 0 1 0 0


x 2F u

x 2F v

x+ ∗ ∗

F : {u}

r

0 1 1

F u

x ∃ xF x F v

x+

F : {v}

a

0 1 1

F v

x ∃ xF x F u

x+

F : {u}

b

(Notice that we needn’t have made another world b—we could simply have

discharged the understar on r.)

Ok, here’s the of�cial model:

W = {r,a,b}D = {u,v}I (F ) = {⟨u, r⟩, ⟨u,b⟩, ⟨v,a⟩}

Another example: I’ll do a validity proof for the following formula: 3∃x(x =a∧2F x)→F a:

i) suppose for reductio that (for some model, some world r , and

some variable assignment g ,) Vg (3∃x(x=a∧2F x)→F a, r ) =0. Thus Vg (3∃x(x=a∧2F x), r ) = 1 and …

ii) …Vg (F a, r ) = 0iii) From i), for some w ∈W , Vg (∃x(x=a∧2F x), w) = 1iv) so for some u ∈D, Vgu/x

(x=a∧2F x, w) = 1)

v) Thus, Vgu/x(x=a, w) = 1 and …

vi) …Vgu/x(2F x, w) = 1

vii) from vi), Vgu/x(F x, r ) = 1

viii) Thus, ⟨[x]gu/x, r ⟩ ∈ I (F )—that is, ⟨u, r ⟩ ∈ I (F )


ix) from v,[x]gu/x= [a]gu/x

x) By the de�nition of denotation plus facts about variable as-

signments, u =I (a)

xi) By viii) and x), ⟨I (a), r ⟩ ∈ I (F )xii) Thus, Vg (F a, r ) = 1. Contradicts line ii)

Notice that in line xii) I inferred that Vg assigned “F a” truth at r . I could have

subscripted V with any variable assignment, since the truth condition for the

formula “F a” is the same, regardless of the variable assignment; I picked gbecause that’s what I needed to get the contradiction.

9.5 Philosophical questions about SQMLOur semantics for quanti�ed modal logic faces philosophical challenges. In

each case we will be able to locate a particular feature of our semantics that

gives rise to the alleged problem. In response, one can stick with the simple

semantics and give it a philosophical defense, or one can revise the semantics.

9.5.1 The necessity of identityLet’s try to come up with a countermodel for the following formula:

∀x∀y(x=y→2(x=y))

When we try to make the formula false by putting a 0 over the initial ∀, we get

an under-plus. So we’ve got to make the inside part, ∀y(x=y→2x=y), false

for some value of x. We do this by putting some object u in the domain, and

letting that be the value of x for which ∀y(x=y→2x=y) is false. We get:

0 0

∀x∀y(x=y→2x=y) ∀y( ux=y→2( ux=y))+ +

r

Now we need to do the same thing for our new false universal: ∀y(x=y→2x=y).For some value of y, the inside conditional has to be false. But that means that

the antecedent must be true. So the value for y has to be u again. We get:


0 0 1 0 0

∀x∀y(x=y→2x=y) ∀y( ux=y→2( ux=y)) u

x=u

y→2( ux=u

y )+ + ∗

r

The understar now requires creation of a world in which x=y is false, when

both x and y are assigned u. But there cannot be any such world! An identity

sentence is true (at any world) if the denotations of the terms are identical.

Our attempt to �nd a countermodel has failed; we must do a validity proof.

Consider any QML model ⟨W ,D,I ⟩, any r ∈W , and any variable assignment

g ; we’ll show that Vg (∀x∀y(x=y→2x=y), r ) = 1:

i) suppose for reductio that Vg (∀x∀y(x=y→2x=y), r ) = 0.

ii) Then, for some u ∈D, Vgu/x(∀y(x=y→2x=y), r ) = 0

iii) So, for some v ∈D, Vgu/x v/y(x=y→2(x=y), r ) = 0.

iv) Thus, Vgu/x v/y(x=y, r ) = 1, and …

v) …Vgu/x v/y(2(x=y), r ) = 0

vi) from iv) [x]gu/x v/y= [y]gu/x v/y

vii) From v), at some world, w, Vgu/x v/y(x=y, w) = 0

viii) And so, [x]gu/x v/y6= [y]gu/x v/y

. Contradicts vi).

Notice at the end how the particular world at which the identity sentence was

false didn’t matter. The truth condition for an identity sentence is simply that

the terms denote the same thing; it doesn’t matter what world this is evaluated

relative to.2

2A note about variables. In validity proofs, I’m using italicized ‘u’ and ‘v’ as variables to

range over objects in the domain of the model I’m considering. So, a sentence like ‘u = v’

might be true, just as the sentence ‘x=y’ of our object language can be true. But when I’m

doing countermodels, I’m using the roman letters ‘u’ and ‘v’ as themselves being members of

the domain, not as variables ranging over members of the domain. Since the letters ‘u’ and ‘v’

are different letters, they are different members of the domain. Thus, in a countermodel with

letters in the domain, if the denotation of a name ‘a’ is the letter ‘u’, and the denotation of the

name ‘b ’ is the letter ‘v’, then the sentence ‘a=b ’ has got to be false, since ‘u’6=‘v’. If I were

using ‘u’ and ‘v’ as variables ranging over members of the domain, then the sentence ‘u = v’

might be true! This just goes to show that it’s important to distinguish between the following

two sentences:


We can think of ∀x∀y(x=y→2(x=y)) as expressing “the necessity of iden-

tity”: it says that whenever identity holds between objects, it necessarily holds.

The necessity of identity is philosophically controversial. On the one hand

it can seem obviously correct. “x=y” says that x and y are one and the same

thing. Now, if there were a world in which x was different from y, since x and

y are the same thing, this would have to be a world in which x was different

from x. How could that be? On the other hand, it was a great discovery that

Hesperus = Phosphorus. Surely, it could have turned out the other way—surely,

Hesperus might not have turned out identical to Phosphorus! But isn’t this

a counterexample to this formula? For a discussion of this example, see Saul

Kripke’s book Naming and Necessity.

It’s worth noting why the necessity of identity turns out valid, given our

semantics. It turns out valid because of the way we de�ned variable assignments:

our variable assignments assign members of the domain to variables absolutely,

rather than relative to worlds. (Similarly: since the interpretation function I ,

according to our de�nition above, assigns referents to names absolutely, rather

than relative to worlds, the formula a=b→2a=b turns out valid.) One could,

instead, de�ne variable assignments as functions that assign members of the

domain to variables relative to worlds. Given appropriate adjustments to the

de�nition of the valuation function, this would have the effect of invalidating

the necessity of identity.3

(Similarly, one could make I assign denotations to

names relative to worlds, thus invaliding a=b→2a=b .)

9.5.2 The necessity of existenceAnother (in)famous valid formula of SQML is the “Barcan Formula”, named

after Ruth Barcan Marcus: ∀x2F x→2∀xF x. (Call the schema ∀α2φ→2∀αφ the “Barcan schema”.) If we try to �nd a countermodel for this formula

we get to the following stage:

u = v‘u’ = ’v’

The �rst could be true, depending on what ‘u’ and ‘v ’ currently refer to, but the second one is

just plain false, since ‘u’ and ‘v’ are different letters.

3See Gibbard (1975).


+1 0 0

∀x2F x→2∀xF x∗

r

0 0

∀xF x F u

x+

F : {}

a

When you have a choice between discharging over-things and under-things,

whether plusses or stars, always do the under things �rst. In this case, this

means discharging the understar and ignoring the over-plus for the moment.

So, discharging the understar gave us world a, in which we made a universal

false. This gave an underplus, and forced us to make an instance false. So I put

object u in our domain, and keep it out of the extension of F in a. This makes

F x false in a, when x is assigned u.

But now, I need to discharge the overplus in r. I must make 2F x true for

every member of the domain, including u, which is now in the domain. But

then this requires F x to be true, when u is assigned to x, in a:

+ ∗1 0 0 1 1

∀x2F x→2∀xF x 2F u

x∗

F : {u}

r

0 0 1

∀xF x F u

x F u

x+

F : {?}

a

So, we fail to get a model. Time for a validity proof; let’s show that every

instance of the Barcan Schema is valid:

i) suppose for reductio that Vg (∀α2φ→2∀αφ, r ) = 0. Then

Vg (∀α2φ, r ) = 1 and …

ii) …Vg (2∀αφ, r ) = 0.

iii) from ii), for some w, Vg (∀αφ, w) = 0

iv) so, for some u in the domain, Vgu/α(φ, w) = 0

v) from i), for every member of the domain, and so for u in

particular, Vgu/α(2φ, r ) = 1.


vi) thus, for every world, and so for w in particular, Vgu/α(φ, w) = 1.

Contradicts iv).

The validity of the Barcan formula in our semantics is infamous because the

Barcan formula seems, intuitively, to be invalid. To see why, we need to think a

bit about the intuitive signi�cance of the relative order of quanti�ers and modal

operators. Consider the difference between the following two sentences:

3∃xF x

∃x3F x

In general, a sentence of the form 3φ says that it’s possible for the component

sentence, φ, to be true. So the �rst of our two sentences, 3∃xF x, says that

it’s possible for “∃xF x” to be true. That is: it’s possible for there to exist

an F . What about the second sentence? In general, a sentence that begins

without a modal operator in front makes a statement about the actual world.

Thus, a statement that begins with “∃x . . .” is saying that there exists, in the

actual world, an object x, such that…. Our second statement, then, says that

there actually exists an object, x, that is possibly F . We have seen that our two

statements have the following meanings:

3∃xF x “it’s possible for there to exist an F ”

∃x3F x “ there actually exists an object, x, that is possibly F ”

It matters, therefore, whether the ∃ comes after or before the 3. If the ∃ comes

�rst, then the statement is saying that there actually exists a certain sort of

object (namely, an object that could have been a certain way.) But if it comes

second, after the 3, then the statement is merely saying that there could haveexisted a certain sort of object. There is a similar contrast with the following

two statements:

2∀xF x

∀x2F x

The �rst says that it’s necessary that: everything is F . That is, in every possible

world, every object that exists in that world is F in that world. The objects

ranged over by the ∀, so to speak, are drawn from the worlds the 2 introduces,

because the ∀ occurs inside the scope of the 2. The second statement, by


contrast, says that: every actual object is necessarily F . That is, every object that

exists in the actual world is F in every possible world. The second statement

concerns just actually existing objects because the ∀ occurs in the front of the

formula, not inside the scope of the 2.

With all this in mind, return to the Barcan formula, ∀x2F x→2∀xF x. It

says:

“If every actually existing thing is F in every possible world, then

in every world, every object in that world is F in that world”

Now we can see why this claim is questionable. Even if every actual thing is

necessarily F , there could still be worlds containing non-F things, so long as

those non-F things don’t exist in the actual world. Suppose, for instance, that

every object in the actual world is necessarily a material object. Then, letting

F stand for “is a material object”, ∀x2F x is true. Nevertheless, 2∀xF x seems

false—it would presumably be possible for there to exist an immaterial object:

a ghost, say. Possible worlds containing ghosts would simply need to contain

objects that do not exist in the actual world (since all the objects in the actual

world are necessarily material.)

This objection to the validity of the Barcan formula is obviously based on

the idea that what objects exist can vary from possible world to possible world.

But this sort of variation is not represented in the SQML de�nition of a model.

Each such model contains a single domain, D, rather than different domains

for different possible worlds. The truth condition we speci�ed for a quanti�ed

sentence ∀αφ, at a world w, was simply that φ is true at w of every member of

D—the quanti�er ranges over the same domain, regardless of which possible

world is being described. That is why the Barcan formula turns out valid under

our de�nition.

This feature of SQML models is problematic for an even more direct reason:

the sentence ∀x2∃y y = x, i.e., “everything necessarily exists”, turns out valid!:

i) Suppose for reductio that Vg (∀x2∃y y=x, w) = 0.

ii) Then Vg (u/x)(2∃y y=x, w) = 0, for some u ∈Diii) So Vg (u/x)(∃y y=x, w ′) = 0, for some w ′ ∈Wiv) So Vg (u/x u ′/y)(y=x, w ′) = 0, for every u ′ ∈Dv) So, since u ∈D, we have Vg (u/x u/y)(y=x, w ′) = 0. But that can’t

be, given the clause for ‘=’ in the de�nition of the valuation

function.


It’s clear that this formula turns out valid for the same reason that the Barcan

formula turns out valid: SQML models have a single domain common to each

possible world.

The Barcan schema is just one of a number of interesting schemas concern-

ing how quanti�ers and modal operators interact (for each schema I also list an

equivalent schema with 3 in place of 2):

∀α2φ→2∀αφ 3∃αφ→∃α3φ (Barcan)

2∀αφ→∀α2φ ∃α3φ→3∃αφ (Converse Barcan)

∃α2φ→2∃αφ 3∀αφ→∀α3φ2∃αφ→∃α2φ ∀α3φ→3∀αφ

We have already discussed the Barcan schema. The third schema raises no philo-

sophical problems for SQML, since, quite properly, it has instances that turn out

invalid. As we saw above, there are SQML models in which 2∃xF x→∃x2F xis false. Let’s look at the other two schemas.

First, the converse Barcan schema. Like the Barcan schema, each of its

instances is valid given the SQML semantics (I’ll leave this to the reader to

demonstrate), and like the Barcan schema, this verdict faces a philosophical

challenge. The antecedent says that in every world, everything that exists inthat world is φ. Existents are thus always φ. It might still be that some object

isn’t necessarily φ: perhaps some object that is φ in every world in which it

exists, fails to be φ in worlds in which it doesn’t exist. This talk of an object

being φ in a world in which it doesn’t exist may seem strange, but consider

the following instance of the converse Barcan schema, substituting “∃y y=x”

(think: “x exists”) for φ:

2∀x∃y y=x→∀x2∃y y=x

This formula seems to be false. Its antecedent is clearly true; but its consequent

says that every object in the actual world exists necessarily, and hence seems

intuitively to be false.

Each instance of the fourth schema, ∃α2φ→2∃αφ, is also validated by

the SQML semantics (again, an exercise for the reader); and again, this is

philosophically questionable. Let’s suppose that physical objects are necessarily

physical. Then, ∃x2P x seems true, letting P mean ‘is physical’. But 2∃xP xseems false—it seems possible that there are no physical objects. This coun-

terexample requires that there be worlds with fewer objects than those that

actually exist, whereas the counterexample to the Barcan formula involved the

possibility that there be more objects than those that actually exist.


9.5.3 Necessary existence defendedThere are various ways to respond to the challenge of the previous section.

From a logical point of view, the simplest is to stick to one’s guns and defend

the SQML semantics. SQML-models accurately model the modal facts. The

Barcan formula, the converse Barcan formula, the fourth schema, and the state-

ment that everything necessarily exists are all logical truths; the philosophical

objections are mistaken. Contrary to appearances it is not contingent what

exists. Each possible world has exactly the same stock of individuals. Call this

the doctrine of Constancy.

One could uphold Constancy either by taking an narrow view of what is

possible, or by taking a broad view of what exists. On the former alternative,

one would claim that it is just not possible for there to be any ghosts, and that

it is just not possible in any sense for an actual object to have failed to exist. On

the latter alternative, which I’ll be discussing for the rest of this section, one

accepts the possibility of ghosts, dragons, and so on, but claims that possible

ghosts and dragons exist in the actual world.

Think of the objects in D as being all of the possible objects. In addition to

normal things—what one would normally think of as the actually existing entities:

people, tables and chairs, planets and electrons, and so on—our defender of

Constancy claims that there also exist objects that, in other possible worlds, are

ghosts, golden mountains, talking donkeys, and so forth; and these are included

in D as well. Call these further objects “merely possible things” (but don’t be

misled by this label; the claim is that merely possible things actually exist.) The

formula “∀xF x” means that every possible object is F in the actual world. It’s

not enough for the normal things to be F , for the normal things are not all

of the things that there are. There are also all the merely possible things, and

each of these must be F as well (must be F here in the actual world, that is), in

order for ∀xF x to be true. Hence, the objection to the Barcan formula from

the previous section fails. That objection assumed that ∀x2F x, the antecedent

of (an instance of) the Barcan formula, was true, when F symbolizes “is a

material object”. But this neglects the merely possible things. It’s true that all

the normal objects are necessarily material objects, but there are some further

things—merely possible things—that are not necessarily material objects.

Further: in ordinary language, when we say “Everything” or “something”,

we typically don’t mean to be talking about all possible objects; we’re typically

talking about just the normal things. Otherwise we would be speaking falsely

when we say, for example, “everything has mass”: merely possible unicorns


presumably have no mass (nor any spatial location, nor any other physical

feature.) Ordinary quanti�cation is restricted to normal things. So if we want

to translate an ordinary claim into the language of QML, we must introduce

a predicate for the normal things, “N”, and use it to restrict quanti�ers. But

now, consider the following ordinary English statement:

If everything is necessarily a material object, then necessarily: ev-

erything is a material object

If we mindlessly translate this into the language of QML, we would get

∀x2F x→2∀xF x—an instance of the Barcan schema. But since in every-

day usage, quanti�ers are restricted to normal things, the thought in the mind

of an ordinary speaker who utters this sentence is more likely the following:

∀x(N x→2F x)→2∀x(N x→F x)

which says:

If every normal thing is necessarily a material object, then neces-

sarily: every normal thing is a material object.

And this formula is not an instance of the Barcan schema, nor is it valid, as may

be shown by the following countermodel:

+1 0 0 0 1 0

∀x(N x→2F x)→2∀x(N x→F x) N u

x→2F u

x∗ †

N : {}

r

0 1 0 0

∀x(N x→F x) N u

x→F u

x+

N : {u} F : {}

a

So in a sense, the ordinary intuitions that were alleged to undermine the Barcan

schema are in fact consistent with Constancy.


The defender of Constancy can defend the converse Barcan schema and the

fourth schema in similar fashion. The objection to the converse Barcan schema

assumed the falsity of ∀x2∃y y=x. “Sheer prejudice!”, according to the friend

of constancy. “And recall further that an ordinary utterance of ‘Everything exists

necessarily’ expresses, not ∀x2∃y y=x, but rather ∀x(N x→2∃y(N y∧y=x)),(N for ‘normal’), the falsity of which is is perfectly compatible with Constancy.

It’s possible to fail to be normal; all that’s impossible is to utterly fail to exist.

Likewise for the fourth schema.”

This defense of SQML is hard to take. Let “G” stand for a kind of object

that, in fact, has no members, but which could have had members. Perhaps ghostis such a kind. ∀x2∼Gx→2∀x∼Gx is an instance of the Barcan schema, and

so true according to the defender of Constancy. Since there could have existed

ghosts, the consequent of this conditional is false. Therefore, its antecedent

∀x2∼Gx must be false. That is, there exists something that could have been a

ghost. But this is a very surprising result. The alleged possible ghost couldn’t

be any material object, presumably, assuming it would be impossible for any

material object to be a ghost. The defender of Constancy, then, is committed to

the existence of objects which we wouldn’t otherwise have dreamed of accepting:

things that could have been ghosts, things that been dragons, things that could

have been gods, and so on.

The defender of Constancy might try to defend this conclusion by remind-

ing us that these “possible-ghosts”, “possible-dragons”, and so on, are not

normal objects. They aren’t in space and time, presumably, which explains why

no one has ever seen, heard, felt, or smelled one. He might even say that they

are non-actual, or even that they do not exist (though they are). We are quite cor-

rect, she might say, to scoff at the idea that some normal/actual/existing objects

are capable of being ghosts; but what’s the big deal about saying that some non-

normal/non-actual/non-existing objects have these capabilities? This move,

too, will be considered philosophically suspect by many. Many philosophers

regard the idea that there are some non-existent things, or some non-actual

things, as being anywhere from obviously false to conceptually incoherent, or

subversive, or worse.4

And how does it help to point out that the objects aren’t

normal? The postulation of non-normal objects—objects above and beyond

the objects that the rest of us believe in—was exactly what I was claiming is

philosophically suspect!

On the other hand, Constancy’s defenders can point to certain powerful

4See Quine (1948); Lycan (1979).


arguments in its favor. Here’s a quick sketch of one such argument. First, the

following seems to be a logical truth:

Ted=Ted

But it follows from this that:

∃y y =Ted

This latter formula, too, is therefore a logical truth. But if φ is a logical truth

then so is 2φ (recall the rule of necessitation from chapter 6). So we may infer

that the following is a logical truth:

2∃y y =Ted

Next, notice that nothing in the argument for 2∃y y =Ted depended on any

special features of me. We may therefore conclude that the reasoning holds

good for every object; and so ∀x2∃y y = x is indeed a logical truth. Since,

therefore, every object exists necessarily, it should come as no surprise that

there are things that might have been ghosts, dragons, and so on—for if there

had been a ghost, it would have necessarily existed, and thus must actually exist.

This and other related arguments have apparently wild conclusions, but they

cannot be lightly dismissed, for it is no mean feat to say exactly where they go

wrong (if they go wrong at all!).5

9.6 Variable domainsWe now consider a way of dealing with the problems discussed in section 9.5.2

above that does not require embracing Constancy.

SQML models contain a single domain,D, over which the quanti�ers range

in each possible world. Since it was this feature that led to the problems of

section 9.5.2, let’s introduce a new semantics that instead provides different

domains for different possible worlds. And let’s also reinstate the accessibility

relation, for reasons to be made clear below:6

The new semantics is called

VDQML (“variable-domains quanti�ed modal logic”):

5On this topic see Prior (1967, 149-151); Plantinga (1983); Fine (1985); Linsky and Zalta

(1994, 1996); Williamson (1998, 2002).

6More care than I take is needed in converting the earlier de�nition of validity if one is

worried about the validity of formulas with free variables. See ?, p. 275.


A VDQML-model is an ordered 5-tuple ⟨W ,R ,D,Q,I ⟩ such that:

i) W is a nonempty set (“possible worlds”)

ii) R is a binary relation onW (“accessibility relation”)

iii) D is a set (“super-domain”)

iv) Q is a function that assigns to any w ∈ W a non-empty7

subset of D. Let us refer to Q(w) as “Dw”. Think of Dw as

w’s “sub-domain”—the set of objects that exist at w.

v) I is a function such that: (“interpretation function”)

a) if α is a constant then I (α) ∈Db) if Π is an n-place predicate then I (Π) is a set of ordered

n+1-tuples ⟨u1, . . . , un, w⟩, where u1, . . . , un are members

of D, and w ∈W .

Denotation is de�ned as before. Truth in a model may then be de�ned thus:

The valuation function VM ,g , for VDQML-modelM (= ⟨W ,R ,D,Q,I ⟩)and variable assignment g , is de�ned as the function that assigns

either 0 or 1 to each wff relative to each member ofW , subject to

the following constraints:

i) for any terms α andβ, VM ,g (α=β, w) = 1 iff [α]M ,g = [β]M ,g

ii) for any n-place predicate, Π, and any terms α1, . . . ,αn,

VM ,g (Πα1 . . .αn, w) = 1 iff ⟨[α1]M ,g , . . . ,[αn]M ,g , w⟩ ∈ I (Π)iii) for any wffs φ and ψ, and any variable, α,

a) VM ,g (∼φ, w) = 1 iff VM ,g (φ, w) = 0b) VM ,g (φ→ψ, w) = 1 iff either VM ,g (φ, w) = 0 or VM ,g (ψ, w) =

1c) VM ,g (∀αφ, w) = 1 iff for each u ∈Dw ,VM ,gu/α

(φ, w) = 1

7One could drop this assumption. But if subdomains can be empty then 2∃x(F x→F x) will

be invalid. Since ∃x(F x→F x) is valid given the chapter 4 semantics for non-modal predicate

logic, we would have the odd result of a logical truth whose necessitation isn’t a logical truth.

One could modify the chapter 4 semantics by allowing predicate logic models with empty

domains, thus invalidating ∃x(F x→F x). This approach is known as free logic.


d) VM ,g (2φ, w) = 1 iff for each v ∈W , ifRwv then VM ,g (φ, v) =1

The de�nitions of validity and semantic consequence remain unchanged. The

obvious derived clauses for ∃ and 3 are as follows:

VM ,g (∃αφ, w) = 1 iff for some u ∈Dw ,VM ,gu/α(φ, w) = 1

VM ,g (3φ, w) = 1 iff for some v ∈W ,Rwv and VM ,g (φ, v) = 1

Thus, we have introduced introduced subdomains. We still have D, a set

that contains all of the possible individuals. But for each possible world w,

we introduce a subset of the domain, Dw , to be the domain for w. When

evaluating a quanti�ed sentence at a world w, the quanti�er ranges only over

Dw . Notice that we also reinstated the accessibility relation. This isn’t necessary

for introducing subdomains; I did this in order to be able to make a certain

point about subdomains in section 9.6.2.

9.6.1 Countermodels to the Barcan and related formulasin VDQML

What is the effect of this new truth de�nition on the Barcan formula and related

formulas? All of the following formulas come out invalid:

2∀xF x→∀x2F x (Converse Barcan) ∃x3F x→3∃xF x∀x2F x→2∀xF x (Barcan) 3∃xF x→∃x3F x2∃xF x→∃x2F x ∀x3F x→3∀xF x∃x2F x→2∃xF x 3∀xF x→∀x3F x

The third one on the list was invalid before, and so is still invalid now. For

note: ifM is a SQML model, then we can construct a corresponding VDQML

model with the same set of worlds, (super-) domain, and interpretation function,

in which every world is accessible from every other, and in whichQ is a constant

function assigning the whole super-domain to each world. It is intuitively clear

that the same sentences are true in this corresponding model as are true inM .

Hence, whenever a sentence is SQML-invalid, it is VDQML-invalid.

As for the Barcan formula∀x2F x→2∀xF x, consider the following VDQML-

model:


Dr: {u} F : {u}r

��

00

Da

: {u,v} F : {u}a

00

Of�cial model:

W : {r,a}R : {⟨r, r⟩, ⟨r,a⟩, ⟨a,a⟩}D : {u,v}D

r: {u} D

a: {u,v}

I (F ) : {⟨u, r⟩, ⟨u,a⟩}

As for the fourth formula on the list, ∃x2F x→2∃xF x, we can construct a

countermodel as follows:

Dr: {u,v} F : {u}r

��

00

Da

: {v} F : {u}a

00

W : {r,a}R : {⟨r, r⟩, ⟨r,a⟩, ⟨a,a⟩}D : {u,v}D

r: {u,v} D

a: {v}

I (F ) : {⟨u, r⟩, ⟨u,a⟩}

And for the converse Barcan formula, 2∀xF x→∀x2F x, we have the fol-

lowing countermodel:

Dr: {u,v} F : {u,v}r

��

00

Da

: {v} F : {v}a

00


W : {r,a}R : {⟨r, r⟩, ⟨r,a⟩, ⟨a,a⟩}D : {u,v}D

r: {u,v} D

a: {v}

I (F ) : {⟨u, r⟩, ⟨v, r⟩, ⟨v,a⟩}

Note also that ∀x2∃y y=x is false at world r in this VDQML-model.

9.6.2 Expanding, shrinking domainsThere are several comments worth making about VDQML-models. First,

note that if we made certain restrictions on variable-domains models, then the

countermodels of the previous section would no longer be legal models. For

example, the �rst example, the counterexample to the Barcan formula, required

a model in which the domain expanded; world a was accessible from world r,

and had a larger domain. But suppose we made the:

Decreasing domains requirement: ifRwv, then Dv ⊆Dw

The counterexample would then go away. Indeed, every instance of the Barcan

schema would then become VDSQML-valid, which may be proved as follows:

i) suppose for reductio that Vg (∀α2φ→2∀αφ, w) = 0. Then

Vg (∀α2φ, w) = 1 and…

ii) …Vg (2∀αφ, w) = 0

iii) by ii), for some v,Rwv and Vg (∀αφ, v) = 0

iv) and so, for some u ∈Dv ,Vgu/α(φ, v) = 0

v) given decreasing domains, Dv ⊆Dw , and so u ∈Dw

vi) by i), for every object inDw , and so for u in particular, Vgu/α(2φ, w) =

1

vii) so, Vgu/α(φ, v) = 1. Contradicts iv)

Similarly, notice that the counterexamples to ∃x2F x→2∃xF x and the

converse Barcan formula assumed that domains can shrink. Just as above, an

added requirement on models will validate these formulas:


Increasing domains requirement: ifRwv then Dw ⊆Dv

Any instance of the schema ∃α2φ→2∃αφ, for example, may be shown to be

validated as follows:

i) suppose for reductio that Vg (∃α2φ→2∃αφ, w) = 0, Then

Vg (∃α2φ, w) = 1 and …

ii) …Vg (2∃αφ, w) = 0

iii) by i), for some u ∈Dw ,Vgu/α(2φ, w) = 1

iv) by ii), for some world v,Rwv and Vg (∃αφ, v) = 0

v) by the increasing domain requirement, Dw ⊆Dv , and so u ∈Dv

vi) by iv), for every object in Dv , and so for u in particular,

Vgu/α(φ, v) = 0

vii) by iii), Vgu/α(φ, v) = 1. Contradicts vi)

Note further that even after imposing the increasing domains requirement,

the Barcan formula remains VDQML-invalid; and after imposing the decreas-

ing domains requirement, the converse Barcan formula and also ∃x2F x→2∃xF xremain VDQML-invalid; this can be seen by the original countermodels for

these formulas above, each of which obeys the requirements in question. How-ever, it should also be noted that in systems in which the accessibility relation

is symmetric, this collapses: imposing either of these requirements results in

imposing the other. That is, in B or S5, imposing either the increasing or the

decreasing domains requirement results in imposing both, and hence results in

all three formulas being validated.

9.6.3 Strong and weak necessityIn order for 2φ to be true at a world, the VDQML semantics requires that φbe true at every accessible world. It might be thought that this requirement is

too strong. In order for 2F a, say, to be true, our de�nition requires F a to be

true in all possible worlds. But what if a fails to exist in some worlds? In order

for “Necessarily, I am human” to be true, must I be human in every possible

world? Isn’t it enough for me to be human in all the worlds in which I exist?


This argument goes by a little too quickly. The main worry of its proponent

is that our semantics requires a to exist necessarily, in order for 2F a to come

out true. But our semantics doesn’t require this. It does require F a to be

true in every world, in order for 2F a to be true; but it does not require ato exist in every world in which F a is true. The clause in the de�nition of a

VDQML-model for the interpretation of predicates was this:

b) if Π is an n-place predicate then I (Π) is a set of ordered n+1-

tuples ⟨u1, . . . , un, w⟩, where u1, . . . , un are members of D, and

w ∈W .

This allows I (F ) to contain pairs ⟨u, w⟩, where u is not a member of Dw . So

one could say that 2F a is consistent with a’s failing to necessarily exist; it’s just

that a has to be F even in worlds where it doesn’t exist.

I doubt this really addresses the worry, since it looks like bad metaphysics

to say that a person could be human at a world where he doesn’t exist. One

could hard-wire a prohibition of this sort of bad metaphysics into VDQML

semantics, by replacing b) with:

b′) if Π is an n-place predicate then I (Π) is a set of ordered n+1-

tuples ⟨u1, . . . , un, w⟩, where u1, . . . , un are members ofDw , and

w ∈W .

thus barring objects from having properties at worlds where they don’t ex-

ist. But some would argue that this goes too far. Given b′), the sentence

∀x2(F x→∃y y=x) turns out valid. “An object must exist in order to be F ”—

sounds clearly true if F stands for ‘is human’, but what if F stands for ‘is famous’?

Suppose John Lennon had never existed (it was all a communist plot); some

would argue that in such a circumstance Lennon still would have been famous.

The issues here are complex.8

But whether or not b) should be replaced

with b′), it looks as though there are some existence-entailing English predicates

π: predicates π such that nothing can be a π without existing. ‘Is human’ seems

to be such a predicate. So we’re back to our original worry about VDQML-

semantics: its truth condition for 2φ requires truth of φ at all worlds, which is

allegedly too strong, at least when φ is a sentence like πa, where π is existence-

entailing.

8The question is that of so-called “serious actualism” (Plantinga, 1983).


One could modify the clause for the 2 in the de�nition of the valuation

function, so that in order for 2F a to be true, a only needs to be F in worlds in

which it exists:

d′) VM ,g (2φ, w) = 1 iff for each v ∈W , ifRwv , and if [α]M ,g ∈Dw for each name or free

9variable α occurring in φ, then

VM ,g (φ, v) = 1

This would indeed have the result that 2F a gets to be true provided a is F in

every world in which it exists. But be careful what you wish for. Along with

this result comes the following: even if a doesn’t necessarily exist, the sentence

2∃x x=a comes out true. For according to d′), in order for 2∃x x=a to be

true, it must merely be the case that ∃x x=a is true in every world in which aexists, and of course this is indeed the case.

If 2∃x x=a comes out true even if a doesn’t necessarily exist, then 2∃x x=adoesn’t say that a necessarily exists. Indeed, it doesn’t look like we have any way

of saying that a necessarily exists, using the language of QML, if the 2 has the

meaning provided for it by d′).

A notion of necessity according to which “Necessarily φ” requires truth in

all possible worlds is sometimes called a notion of strong necessity. In contrast,

a notion of weak necessity is one according to which “Necessarily φ” requires

merely that φ be true in all worlds in which objects named within φ exist. d′)

corresponds to weak necessity, whereas our original de�nition d) corresponds

to strong necessity.

As we saw, if the 2 expresses weak necessity, then one cannot even express

the idea that a thing necessarily exists. That’s because one needs strong necessity

to say that a thing necessarily exists: in order to necessarily exist, you need to

exist at all worlds, not just at all worlds at which you exist! So this is a serious

de�ciency of having the 2 of QML express weak necessity. But if we allow the

2 to express strong necessity instead, there is no corresponding de�ciency, for

one can still express weak necessity using the strong 2 and other connectives.

For example, to say that a is weakly necessarily F (that is, that a is F in every

world in which it exists), one can say: 2(∃x x=a→F a).So it would seem that we should stick with our original truth condition d)

for the 2, and live with the fact that statements like 2F a turn out false if a fails

to be F at worlds in which it doesn’t exist. Those who think that “Necessarily,

9I.e., not bound to any quanti�er in φ.


Ted is human” is true despite Ted’s possible nonexistence can always translate

this natural language sentence into the language of QML as 2(∃x x=a→F a)(which requires a to be F only at worlds at which it exists) rather than as 2F a(which requires a to be F at all worlds).

Chapter 10

Two-dimensional modal logic

In this chapter we consider an extension to modal logic with considerable

philosophical interest.

10.1 ActualityThe word ‘actually’, in one of its senses anyway, can be thought of as a one-place

sentence operator: “Actually, φ.”

‘Actually’ might at �rst seem redundant. “Actually, snow is white” basically

amounts to: “snow is white”. But the actuality operator interacts with modal

operators in interesting ways. The following two sentences, for example, clearly

have different meanings:

Necessarily, if snow is white then snow is white

Necessarily, if snow is white then snow is actually white

The �rst sentence expresses the triviality that snow is white in any possible

world in which snow is white. But the second sentence makes the nontrivial

statement that if snow is white in any world, then snow is white in the actualworld.

So, ‘actually’ is nonredundant, and consequently, worth thinking about.

Let’s add a symbol to modal logic for it. “@φ” will symbolize “Actually, φ”.

We can now symbolize the pair of sentences above as 2(S→S) and 2(S→@S),

253

CHAPTER 10. TWO-DIMENSIONAL MODAL LOGIC 254

respectively. For some further examples of sentences we can symbolize using

‘actually’, consider:1

It might have been that everyone who is actually rich is poor

3∀x(@Rx→P x)

There could have existed something that does not actually exist

3∃x@∼∃y y=x

10.1.1 Kripke models with actual worldsFor the purposes of this chapter, the logic of iterated boxes and diamonds isn’t

relevant, so let’s simplify things by dropping the accessibility relation from

models; we will thereby treat every world as being accessible from every other.

Before laying out the semantics of @, let’s examine a slightly different way

of laying out standard modal logic. For propositional modal logic, instead of

de�ning a model as an ordered pair ⟨W ,I ⟩ (no accessibility relation, remem-

ber), one could instead de�ne a model as a triple ⟨W , w@,I ⟩, whereW and Iare as before, and w@ is a member ofW , thought of as the actual, or designatedworld of the model. The designated world w@ plays no role in the de�nition of

the valuation for a given model; it only plays a role in the de�nitions of truth

in a model and validity:

φ is true in modelM (= ⟨W , w@,I ⟩) iff VM(φ, r) = 1

φ is valid in system S iff φ is true in all models for system S

The old de�nition of validity for a system, recall, never employed the notion

of truth in a model; rather, it proceeded via the notion of validity in a frame.

The nice thing about the new de�nition is that it’s parallel to the way validity

is usually de�ned in model theory: one �rst de�nes truth in a model, and then

de�nes validity as truth in all models. But the new de�nition doesn’t differ in

any substantive way from the old de�nition, in that it yields exactly the same

class of valid formulas:

Proof: It’s obvious that everything valid on the old de�nition is

valid on the new de�nition (the old de�nition says that validity

1In certain special cases, we could do without the new symbol @. For example, instead of

symbolizing “Necessarily, if snow is white then snow is actually white” as 2(S→@S), we could

symbolize it as 3S→S. But the @ is not in general eliminable; see Hodes (1984b,a).


is truth in all worlds in all models; the addition of the designated

world w@ doesn’t play any role in de�ning truth at worlds, so each

of the new models has the same distribution of truth values as one

of the old models.) Moreover, suppose that a formula is invalid on

the old de�nition—i.e., suppose that φ is false at some world, w, in

some modelM . Now construct a model of the new variety that’s

just likeM except that its designated world is w. φ will be false in

this model, and so φ turns out invalid under the new de�nition.

In a parallel way, one can also add a designated world to models for quanti-

�ed modal logic.

10.1.2 Semantics for @Now for the semantics of @. We can give @ a very simple semantics using

models with designated worlds. Further, the designated world will now be

involved in the notion of truth in a model, not just in the de�nition of validity.

We’ll move straight to quanti�ed modal logic, bypassing propositional logic. To

keep things simple, let the models have a constant domain and no accessibility

relation. (It will be obvious how to add these complications back in, if they are

desired.) De�ne a Designated-world QML-model as a four-tuple ⟨W , w@,D,I ⟩,where:

i) W is a non-empty set (“worlds”)

ii) w@ is a member ofW (“designated/actual world”)

iii) D is a non-empty set (“domain”)

iv) I is a function that assigns semantic values (as before—names

are assigned members ofD; predicates are assigned extensions

relative to worlds)

In the de�nition of the valuation for such a model, the semantic clauses for the

old logical constants run as before, with the exception that the clause for the 2

no longer mentions accessibility:

VM ,g (2φ, w) = 1 iff for every v ∈W ,VM ,g (φ, v) = 1

And we now add a clause for the new operator @:

VM ,g (@φ, w) = 1 iff VM ,g (φ, w@) = 1

i.e., @φ is true at any world iff φ is true in the designated world of the model.


10.1.3 ExamplesExample 1: Show that � ∀x(F x∨2Gx)→2∀x(Gx∨@F x)

i) Suppose for reductio that this formula is not valid. Then for

some model and some variable assignment g , Vg (∀x(F x∨2Gx)→2∀x(Gx∨@F x), w@) = 0.

ii) Then Vg (∀x(F x∨2Gx), w@) = 1 and…

iii) …Vg (2∀x(Gx∨@F x), w@) = 0

iv) Given the latter, there is some world, call it “a”, such that

Rw@a and Vg (∀x(Gx∨@F x),a) = 0. And so, there is some

object, call it “u”, in the model’s domain, D, such that

Vgu/x(Gx∨@F x,a) = 0

v) And so Vgu/x(Gx,a) = 0 and…

vi) …Vgu/x(@F x,a) = 0

vii) Given the latter, Vgu/x(F x, w@) = 0 (by the clause in the truth

de�nition for @)

viii) Given ii), for every object in D, and so for u in particular,

Vgu/x(F x∨2Gx, w@) = 1.

ix) And so, either Vgu/x(F x, w@) = 1 or Vgu/x

(2Gx, w@) = 1

x) From ix and vii, Vgu/x(2Gx, w@) = 1

xi) And so, Vgu/x(Gx,a) = 1—contradicts v)

Example 2: show that 2 2∀x(Gx∨@F x)→2∀x(Gx∨F x): here is a model in

which this formula is false at the actual world, w@:

W = {w@,a}R = {⟨w@,a⟩}D = {u}I (F ) = {⟨u, w@⟩}I (G) =∅

The formula turns out false in this model: the consequent is false because at

world a, something (namely, u) is neither G nor F ; but the antecedent is true:

since u is F at w@, it’s necessary that u is either G or actually F .


10.2 ×Adding @ to the language of quanti�ed modal logic is a step in the right

direction, since it allows us to express certain kinds of comparisons between

possible worlds that we couldn’t express otherwise. But it doesn’t go far enough;

we need a further addition.2

Consider this sentence:

It might have been the case that, if all those then rich might all

have been poor, then someone is happy

What it’s saying, in possible worlds terms, is this:

For some world w, if there’s a world v such that (everyone who is

rich in w is poor in v), then someone is happy in w.

This is a bit like “It might have been that everyone who is actually rich is poor”;

in this new sentence the word ‘then’ plays a role a bit like the role ‘actually’

played in the earlier sentence. But the ‘then’ does not take us back to the actual

world of the model; it rather takes us back to the world, w, that is introduced

by the �rst possibility operator, ‘it might have been the case that’. We cannot,

therefore, symbolize our new sentence thus:

3(3∀x(@Rx→P x)→∃xH x)

for this has the truth condition that there is world w such that, if there’s a world

v such that (everyone who is rich in w@ is poor in v), then someone is happy in

w. The problem is that the @, as we’ve de�ned it, always takes us back to the

model’s designated world, whereas what we need to do is to “mark” a world,

and have @ take us back to the “marked” world:

3×(3∀x(@Rx→P x)→∃xH x)

× marks the spot: it is a point of reference for subsequent occurrences of @.

2See Hodes (1984a) on the limitations of @; see Cresswell (1990) on× (his symbol is “Ref”),

and further related additions.


10.2.1 Two-dimensional semantics for ×So let’s further augment the language of QML with another one-place sentence

operator, ×. The idea is that ×φ means the same thing as φ, except that

subsequent occurrences of @ inφ are to be interpreted as picking out the world

that was the “current world of evaluation” when the × was encountered. (This

will become clearer once we lay out the semantics for × and @.)

To lay out this semantics, let’s return to the old QML models (i.e., without

a designated world; and let’s continue to omit the accessibility relation). Thus,

a model is a triple ⟨W ,D,I ⟩, W a non-empty set, D a non-empty set, I a

function assigning referents to names and extensions to predicates at worlds as

before.

But now we change the de�nition of truth. We no longer evaluate formulas

at worlds. Instead we evaluate a formula at a pair of worlds (hence: “two-

dimensional semantics”). One world is the world we’re used to; it’s the world

that we’re evaluating the formula for truth in. Call this the “world of evaluation”.

The other world is a “reference world”—it’s the world that we’re currently

thinking of as the actual world, and the world that will be relevant to the

evaluation of @. Thus,

VM ,g (φ, w1, w2)

will mean thatφ is true at world w2, with reference world w1. We de�ne [α]M ,g ,

the denotation of term α relative to modelM and variable assignment g , as

before. And VM ,g is de�ned as the function that assigns to each wff, relative to

each pair of worlds, either 0 or 1 subject to the following constraints:

i) VM ,g (Πα1 . . .αn, v, w) = 1 iff ⟨[α1]M ,g , . . . ,[αn]M ,g , w⟩ ∈ I (Π)ii) VM ,g (∼φ, v, w) = 1 iff VM ,g (φ, v, w) = 0

iii) VM ,g (φ→ψ, v, w) = 1 iff VM ,g (φ, v, w) = 0 or VM ,g (ψ, v, w) =1

iv) VM ,g (∀αφ, v, w) = 1 iff for all u ∈D,VM ,gu/α(φ, v, w) = 1

v) VM ,g (2φ, v, w) = 1 iff for all w ′ ∈W ,VM ,g (φ, v, w ′) = 1

vi) VM ,g (@φ, v, w) = 1 iff VM ,g (φ, v, v) = 1

vii) VM ,g (×φ, v, w) = 1 iff VM ,g (φ, w, w) = 1


Note what the× does: change the reference world. When evaluating a formula,

it says to forget about the old reference world, and make the new reference

world whatever the current world of evaluation happens to be.

We can de�ne validity and consequence thus:

φ is 2D-valid (“�2Dφ”) iff for every modelM , every world w in that

model, and every assignment g based on that model, VM ,g (φ, w, w) =1

φ is a 2D-semantic consequence of Γ (“Γ �2Dφ”) iff for every model

M , every assignment g based on that model, and every world w in

that model, if VM ,g (γ , w, w) = 1 for each γ ∈ Γ, then VM ,g (φ, w, w) =1

Valid formulas are thus de�ned as those that are true at every pair of worlds of

the form ⟨w, w⟩; semantic consequence is truth-preservation at every pair of

worlds ⟨w, w⟩.Notice, however, that these aren’t the only notions of validity and con-

sequence that one could introduce. There is also the notion of truth, and

truth-preservation, at every pair of worlds:3

φ is generally 2D-valid (“�G2D

φ”) iff for every modelM , any worlds

v and w in that model, and every assignment g based on that model,

VM ,g (φ, v, w) = 1

φ is a general 2D-semantic consequence of Γ (“Γ �G2D

φ”) iff for every

modelM , every assignment g based on that model, and any worlds

v and w in that model, if VM ,g (γ , v, w) = 1 for each γ ∈ Γ, then

VM ,g (φ, v, w) = 1

Validity and general validity, and consequence and general consequence, come

apart in various ways, as we’ll see below.

As we saw, moving to this new language increases the �exibility of the @;

we can symbolize

It might have been the case that, if all those then rich might all

have been poor, then someone is happy

3The term ‘general validity’ is from Davies and Humberstone (1980); the �rst de�nition of

validity corresponds to their “real-world validity”.


as

3×(3∀x(@Rx→P x)→∃xH x)

Moreover, it costs us nothing. For we can replace any sentence φ of the old

language with ×φ in the new language (i.e. we just put the × operator at the

front of the sentence.)4

For example, instead of symbolizing

It might have been that everyone who is actually rich is poor

as 3∀x(@Rx→P x) as we did before, we symbolize it now as×3∀x(@Rx→P x).

10.2.2 ExamplesExample 1: show that if �φ then �@φ:

Suppose for reductio that φ is valid but @φ is not. That means

that in some model and some world, w (and some assignment g ,

but I’ll suppress this since it isn’t relevant here), V(@φ, w, w) = 0.

Thus, given the truth condition for @, V(φ, w, w) = 0. But that

violates the validity of φ.

Example 2: � φ↔@φ, but 2 2(φ↔@φ). (Moral: any proof theory for this

logic had better not include the rule of necessitation!)

The truth condition for @ insures that for any world w in any model

(and any variable assignment), V(@φ, w, w) = 1 iff V(φ, w, w) = 1,

and so V(φ↔@φ, w, w) = 1. Thus, �φ↔@φ.

But some instances of 2(φ↔@φ) aren’t valid. Let φ be ‘Fa’; here’s

a countermodel:

W = {c,d}D = {u}I (a) = u

I (F ) = {⟨u,c⟩}4This amounts to the same thing as the old symbolization in the following sense. Let

φ be any wff of the old language. Thus, φ may have some occurrences of @, but it has no

occurrences of ×. Then, for every QML-modelM = ⟨W ,D,I ⟩, and any v, w ∈ W ,×φ is

true at ⟨v, w⟩ inM iff φ is true in the designated-world QML model ⟨W , w,D,I ⟩.


In this model, V(2(F a↔@F a), c, c) = 0, because V(F a↔@F a, c,d) =0. For ‘F a’ is true at ⟨c,d⟩ iff the referent of ‘a’ is in the extension of

‘F ’ at world d (it isn’t) whereas ‘@F a’ is true at ⟨c,d⟩ iff the referent

of ‘a’ is in the extension of ‘F ’ at world c (it is).

Note that this same model shows that φ↔@φ is not generally valid.

General validity is truth at all pairs of worlds, and the formula

F a↔@F a, as we just showed, is false at the pair ⟨c,d⟩.

Example 3: �φ→2@φ

Consider any model, world w (and variable assignment), and sup-

pose for reductio that V(φ, w, w) = 1 but V(2@φ, w, w) = 0. Given

the latter, there is some world, v, such that V(@φ, w, v) = 0. And

so, given the truth condition for @, V(φ, w, w) = 0. Contradiction.

Example 4: �2×∀x3@F x→2∀xF x

i) Suppose for reductio that for some world w, some variable

assignment g , and some model, Vg (2×∀x3@F x, w, w) = 1and …

ii) …Vg (2∀xF x, w, w) = 0.

iii) Given the latter, for some world, call it “a”, Vg (∀xF x, w,a) =0.

iv) And so for some u ∈D (call it “u”), Vgu/x(F x, w,a) = 0.

v) Given i), Vg (×∀x3@F x, w,a) = 1

vi) Given the truth condition for ×, Vg (∀x3@F x,a,a) = 1

vii) Thus, for every object in the domain, and so for u in particular,

Vgu/x(3@F x,a,a) = 1

viii) Thus, for some world, call it b , Vgu/x(@F x,a, b ) = 1

ix) Given the truth condition for @, Vgu/x(F x,a,a) = 1

x) Given the truth condition for atomics, ⟨[x]gu/x,a⟩ ∈ I (F )

xi) But given iv, ⟨[x]gu/x,a⟩ /∈I (F ). Contradiction


10.3 FixedlyThe two-dimensional approach to semantics—evaluating formulas at pairs of

worlds rather than single worlds—raises an intriguing possibility. The 2 is a

universal quanti�er over the world of evaluation; we might, by analogy, follow

Davies and Humberstone (1980) and introduce an operator that is a universal

quanti�er over the reference world. Davies and Humberstone call this operator

F, and read “Fφ” as “�xedly, φ”. Grammatically, F is a one-place sentential

operator. Its semantic clause is this:

VM ,g (Fφ, v, w) = 1 iff for every v ′,VM ,g (φ, v ′, w) = 1

Everything else, including the de�nitions of validity and consequence, stays

the same.

Humberstone and Davies point out that given F, @, and 2, we can introduce

two new operators: F@ and F2. It’s easy to show that:

VM ,g (F@φ, v, w) = 1 iff for every v ′ ∈W ,VM ,g (φ, v ′, v ′) = 1

VM ,g (F2, v, w) = 1 iff for v ′, w ′ ∈W ,VM ,g (φ, v ′, w ′) = 1

Thus, we can think of F@ and F2, as well as 2 and F themselves, as being

“kinds of necessities”, since their truth conditions introduce universal quanti�ers

over worlds of evaluation and reference worlds. (What about 2F? It’s easy to

show that 2F is just equivalent to F2.)

Humberstone and Davies don’t use two-dimensional semantics; they in-

stead use designated-world QML models (and they don’t include ×). Say that

designated-world QML models are variants iff they are alike except perhaps

for the designated world. The truth condition for F is then this:

VM ,g (Fφ, w) = 1 iff for every model M ′that is a variant of M ,

VM ′,g (φ, w) = 1

But this isn’t signi�cantly different from the present two-dimensional approach.

10.3.1 ExamplesExample 1: for any wff, φ, �

2DF@φ→φ


Suppose otherwise: suppose V(F@φ, w, w) = 1 but V(φ, w, w) = 0.

Given the former, V(@φ, w, w) = 1 (given the truth condition for

‘F’); but then V(φ, w, w) = 1 (given the truth condition for @).

Contradiction.

Example 2: For some wffs φ, 2G2D

F@φ→φ

Ourφwill be @Ga↔Ga. General validity requires truth at all pairs

⟨v, w⟩ in all models. But in the following model, Vg (F@(@Ga↔Ga)→(@Ga↔Ga), c,d) = 0 (for any g ):

W = {c,d}D = {u}I (a) = u

I (G) = {⟨u,c⟩}

In this model, the referent of ‘a’ is in the extension of ‘G’ in world

c, but not in world d. That means that @Ga is true at ⟨c,d⟩ whereas

Ga is false at ⟨c,d⟩, and so @Ga↔Ga is false at ⟨c,d⟩. But F@φmeans that φ is true at all pairs of the form ⟨v, v⟩, and the for-

mula @Ga↔Ga is true at any such pair (in any model). Thus,

F@(@Ga↔Ga) true at ⟨c,d⟩ in this model.

Example 3: for some φ, 22Dφ→Fφ

In the model of the previous problem, the formula @Ga→F@Gais false at ⟨c, c⟩. The antecedent is true because the referent of ‘a’ is

in the extension of ‘G’ at c. The consequent is false because F@Gameans that ‘Ga’ is true at all pairs of the form ⟨v, v⟩, whereas ‘Ga’

is not true at ⟨d,d⟩ (since the referent of ‘a’ is not in the extension

of ‘G’ at d).

Example 4: If φ has no occurrences of @, then �2Dφ→Fφ

Let’s prove by induction that if φ has no occurrences of @, then

φ→Fφ is generally valid (i.e., true in any model at any world pair

⟨v, w⟩ under any variable assignment). The result then follows,

because general validity (truth at all pairs) obviously implies validity

(truth at pairs ⟨w, w⟩).


First, letφ be atomic. Then we’re trying to show that for any worlds

v, w, and any variable assignment g , Vg (Πα1 . . .αn→FΠα1 . . .αn, v, w) =1. Suppose otherwise—suppose that (i) Vg (Πα1 . . .αn, v, w) = 1and (ii) Vg (FΠα1 . . .αn, v, w) = 0. Given (ii), for some world, call

it v ′,Vg (Πα1 . . .αn, v ′, w) = 0, and so the ordered n-tuple of the

denotations of α1 . . .αn is not in the extension of Π at w, which

contradicts (i).

Now the inductive step. We must assume that φ and ψ obey our

statement, and show that complex formulas built from φ and ψ also

obey our statement. That is, we assume the inductive hypothesis:

(ih) φ and ψ have no occurrences of @, �G2D

φ→Fφand �

G2Dψ→Fψ

and we must show that the following are also generally valid:

∼φ→F∼φ(φ→ψ)→F(φ→ψ)∀αφ→F∀αφ2φ→F2φFφ→FFφ×φ→F×φ

∼ : Suppose otherwise—suppose V(∼φ→F∼φ, v, w) = 0 for some

v, w. So V(∼φ, v, w) = 1 and V(F∼φ, v, w) = 0. So V(φ, v, w) = 0,

and for some v ′,V(∼φ, v ′, w) = 0; and so V(φ, v ′, w) = 1. By (ih),

V(φ→Fφ, v ′, w) = 1, and so V(Fφ, v ′, w) = 1, and so V(φ, v, w) =1—contradiction.

→ : Suppose for some v, w,V(φ→ψ)→ F(φ→ψ), v, w) = 0. So (i)

V(φ→ψ, v, w) = 1 and V(F(φ→ψ), v, w) = 0. So, for some world,

call it u, V(φ→ψ, u, w) = 0, and so V(φ, u, w) = 1 and V(ψ, u, w) =0. Given the former and the inductive hypothesis, V(Fφ, u, w) = 1,

and so V(φ, v, w) = 1. And so, given (i), V(ψ, v, w) = 1, and so,

given the inductive hypothesis, V(Fψ, v, w) = 1, and so V(ψ, u, w) =1, which contradicts (ii).

∀ : Suppose for some v, w, Vg (∀αφ, v, w) = 1, but Vg (F∀αφ, v, w) =0. Given the latter, for some v ′, Vg (∀αφ, v ′, w) = 0; and so, for


some u in the domain, Vgu/α(φ, v ′, w) = 0. Given the former, Vgu/α

(φ, v, w) =1; given (ih) it follows that Vgu/α

(Fφ, v, w) = 1, and so, Vgu/α(φ, v ′, w) =

1. Contradiction.

2 : suppose (i) V(2φ, v, w) = 1 and (ii) V(F2φ, v, w) = 0, for some

v, w. From (ii), V(2φ, v ′, w) = 0 for some v ′, and so V(φ, v ′, w ′) =0 for some w ′. Given (i), V(φ, v, w ′) = 1; and so, given (ih),

V(Fφ, v, w ′) = 1, and so V(φ, v ′, w ′) = 1. Contradiction.

F: suppose V(Fφ, v, w) = 1 and V(FFφ, v, w) = 0, for some v, w.

From the latter, V(Fφ, v ′, w) = 0 for some v ′, and so V(φ, v ′′, w) =0 for some v ′′, which contradicts the former.

×: suppose Vg (×φ, v, w) = 1 but Vg (F×φ, v, w) = 0, for some

v, w. Given the latter, Vg (×φ, v ′, w) = 0 for some v ′, and so

Vg (φ, w, w) = 0, which contradicts the former.

10.4 A philosophical application: necessity and apriority

The two-dimensional modal framework has been put to signi�cant philosophi-

cal use in the past twenty-�ve or so years.5

This is not the place for an extended

survey; rather, I will brie�y present the two-dimensional account of just one

philosophical issue: the relationship between necessity and a priority.

In Naming and Necessity, Saul Kripke famously presented putative examples

of necessary a posteriori statements and of contingent a priori statements:

Hesperus = Phosphorus

B (the standard meter bar) is one meter long

The �rst statement, Kripke argued, is necessary because whenever we try to

imagine a possible world in which Hesperus is not Phosphorus, we �nd that

we have in fact merely imagined a world in which ‘Hesperus’ and ‘Phosphorus’

denote different objects than they in fact denote. Given that Hesperus and

5For work in this tradition, see Stalnaker (1978, 2003a, 2004); Evans (1979); Davies and

Humberstone (1980); Hirsch (1986); Chalmers (1996, 2006); Jackson (1998); see Soames (2004)

for an extended critique.


Phosphorus are in fact one and the same entity—namely, the planet Venus—

there is no possible world in which Hesperus is different from Phosphorus, for

such a world would have to be a world in which Venus is distinct from itself.

Thus, the statement is necessary, despite its a posteriority: it took astronomical

evidence to learn that Hesperus and Phosphorus were identical; no amount

of pure rational re�ection would have suf�ced. As for the second statement,

Kripke argues that one can know its truth as soon as one knows that the phrase

‘one meter’ has its reference �xed by the description: “the length of bar B”.

Thus it is a priori. Nevertheless, he argues, it is contingent: bar B does not

have its length essentially, and thus could have been longer or shorter than one

meter.

On the face of it, the existence of necessary a priori or contingent a posterioristatements is paradoxical. How can a statement that is true in all possible worlds

be in principle resistant to a priori investigation? Worse, how can a statement

that might have been false be known a priori?The two-dimensional framework has been thought by some to shed light

on all this. Let’s consider the contingent a priori �rst. Let’s de�ne the following

notion of contingency:

φ is super�cially contingent in modelM at world w iff, for every

variable assignment g ,VM ,g (2φ, w, w) =VM ,g (2∼φ, w, w) = 0.

This corresponds, intuitively, to this: if we were sitting at w, and we uttered

3φ∧3∼φ, we’d speak the truth.

How should we formalize the notion of a priority? As a rough and ready

guide, let’s think of a sentence as being a priori iff it is 2D-valid—i.e., true at

every pair ⟨w, w⟩ of every model. In defense of this guide: we can think of the

truth value of an utterance of a sentence as being the valuation of that sentence

at the pair ⟨w, w⟩ in a model that accurately models the genuine possibilities,

and in which w accurately models the (genuine) possible world of the speaker.

So any 2D-valid sentence is invariably true whenever uttered; hence, if φ is

2D-valid, any speaker who understands his or her language is in a position to

know that an utterance of φ would be true.

Under these de�nitions, there are sentences that are super�cially contingent

but nevertheless a priori. Consider any sentence of the form: φ↔@φ. In any

model in which φ is true at w and false at some other world, the sentence

is super�cially contingent. But it is a priori, since, as we showed above, it is

2D-valid (though it’s not generally 2D-valid, as we also showed above.)


That was a relatively simple example; but one can give other examples that

are similar in spirit both to Kripke’s example of the meter bar, and to a related

example due to Gareth Evans (1979):

Bar B is one meter

Julius invented the zip

where bar B is the standard meter bar, and the “descriptive names” ‘one meter’

and ‘Julius’ are said to be “rigid designators” whose references are “�xed” by the

descriptions ‘the length of bar B ’ and ‘the inventor of the zip’, respectively. Now,

whether or not these sentences, understood as sentences of everyday English,

are indeed genuinely contingent and a priori depends on delicate issues in the

philosophy of language concerning descriptive names, rigid designation, and

reference �xing. Rather than going into all that, let’s construct some examples

that are similar to Kripke’s and Evans’s. Let’s simply stipulate that ‘one meter’

and ‘Julius’ are to abbreviate “actualized descriptions”: ‘the actual length of bar

B ’ and ‘the actual inventor of the zip’. With a little creative reconstruing in the

�rst case, the sentences then have the form: “the actual G is G”:

the actual length of bar B is a length of bar B

the actual inventor of the zip invented the zip

Now, these sentences are not quite a priori, since for all one knows, the G might

not exist—there might exist no unique length of bar B , no unique inventor of

the zip. So suppose we consider instead the following sentences:

If there is exactly one length of bar B , then the actual length of bar

B is a length of bar B

If there is exactly one inventor of the zip, then the actual inventor

of the zip invented the zip

Each has the form:

If there is exactly one G, then the actual G is G

∃x(Gx∧∀y(Gy→y=x))→∃x(@Gx∧∀y(@Gy→y=x)∧Gx)


Any sentence of this form is 2D-valid (though not generally 2D-valid), and is

super�cially contingent. So we have further examples of the contingent a prioriin the neighborhood of the examples of Kripke and Evans.

Various philosophers want to concede that these sentences are contingent

in one sense—namely, in the sense of super�cial contingency. But, they claim,

this is a relatively unimportant sense (hence the term ‘super�cial contingency’,

which was coined by Evans). In another sense, they’re not contingent at all.

Evans calls the second sense of contingency “deep contingency”, and de�nes it

thus (1979, p. 185):

If a deeply contingent statement is true, there will exist some state of

affairs of which we can say both that had it not existed the statement

would not have been true, and that it might not have existed.

The intended meaning of ‘the statement would not have been true’ is that the

statement, as uttered with its actual meaning, would not have been true. The

idea is supposed to be that ‘Julius invented the zip’ is not deeply contingent,

because we can’t locate the required state of affairs, since in any situation in

which ‘Julius invented the zip’ is uttered with its actual meaning, it is uttered

truly. So the Julius example is not one of a deeply contingent a priori truth.

Evans’s notion of deep contingency is far from clear. One of the nice things

about the two-dimensional modal framework is that it allows us to give a

clear de�nition of deep contingency. Davies and Humberstone (1980) give a

de�nition of deep contingency which is parallel to the de�nition of super�cial

contingency, but with F@ in place of 2:

φ is deeply contingent inM at w iff (for all g ) VM ,g (F@φ, w, w) = 0and VM ,g (F@∼φ, w, w) = 0.

Under this de�nition, the examples we have given are not deeply contingent.

To be sure, this de�nition is only as clear as the two-dimensional notions of

�xedness and actuality. The formal structure of the two-dimensional framework

is of course clear, but one can raise philosophical questions about how that

formalism is to be interpreted. But at least the formalism provides a clear

framework for the philosophical debate to occur.

Our discussion of the necessary a posteriori will be parallel to that of the

contingent a priori. Just as we de�ned super�cial contingency as the falsity of

the 2, so we can de�ne super�cial necessity as the truth of the 2:

φ is super�cially necessary inM at w iff (for all g ) VM ,g (2φ, w, w) = 1


How shall we construe a posteriority? Let’s follow our earlier strategy, and take

the failure to be 2D-valid as our guide.

But here we must take a bit more care. It’s quite a trivial matter to construct

models in which 2D-invalid sentences are necessarily true; and we don’t need

the two-dimensional framework to do it. We clearly don’t want to say that

‘Everything is a lawyer” is an example of the necessary a posteriori. But let Fstand for ‘is a lawyer’; we can construct a model in which the predicate F is

true of every member of the domain at any world, ∀xF x is true, and so is

super�cially necessary at every world, despite the fact that it is not 2D-valid.

But this is too cheap. We began by letting the predicate F stand for a predicate

of English, but then constructed our model without attending to the modal

fact that it’s simply not the case that it’s necessarily true that everything is a

lawyer. If F is indeed to stand for ‘is a lawyer’, we would need to include in any

realistic model—any model faithful to the modal facts—worlds in which not

everything is in the extension of F .

To provide nontrivial models of the necessary a posteriori, when we have

chosen to think of the nonlogical expressions of the language of QML as

standing for certain expressions of English, our strategy will be provide realisticmodels—models that are faithful to the real modal facts in relevant respects,

given the choice of what the nonlogical expressions stand—in which 2D-invalid

sentences are necessarily true. Now, since the notion of a “realistic model”

has not been made precise, the argument here will be imprecise; but in the

circumstances this imprecision is inevitable.

So: consider now, as a schematic example of an a posteriori and super�cially

necessary sentence:

If the actual F and the actual G exist, then they are identical

[∃x(@F x∧∀y(@F y→x=y))∧∃z(@Gz∧∀y(@Gy→z=y)]→∃x[@F x∧∀y(@F y→x=y)∧∃z(@Gz∧∀y(@Gy→z=y)∧ z=x)]

This sentence isn’t 2D-valid. Nevertheless, it is super�cially necessary in any

model and any world w in which F and G each have a single object in their

extension, no matter what the extensions of F and G are in other worlds in the

model. So whenever such a model is realistic (given what we let F and G stand

for), we will have our desired example.

We can �ll in this schema and construct an example similar to Kripke’s

Hesperus and Phosphorus example. Set aside controversies about the semantics

of proper names in natural language; let’s just stipulate that ‘Hesperus’ is to


be short for ‘the actual F ’, and that Phosphorus is to be short for ‘the actual

G’. And let’s think of F as standing for ‘the �rst heavenly body visible in the

evening’, and G for ‘the last heavenly body visible in the morning’. Then

(HP) If Hesperus and Phosphorus exist then they are identical

has the form ‘If the actual F and the actual G exist then they are identical’,

which was discussed in the previous paragraph. We may then construct a

realistic model in which F and G each have a single object in their extension in

some world, w, but in which they have different objects in their extensions in

other worlds. In such a model, the sentence

(2HP) 2(If Hesperus and Phosphorus exist then they are identical)

is true at ⟨w, w⟩, and so we again have our desired example: (HP) is super�cially

necessary, despite the fact that it is a posteriori (2D-invalid).

Isn’t it strange that (HP) is both a posteriori and necessary? The two-

dimensional response is: no, it’s not, since although it is super�cially necessary,

it isn’t deeply necessary in the following sense:

φ is deeply necessary inM at w iff (for all g ) VM ,g (F@φ, w, w) = 1

It isn’t deeply necessary because in any realistic model (given what F and Gcurrently stand for), there must be worlds and objects other than c and u, that

are con�gured as they are in the model below:

W = {c,d}D = {u,v}I (F ) = {⟨u,c⟩, ⟨u,d⟩}I (G) = {⟨u,c⟩, ⟨v,d⟩}

In this model, even though (2HP) is false at ⟨c, c⟩, still, F@(HP), i.e.:

F@{[∃x(@F x∧∀y(@F y→x=y))∧∃z(@Gz∧∀y(@Gy→z=y)]→∃x[@F x∧∀y(@F y→x=y)∧∃z(@Gz∧∀y(@Gy→z=y)∧ z=x)]}


is false at ⟨c, c⟩ (and indeed, at every pair of worlds), since (HP) is false at ⟨d,d⟩.And so, (HP) is not deeply necessary in this model.

One might try to take this two-dimensional line further, and claim that in

every case of the necessary a posteriori (or the contingent a priori), the necessity

(contingency) is merely super�cial. But defending this stronger line would

require more than we have in place so far. To take one example, return again

to ‘Hesperus = Phosphorus’, but now, instead of thinking of ‘Hesperus’ and

‘Phosphorus’ as abbreviations for actualized descriptions, let us represent them

by names in the logical sense (i.e., the expressions called “names” in the de�ni-

tion of well-formed formulas, which are assigned denotations by interpretation

functions in models). Thus, ‘Hesperus=Phosphorus’ is now represented as:

a=b . Consider the following model:

W = {c,d}D = {u,v}I (a) = u

I (b ) = u

The model is apparently realistic; it falsi�es no relevant modal facts. But the

sentence a=b is deeply necessary (at any world in the model). And yet it is aposteriori (2D-invalid).

Bibliography

Benacerraf, Paul and Hilary Putnam (eds.) (1983). Philosophy of Mathematics.2

ndedition. Cambridge: Cambridge University Press.

Boolos, George (1975). “On Second-Order Logic.” Journal of Philosophy 72:

509–527. Reprinted in Boolos 1998: 37–53.

— (1984). “To Be Is to Be the Value of a Variable (or to Be Some Values of

Some Variables).” Journal of Philosophy 81: 430–49. Reprinted in Boolos 1998:

54–72.

— (1985). “Nominalist Platonism.” Philosophical Review 94: 327–44. Reprinted

in Boolos 1998: 73–87.

— (1998). Logic, Logic, and Logic. Cambridge, MA: Harvard University Press.

Boolos, George and Richard Jeffrey (1989). Computability and Logic. 3rd

edition.

Cambridge: Cambridge University Press.

Chalmers, David (1996). The Conscious Mind. Oxford: Oxford University Press.

— (2006). “Two-Dimensional Semantics.” In Ernest Lepore and Barry C.

Smith (eds.), Oxford Handbook of Philosophy of Language, 574–606. New York:

Oxford University Press.

Cresswell, M. J. (1990). Entities and Indices. Dordrecht: Kluwer.

Cresswell, M.J. and G.E. Hughes (1996). A New Introduction to Modal Logic.London: Routledge.

Davies, Martin and Lloyd Humberstone (1980). “Two Notions of Necessity.”

Philosophical Studies 38: 1–30.

272

BIBLIOGRAPHY 273

Dummett, Michael (1973). “The Philosophical Basis of Intuitionist Logic.” In

H. E. Rose and J. C. Shepherdson (eds.), Proceedings of the Logic Colloquium,Bristol, July 1973, 5–49. Amsterdam: North-Holland. Reprinted in Benacerraf

and Putnam 1983: 97–129.

Enderton, Herbert (1977). Elements of Set Theory. New York: Academic Press.

Evans, Gareth (1979). “Reference and Contingency.” The Monist 62: 161–189.

Reprinted in Evans 1985.

— (1985). Collected Papers. Oxford: Clarendon Press.

Fine, Kit (1985). “Plantinga on the Reduction of Possibilist Discourse.” In

J. Tomberlin and Peter van Inwagen (eds.), Alvin Plantinga, 145–186. Dor-

drecht: D. Reidel.

Gamut, L. T. F. (1991a). Logic, Language, and Meaning, Volume 1: Introductionto Logic. Chicago: University of Chicago Press.

— (1991b). Logic, Language, and Meaning, Volume 2: Intensional Logic and LogicalGrammar. Chicago: University of Chicago Press.

Gibbard, Allan (1975). “Contingent Identity.” Journal of Philosophical Logic 4:

187–221. Reprinted in Rea 1997: 93–125.

Glanzberg, Michael (2006). “Quanti�ers.” In Ernest Lepore and Barry C.

Smith (eds.), The Oxford Handbook of Philosophy of Language, 794–821. Oxford

University Press.

Harper, William L., Robert Stalnaker and Glenn Pearce (eds.) (1981). Ifs: Con-ditionals, Belief, Decision, Chance, and Time. Dordrecht: D. Reidel Publishing

Company.

Hirsch, Eli (1986). “Metaphysical Necessity and Conceptual Truth.” In

Peter French, Theodore E. Uehling, Jr. and Howard K. Wettstein (eds.),

Midwest Studies in Philosophy XI: Studies in Essentialism, 243–256. Minneapolis:

University of Minnesota Press.

Hodes, Harold (1984a). “On Modal Logics Which Enrich First-order S5.”

Journal of Philosophical Logic 13: 423–454.

BIBLIOGRAPHY 274

— (1984b). “Some Theorems on the Expressive Limitations of Modal Lan-

guages.” Journal of Philosophical Logic 13: 13–26.

Jackson, Frank (1998). From Metaphysics to Ethics: A Defence of Conceptual Analysis.Oxford: Oxford University Press.

Kripke, Saul (1972). “Naming and Necessity.” In Donald Davidson and Gilbert

Harman (eds.), Semantics of Natural Language, 253–355, 763–769. Dordrecht:

Reidel. Revised edition published in 1980 as Naming and Necessity (Cambridge,

MA: Harvard University Press).

Lemmon, E. J. (1965). Beginning Logic. London: Chapman & Hall.

Lewis, C. I. (1918). A Survey of Symbolic Logic. Berkeley: University of California

Press.

Lewis, C. I. and C. H. Langford (1932). Symbolic Logic. New York: Century

Company.

Lewis, David (1973). Counterfactuals. Oxford: Blackwell.

— (1977). “Possible-World Semantics for Counterfactual Logics: A Rejoinder.”

Journal of Philosophical Logic 6: 359–363.

— (1979). “Scorekeeping in a Language Game.” Journal of Philosophical Logic 8:

339–59. Reprinted in Lewis 1983: 233–249.

— (1983). Philosophical Papers, Volume 1. Oxford: Oxford University Press.

Linsky, Bernard and Edward N. Zalta (1994). “In Defense of the Simplest

Quanti�ed Modal Logic.” In James Tomberlin (ed.), Philosophical Perspectives8: Logic and Language, 431–458. Atascadero: Ridgeview.

— (1996). “In Defense of the Contingently Nonconcrete.” Philosophical Studies84: 283–294.

Loewer, Barry (1976). “Counterfactuals with Disjunctive Antecedents.” Journalof Philosophy 73: 531–537.

Lycan, William (1979). “The Trouble with Possible Worlds.” In Michael J.

Loux (ed.), The Possible and the Actual, 274–316. Ithaca: Cornell University

Press.

BIBLIOGRAPHY 275

Mendelson, Elliott (1987). Introduction to Mathematical Logic. Belmont, Cali-

fornia: Wadsworth & Brooks.

Plantinga, Alvin (1983). “On Existentialism.” Philosophical Studies 44: 1–20.

Priest, Graham (2001). An Introduction to Non-Classical Logic. Cambridge:

Cambridge University Press.

Prior, A. N. (1967). Past, Present, and Future. Oxford: Oxford University Press.

— (1968). Papers on Time and Tense. London: Oxford University Press.

Quine, W. V. O. (1948). “On What There Is.” Review of Metaphysics 2: 21–38.

Reprinted in Quine 1953a: 1–19.

— (1953a). From a Logical Point of View. Cambridge, Mass.: Harvard University

Press.

— (1953b). “Mr. Strawson on Logical Theory.” Mind 62: 433–451.

Rea, Michael (ed.) (1997). Material Constitution. Lanham, Maryland: Rowman

& Little�eld.

Russell, Bertrand (1905). “On Denoting.” Mind 479–93. Reprinted in Russell

1956: 41–56.

— (1956). Logic and Knowledge. Ed. Robert Charles Marsh. New York: G.P.

Putnam’s Sons.

Sher, Gila (1991). The Bounds of Logic: A Generalized Viewpoint. Cambridge,

Mass.: MIT Press.

Sider, Theodore (2003). “Reductive Theories of Modality.” In Michael J. Loux

and Dean W. Zimmerman (eds.), Oxford Handbook of Metaphysics, 180–208.

Oxford: Oxford University Press.

Soames, Scott (2004). Reference and Description: The Case against Two-Dimensionalism. Princeton: Princeton University Press.

Stalnaker, Robert (1968). “A Theory of Conditionals.” In Studies in LogicalTheory: American Philosophical Quarterly Monograph Series, No. 2. Oxford:

Blackwell. Reprinted in Harper et al. 1981: 41–56.

BIBLIOGRAPHY 276

— (1978). “Assertion.” In Peter Cole and Jerry Morgan (eds.), Syntax and Se-mantics, Volume 9: Pragmatics, 315–332. New York: Academic Press. Reprinted

in Stalnaker 1999: 78–95.

— (1981). “A Defense of Conditional Excluded Middle.” In Harper et al.

(1981), 87–104.

— (1999). Context and Content: Essays on Intentionality in Speech and Thought.Oxford: Oxford University Press.

— (2003a). “Conceptual Truth and Metaphysical Necessity.” In Stalnaker

(2003b), 201–215.

— (2003b). Ways a World Might Be. Oxford: Oxford University Press.

— (2004). “Assertion Revisited: On the Interpretation of Two-Dimensional

Modal Semantics.” Philosophical Studies 118: 299–322. Reprinted in Stalnaker

2003b: 293–309.

von Fintel, Kai (2001). “Counterfactuals in a Dynamic Context.” In Ken Hale:A Life in Language, 123–152,. Cambridge, MA: MIT Press.

Westerståhl, Dag (1989). “Quanti�ers in Formal and Natural Languages.” In

D. Gabbay and F. Guenther (eds.), Handbook of Philosophical Logic, volume 4,

1–131. Dordrecht: Kluwer.

Williamson, Timothy (1998). “Bare Possibilia.” Erkenntnis 48: 257–273.

— (2002). “Necessary Existents.” In A. O’Hear (ed.), Logic, Thought andLanguage, 233–51. Cambridge: Cambridge University Press.

Documents

Ted Draft] Logic for Philosophers