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Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Truth and Paradox
Leon HorstenUniversity of Bristol
Summer School onSet Theory and Higher-Order Logic
London, 1–6 August 2011
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Structure of the Tutorial
1. Introduction: Truth and the Liar
2. Definition, Models, Axioms
3. Typed Disquotational Theories
4. Typed Compositional Theories
5. Type-free Disquotational Theories
6. Type-free Compositional Theories
7. The Revision Theory of Truth
8. Selected References
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Tarski-biconditionals
The disquotational intuition: supposing φ is tantamount tosupposing the truth of φ, and conversely.
A plausible axiom scheme for truth:
‘φ’ is true if and only if φ.
I This scheme has infinitely many instances
I Instances of this scheme are called Tarski-biconditionals.
(after the logician Alfred Tarski)
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Paradoxical sentences
The liar sentence L:
This sentence is not true.
The truth teller:
This sentence is true.
I L will be our paradigmatic test case
I but we should be aware that it is not the onlyparadoxical sentence
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Semantic paradox
The argument of the liar paradox:
Proof.It is an instance of the Tarski biconditional scheme that L istrue if and only if L. But L if and only if L is not true —forthis is what L says of itself. So L is true if and only if L isnot true: a short truth table calculation convinces us that wehave lapsed into inconsistency.
How credible is this argument?
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
The framework
For simplicity, we want a toy model for English.
Language: The language of truth (LT ): the language ofarithmetic plus a truth predicate T .
Background theory: First-order Peano arithmetic (PAT ).[The truth predicate T is allowed in the induction scheme.]
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
The arithmetical background
Why arithmetic?
Coding
I PA can serve as a theory of syntax (via coding);
I PA allows the formation of self-referential sentences (viacoding).
Note: We will be sloppy about notation.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Defining truth
Tarski showed us how to give a definition of truth for aformal language in purely logical and mathematical terms.
material adequacy condition: a definition of truth fora language L should imply all the Tarski-biconditionals forsentences of L.
⇒ Metawissenschaft
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Defining first-order arithmetical truth insecond-order arithmetic
There is a first-order formula val+ that defines the atomicarithmetical truths.Consider the condition Ψ(X , φ):
φ atomic closed→ [X (φ)↔ val+(φ)] ∧∃ψ, λ : φ = ψ ∧ λ→ [X (φ)↔ X (ψ) ∧ X (λ)] ∧ . . .
Definitiontrue(φ) ≡ ∀X : ∀λΨ(X , λ)→ X (φ).
The second-order arithmetical predicate true(x) satisfiesTarski’s material adequacy condition.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Models for truth
A theory of truth for a language L should describe a class ofintended models for L.
[This thesis was popular in the 1970s and 1980s.]
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Universality
We want a theory of truth for our language:
We want a truth theory for English in English.
Metalanguage = Objectlanguage
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Definitions and universality
Problem for the definitional approach: A truthdefinition for L (objectlanguage) can only be given in alanguage (metalanguage) L′ that is essentially richer than L:
Theorem (Undefinability theorem I)
There is no first-order arithmetical formula that definesfirst-order arithmetical truth.
But perhaps a theory of truth for English as it now is canonly be given in future English . . .
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Models and universality
A class of models for a language L is described in anessentially richer metalanguage (incompleteness theorem).
Additional problem for the model-theoreticapproach: the domain of a model is a set
The domain of discourse of English does not form a set.
But perhaps the techniques can be adapted to proper classdomains . . .
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Axioms for Truth
ThesisA theory of truth for a language L should posit rules ofinference and / or axioms for L.
The problems of the definitional and the model-theoreticapproach are not applicable.
⇒ there is no immediately apparent obstacle to our universalambitions
DefinitionLet LT be LPA plus a primitive truth predicate T .
Aim: Formulate a truth theory for LT in LT .
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
The role of models and definitions
Models can have a strong heuristic force: they give pictures.
We can give truth definitions for fragments of our language.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Soundness and completeness
Desiderata for axiomatic truth theories:
I soundness
I truth theoretic completeness
More specific desiderata will be discussed at the end of thelectures.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
The naive theory of truth
Recall the disquotational intuition.
⇒ Take all the Tarski-biconditionals as your theory of truth.
The formal theory NT :
1. PAT
2. T (φ)↔ φ for all φ ∈ LT
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Godel’s diagonal lemma
Theorem (Godel)
For each formula φ(x) ∈ LT , there is a sentence λ ∈ LT
such that PAT proves
λ↔ φ(λ).
This sentence λ is self-referential.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Tarski’s undefinability theorem
Theorem (Undefinability theorem 2)
No consistent extension S of PAT proves T (φ)↔ φ for allφ ∈ LT .
Proof.Use the diagonal lemma to produce a (liar) sentence λ:
PAT ` λ↔ ¬T (λ).
If the theory S in question indeed proves T (φ)↔ φ for allφ ∈ LT , then in particular S proves T (λ)↔ λ. Puttingthese equivalences together, we obtain a contradiction in S:
S ` T (λ)↔ ¬T (λ).
Consequence: NT is inconsistent.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
The typed disquotational theory
Tarski’s diagnosis: the root of the disease lies inallowing the Tarski-biconditionals to regulate thetruth-conditions of sentences that themselves contain thetruth predicate (such as L).
⇒ typed truth theories
The disquotational theory DT :
DT1 PAT ;
DT2 T (φ)↔ φ for all φ ∈ LPA.
Tarski’s strictures are respected.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Soundness
Proposition
DT has a nice model.
Proof.Consider the model
M =: 〈N, {φ | φ ∈ LPA ∧ N |= φ}〉,
i.e., the model in which as the extension of the truthpredicate we take all arithmetical truths. An induction onthe length of proofs in DT verifies that M |= DT .
Consequence: DT is arithmetically sound.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
The Tarskian hierarchy 1
Conversation:
A: It is true that 0 = 0.B: What you have just said is true.
Proposition
DT 6` T (T (0 = 0))
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
The Tarskian hierarchy 2
The theory DT1:
1. PAT ,T1 ;
2. T (φ)↔ φ for all φ ∈ LPA;
3. T1(φ)↔ φ for all φ ∈ LT .
TheoremDT1 has a nice model.
Proposition
DT1 ` T1(T (0 = 0))
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
The Tarskian hierarchy 3
Extended conversation:
A: It is true that 0 = 0.B: What you have just said is true.C: Yes, B, that is very true.
⇒ the Tarskian hierarchy DT , DT1, DT2,. . .
⇒ the notion of truth is irrevocably fragmented
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Contextualist theories
(Burge, Gaifman, Barwise & Perry, Glanzberg, . . . )
ThesisTruth is a uniform but indexical concept.
Sentence L is not true0.Sentence L is true1.
Because of the indexical shift in extension of the truthpredicate between context 0 and context 1, this is not acontradiction.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
The strengthened liar paradox
(S) Sentence S is not true in any context.
I this is called a strengthened liar sentence
I it is not hard to figure out that there is no context inwhich S can be coherently evaluated.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
The weakness of disquotationalism
The compositional intuition: truth distributes over thelogical connectives.
Proposition
For all φ ∈ LPA: DT ` T (φ) ∨ T (¬φ)
Proof.Already propositional logic alone proves φ ∨ ¬φ. Tworestricted Tarski-biconditionals are T (φ)↔ φ andT (¬φ)↔ ¬φ. Combining these facts yields the result.
Proposition (Tarski)
DT 0 ∀φ ∈ LPA : T (φ) ∨ T (¬φ)
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
The typed compositional theory
(Davidson...)
The compositional theory TC :
TC1 PAT ;
TC2 ∀ atomic φ ∈ LPA : T (φ)↔ val+(φ);
TC3 ∀φ ∈ LPA : T (¬φ)↔ ¬T (φ);
TC4 ∀φ, ψ ∈ LPA : T (φ ∧ ψ)↔ (T (φ) ∧ T (ψ));
TC5 ∀φ(x) ∈ LPA : T (∀xφ(x))↔ ∀xT (φ(x)).
TheoremTC has nice models, and DT ⊆ TC .
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Substitution
For a moment we have to be sticklers for notation...
A sentence of the form ∃xT (φ(x)) is really expressed alongthe following lines:
“There is a number x such that when the standardnumeral for x is substituted for the variable x inφ(x), a true sentence results.”
Thus a substitution function (expressible in the language ofarithmetic) appears in formulae such as TC5, but also in allthe other compositional axioms.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
True and true of
What if we do not have standard names for all elements ofthe language of discourse (R,. . . )?
⇒ we work with a satisfaction relation (‘true of’) and definethe truth predicate in terms of it.
The axiom TC5 then becomes (roughly):
∀x ,∀φ(y) : Sat(x ,∀yφ(y))↔ ∀zSat(z , φ(y)))
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Deflationism and conservativeness
Deflationism: The concept of truth does not play asubstantial role in philosophical, mathematical, scientificdebates.
DefinitionA theory of truth S is arithmetically conservative over PA iffor every sentence φ ∈ LPA, if S ` φ, then already PA ` φ.
Proposition
DT is arithmetically conservative over PA.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
The non-conservativeness of truth 1
TheoremTC ` ∀φ ∈ LPA : BewPA (φ)→ T (φ), where BewPA(...) isan arithmetical predicate that expresses provability in Peanoarithmetic in a natural way.
Proof.This is proved by an induction, inside TC , on the length ofproofs. [Here we need that T is allowed in the inductionscheme.]For the basis case, we have to prove that all the axioms oftrue. [For the subcase of mathematical induction we againneed that T is allowed in the induction scheme.]
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
The non-conservativeness of truth 2
Corollary
TC ` ¬BewPA(0 = 1).
Proof.This follows from the previous theorem and the fact thatDT ⊆ TC by instantiating 0 = 1 for φ.
So by Godel’s second incompleteness theorem, TC is notconservative over PA.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
The non-conservativeness of truth 3
More precise information:
DefinitionThe second-order system ACA contains full second-orderinduction but only those instances of Comprehension
∃X∀y : X (y)↔ φ(y)
where φ(y) contains no bound second-order quantifiers (anddoes not contain X free).
TheoremThe first-order arithmetical strength of TC is exactly that ofthe second-order system ACA.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Back to deflationism
Thesis (deflationism)
The concept of truth is not substantial but provides extraconceptual power.
⇒ An axiomatic theory of truth must be conservative butnoninterpretable.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
A deflationist truth theory 1
Fischer proposed a minimally adequate truth theory PT−
Definitiontot(φ(x)) ≡ ∀y : T (φ(y) ∨ T (¬φ(y))
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
A deflationist truth theory 2
PT-1 PAT without the induction axiom;
PT-2 ∀φ(x) ∈ LT : [tot(φ(x)) ∧ T (φ(0)) ∧ ∀y(T (φ(y))→T (φ(y + 1))]→ ∀xT (φ(x));
PT-3 ∀ atomic φ ∈ LPA : T (φ)↔ val+(φ);
PT-4 ∀ atomic φ ∈ LPA : T (¬φ)↔ ¬val+(φ);
PT-5 ∀φ, ψ ∈ LPA : T (φ ∧ ψ)↔ (T (φ) ∧ T (ψ));
PT-6 ∀φ, ψ ∈ LPA : T (¬(φ ∧ ψ))↔ (T (¬φ) ∨ T (¬ψ));
PT-7 ∀φ(x) ∈ LPA : T (∀xφ(x))↔ ∀xT (φ(x));
PT-8 ∀φ(x) ∈ LPA : T (¬∀xφ(x))↔ ∃xT (¬φ(x));
PT-9 ∀φ ∈ LPA : T (¬¬φ)↔ T (φ).
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
A deflationist truth theory 3
Observation: PT− is still highly compositional, but not ascompositional as TC .
TheoremPT− is conservative over PA but is not interpretable in PA.
PT− seems sufficient for capturing the technical use of theconcept of truth that is made in mathematics (such asmodel theory).
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Type-free truth
I Truth theories which are formulated in languages thatcontain truth predicates of different levels and whichprove iterated truth ascriptions only if the hierarchyconstraints are satisfied, are called typed theories oftruth.
I There also exist truth systems which contain a singletruth predicate but which do validate sentences of theform T (T (0 = 0)). These systems are called type-freetheories of truth (reflexive, semantically closed theories).
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
The strength of type-free disquotation
Theorem (McGee)
Any theory extending PA can be reaxiomatized by theaxioms of PA and a set of Tarski-biconditionals.
Proof.Consider an axiom ψ of S . Using the diagonal lemma, wecan find a sentence such that
λ↔ (T (λ)↔ ψ)
is provable in PA. This equivalence is logically equivalent to:
ψ ↔ (T (λ)↔ λ).
So ψ is PA-provably equivalent to the Tarski-biconditionalT (λ)↔ λ.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Positive disquotation
(Halbach)
The theory PUTB is the theory given by PAT and the set ofall sentences
∀x (T (φ(x))↔ φ(x)),
where in the formula φ(x), T must not occur in the scope ofan odd number of negation symbols in the formula φ.
TheoremPUTB ` T (T (0 = 0))
But: PUTB 6` ∀φ ∈ LPA : T (¬φ)↔ ¬T (φ)
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Going partial
(Kripke)
Structure of the argument of the liar paradox:
1. L ∨ ¬L
2. L⇒ ⊥
3. ¬L⇒ ⊥
So, ⊥
Moral: Do not assert the law of excluded third.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Partial models
I The aim is build a model for the language LT in stages
I The arithmetical vocabulary is interpreted throughoutas in the standard model N.
I The truth predicate T will be the only partiallyinterpreted symbol: it will receive, at each ordinal stage,an extension E and an anti-extension A.
M = (E ,A)
Note:
1. E ∩ A = ∅2. E ∪ A does not typically exhaust the domain, for
otherwise T would be a total predicate.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Strong Kleene valuation
I For any atomic formula Fx1...xn :
1. M |=sk Fk1...kn if the n-tuple (k1, ..., kn) belongs to theextension of F ;
2. M |=sk ¬Fk1...kn if the n-tuple (k1, ..., kn) belongs tothe anti-extension of F .
I For any formulae φ, ψ :
1. M |=sk φ ∧ ψ if and only if M |=sk φ and M |=sk ψ;2. M |=sk ¬ (φ ∧ ψ) if and only if either M |=sk ¬φ or M|=sk ¬ψ (or both);
3. M |=sk ∀xφ if and only if for all n, M |=sk φ(n/x);4. M |=sk ¬∀xφ if and only if for at least one n, M|=sk ¬φ(n/x);
5. M |=sk ¬¬φ if and only if M |=sk φ.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
A chain of partial models 1
M0 = (E0,A0) =: (∅, ∅) ;
Eα+1 =: {φ ∈ LT |Mα |=sk φ} and
Aα+1 =: {φ ∈ LT |Mα |=sk ¬φ} ;
For λ limit ordinal, we set:
Eλ =:⋃κ<λ
Eκ,
Aλ =:⋃κ<λ
Aκ.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
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A chain of partial models 2
This inductive definition gives rise to a transfinite chain ofpartial models:
(E0,A0), (E1,A1), . . . , (Eω,Aω), . . .
Theorem (monotonicity)
For any two partial models (Ea,Aa), (Eb,Ab), if Ea ⊆ Eb andAa ⊆ Ab, then
{φ | (Ea,Aa) |=sk φ} ⊆ {φ | (Eb,Ab) |=sk φ}.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
The minimal fixed point model
A consequence of this is that:
Corollary
For all α, β with α < β, we have:
{φ | (Eα,Aα) |=sk φ} ⊆ {φ | (Eβ,Aβ) |=sk φ}.
Proposition
For some ordinal ρ, Eρ = Eρ+1 and Aρ = Aρ+1.
This ordinal ρ is called the (Strong Kleene) minimal fixedpoint, and (Eρ,Aρ) is called the minimal fixed point model.
Theorem (Kripke)
The minimal Strong Kleene fixed point is a Π11-complete set.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
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The RevisionTheory
SelectedReferences
Almost having it all
The minimal fixed point almost makes the unrestrictedTarski-biconditionals true:
TheoremFor all sentences φ in LT :
Mρ |=sk φ⇔Mρ |=sk T (φ).
Proof.First, suppose Mρ |=sk φ. Then by the definition of thesequence of partial models, Mρ+1 |=sk T (φ). But since Mρ
is a fixed point, we have Mρ+1 = Mρ. So Mρ |=sk T (φ).Second, suppose Mρ |=sk T (φ). Then there must be anordinal α < ρ such that Mα |=sk φ. And therefore, bymonotonicity, Mρ |=sk φ.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Paradoxical sentences
DefinitionClosing off a partial model means letting its anti-extensionspill over into all of the complement of its extension.
TheoremMρ 2sk T (L) and Mρ 2sk ¬T (L).
Proof.By closing off the model Mρ we obtain a classical modelMc
ρ. Suppose that Mρ |=sk T (L), and thereby Mρ |=sk L.Then by monotonicity, also Mc
ρ |=sk T (L) and Mcρ |=sk L.
But since Mcρ is just a classical model, the diagonal lemma
holds in it. So we have
Mcρ |=sk L↔ ¬T (L).
But putting these three facts together gives us acontradiction. So we deny our supposition and conclude thatMρ 2sk T (L).
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Other valuation schemes
So far we have been using the Strong Kleene valuationscheme...
I Weak Kleene: “if a component is gappy, then the wholeis gappy.” (Feferman)
I Paraconsistent: “Truth gluts instead of truth gaps ”(Priest)
I . . .
Formally, it does not make a lot of difference . . .
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Supervaluation
In the supervaluation approach, a formula φ ∈ LT isregarded as true in a partial model M = (E ,A) if and only ifφ is true in all total (or classical) models Mc = 〈N, C〉 forwhich the interpretation C of the truth predicate is such thatE ⊆ C and A ⊆ N\C.
Similarly, we say that a formula φ ∈ LT is regarded as falsein a partial model M = (E ,A) if and only if φ is false in alltotal (or classical) models Mc = 〈N, C〉 for which theinterpretation C of the truth predicate is such that E ⊆ Cand A ⊆ N\C.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Properties of supervaluation fixed points
The laws of classical logic are supervaluation-true in theminimal fixed point of the supervaluation scheme.
The supervaluation fixed point is not compositional.
TheoremThe supervaluation fixed point is a complete Π1
1 set.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Axiomatising Strong Kleene Fixed Points
The theory KF :
KF1 ∀ atomic φ ∈ LPA : T (φ)↔ val+(φ);
KF2 ∀ atomic φ ∈ LPA : T (¬φ)↔ val−(φ);
KF3 ∀φ ∈ LT : T (¬¬φ)↔ T (φ);
KF4 ∀φ, ψ ∈ LT : T (φ ∧ ψ)↔ (T (φ) ∧ T (ψ));
KF5 ∀φ, ψ ∈ LT : T (¬ (φ ∧ ψ))↔ (T (¬φ) ∨ T (¬ψ));
KF6 ∀φ (x) ∈ LT : T (∀xφ (x))↔ ∀yT (φ (y));
KF7 ∀φ (x) ∈ LT : T (¬∀xφ (x))↔ ∃yT (¬φ (y));
KF8 ∀φ ∈ LT : T (T (φ))↔ T (φ);
KF9 ∀φ ∈ LT : T (¬T (φ))↔ T (¬φ);
KF10 ∀φ ∈ LT : ¬(T (φ) ∧ T (¬φ)).
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
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Inner logic and outer logic
TheoremKF has nice models.
Proof.The closed off minimal fixed point model.
DefinitionThe inner logic of KF is the collection of sentences φ ∈ LT
such that KF ` T (φ).
Proposition
KF ` L ∧ ¬T (L), where L is the liar sentence.
So the inner logic of KF does not coincide with the externallogic of KF .
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
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SelectedReferences
Ramified Analysis
Let us add new second-order variables X 1,Y 1, . . . to thelanguage of second-order arithmetic. Add a newcomprehension principle
∃X 1∀y : X 1(y)↔ φ(y)
where φ(y) contains no bound second-order quantifiers ofthe new kind (and does not contain X 1 free).
The resulting system is called ACA1.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
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The RevisionTheory
SelectedReferences
The strength of KF
We can go on, and construct ACAn, . . . , ACA<ω...
Definitionε0 is the least upper bound of {ω, ωω, ωωω
, . . .}.
Theorem (Feferman)
The first order arithmetical theorems of KF are exactly thoseof ACAε0 .
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
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The RevisionTheory
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Truth and Classes
Definitionx ∈ y ≡ y ∈ LT ∧ T (y(x))
Then KF can be seen as a theory of definable or predicativeclasses (Feferman).
⇒ KF as a way of ascending from PA to predicative analysis.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
A way out?
What is the source of the divergence between inner andouter logic?
Motivation of Kripke’s theory = partial
Logic of KF = classical
⇒ Mismatch
Solution: Formalise Kripke’s theory in partial logic!
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
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The RevisionTheory
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Logic and mathematics
Restricted conditionalisation:
T (φ) ∨ T (¬φ) φ (Hyp)...ψ (Hyp)
φ→ ψ
Rule of induction:
φ(0) φ(x) (Hyp)...
φ(s(x)) (Hyp)
∀xφ(x)
Here φ(x) ranges over all formulae of LT .
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
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The system PKF
PKF1val+(t1 = t2)
T (t1 = t2)
T (t1 = t2)
val+(t1 = t2)
PKF2T (φ) ∧ T (ψ)
T (φ ∧ ψ)
T (φ ∧ ψ)
T (φ) ∧ T (ψ)
PKF3T (φ) ∨ T (ψ)
T (φ ∨ ψ)
T (φ ∨ ψ)
T (φ) ∨ T (ψ)
PKF4∀xT (φ(x))
T (∀xφ(x))
T (∀xφ(x))
∀xT (φ(x))
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
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The system PKF Ctd
PKF5∃xT (φ(x))
T (∃xφ(x))
T (∃xφ(x))
∃xT (φ(x))
PKF6T (φ)
T (T (φ))
T (T (φ))
T (φ)
PKF7¬T (φ)
T (¬φ)
T (¬φ)
¬T (φ)
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
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Properties of PKF
TheoremInner logic of PKF = Outer logic of PKF
TheoremPKF holds in all fixed point models of the Strong Kleeneconstruction.
TheoremThe first-order arithmetical theorems of PKF are those ofACAωω .
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
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A chain of classical models
M0 =: 〈N, ∅〉
Mα+1 =: 〈N, {φ ∈ LT |Mα |= φ}〉.
For λ a limit ordinal:
Mλ =: 〈N, {φ ∈ LT | ∃β∀γ : (γ ≥ β ∧ γ < λ)⇒Mγ |= φ}〉.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
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Stable truth
Stable truth:A sentence φ ∈ LT is said to be stably true if at someordinal stage α, φ enters in the extension of the truthpredicate of Mα and stays in the extension of the truthpredicate in all later models.
Stable falsehood:A sentence φ ∈ LT is said to be stably false if at someordinal stage α, φ is outside the extension of the truthpredicate of Mα and stays out forever thereafter.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Nearly stable truth
Nearly stable truth:A sentence φ ∈ LT is said to be nearly stably true if forevery stage α after some stage β, there is a natural numbern such that for all natural numbers m ≥ n, φ is in theextension of the truth predicate of Mα+m.
Nearly stable falsehood:A sentence φ ∈ LT is said to be nearly stably false if forevery stage α after some stage β, there is a natural numbern such that for all natural numbers m ≥ n, φ is outside theextension of the truth predicate of Mα+m.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Properties of the (nearly) stable truths
The liar sentence is neither (nearly) stably true nor (nearly)stably false.
Proposition
The chain of revision models is eventually periodic.
TheoremThe (nearly) stable truths form a set that is morecomplicated than a complete Π1
1 set.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
The Friedman-Sheard theory 1
The theory FS :
FS1 PAT ;
FS2 ∀ atomic φ ∈ LPA : T (φ)↔ val+(φ);
FS3 ∀φ ∈ LT : T (¬φ)↔ ¬T (φ);
FS4 ∀φ, ψ ∈ LT : T (φ ∧ ψ)↔ T (φ) ∧ T (ψ);
FS5 ∀φ(x) ∈ LT : T (∀xφ(x))↔ ∀xT (φ(x)).
NEC From a proof of φ, infer T (φ);
CNEC From a proof of T (φ), infer φ.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
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The Friedman-Sheard theory 2
Proposition
Inner logic of FS = outer logic of FS
Proposition
FS is not stably true.
Proof.FS3 is not stably true: both the liar sentence and itsnegation are false at all limit ordinals
Proposition
FS is nearly stably true.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
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The strength of FS
FS only proves finite truth-iterations.
TheoremThe first-order arithmetical consequences of FS are exactlythose of ACAω.
Corollary
FS is arithmetically sound.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Omega-inconsistency
Even though FS is consistent and indeed even arithmeticallysound, it is in some sense “almost inconsistent”:
DefinitionAn arithmetical theory T is ω-inconsistent if for someformula φ(x), the theory T proves ∃xφ(x) while at the sametime for every n ∈ N, T proves ¬φ(n)
Theorem (McGee)
For some formula φ (x) ∈ LT : FS ` ∃xφ (x) andFS ` ¬φ (n) for all n ∈ N.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
A fork in the road
DT
TC
FS-like KF-like
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
Desiderata for axiomatic truth theories
I Coherence
I Tarski-biconditionals
I Compositionality
I Sustaining ordinary reasoning
I Strength
I Capturing a picture
Fulfillment of these norms is a matter of degree.
Truth and Paradox
Leon HorstenUniversity of
Bristol
Truth and the Liar
Definitions,Models, Axioms
TypedDisquotationalTheories
TypedCompositionalTheories
Type-freeDisquotationalTheories
Type-freecompositionaltheories
The RevisionTheory
SelectedReferences
References
I Cantini, A. Logical Frameworks for Truth andAbstraction. North-Holland, 1996.
I Feferman, S. Reflecting on Incompleteness. Journal ofSymbolic Logic 56(1991), p. 1–49.
I Gupta, A. & Belnap, N. The Revision Theory of Truth.MIT Press, 1993.
I Halbach, V. Axiomatic Theories of Truth. CambridgeUniversity Press, 2011.
I Horsten L. The Tarskian Turn. Deflationism andaxiomatic truth. MIT Press, 2011.
I McGee, V. Truth, Vagueness and Paradox. An essay onthe logic of truth. Hackett, 1991.
I Visser, A. Semantics and the Liar Paradox. In: D.Gabbay et al (eds) Handbook of Philosophical Logic.Volume 4. Reidel, 1984, p. 617–706.
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