PL continued:PL continued:Quantifiers and the syntax of PLQuantifiers and the syntax of PL

1.1. Quantifiers of Quantifiers of PLPL1.1. Quantifier symbolsQuantifier symbols2.2. Variables used with quantifiersVariables used with quantifiers

2.2. Truth functional compounds Truth functional compounds of PL3.3. The formal syntax of The formal syntax of PLPL

1.1. VocabularyVocabulary2.2. Quantifier of Quantifier of PLPL3.3. Atomic formula of Atomic formula of PLPL4.4. Recursive definition of ‘formula of Recursive definition of ‘formula of PLPL’’

PL continued:PL continued:Quantifiers and the syntax of PLQuantifiers and the syntax of PL

5. Kinds of formulas of PL1.1. Main logical operatorsMain logical operators

2.2. Subformulas (immediate and otherwise)Subformulas (immediate and otherwise)

6. Variables: bound and free

7.7. Sentences of Sentences of PLPL

PL continued:PL continued:Quantifiers and the syntax of PLQuantifiers and the syntax of PL

Quantity terms: all, each, everyone, everything, Quantity terms: all, each, everyone, everything, someone, something, no one, none, nothingsomeone, something, no one, none, nothing

Quantifier symbols: Quantifier symbols: and and Quantifiers: Quantifiers:

((x): A universal quantifierx): A universal quantifier

‘‘Each x is such that…’Each x is such that…’

((x): An existential quantifierx): An existential quantifier

‘‘There is at least one x such that…’ (or ‘there is There is at least one x such that…’ (or ‘there is some x such that …’)some x such that …’)

PL continued: Quantifiers and the syntax of PLPL continued: Quantifiers and the syntax of PL

UD: people in Michael’s officeUD: people in Michael’s office

Lxy: x likes yLxy: x likes y

m: Michaelm: Michael

r: Ritar: Rita

s: Sues: Sue

‘‘Michael likes everyone’Michael likes everyone’

‘‘Each person is such that Michael likes them’Each person is such that Michael likes them’

((x) Lmxx) Lmx

Note: the variable ‘x’ appears twice: as part of the Note: the variable ‘x’ appears twice: as part of the quantifier and as part of the expression that follows quantifier and as part of the expression that follows it.it.

PL continued: Quantifiers and the syntax of PLPL continued: Quantifiers and the syntax of PL‘‘Michael likes someone’Michael likes someone’((x) Lmxx) Lmx

‘‘Everyone likes Michael’Everyone likes Michael’((x) Lxmx) Lxm

Michael doesn’t like anyone.Michael doesn’t like anyone.~(~(x) Lmxx) Lmx

Michael doesn’t like some people.Michael doesn’t like some people.((x) ~Lmxx) ~Lmx

PL continued: Quantifiers and the syntax of PLPL continued: Quantifiers and the syntax of PL

Variables used in quantifiers and in sentences of Variables used in quantifiers and in sentences of PLPL::

w, x, y and zw, x, y and z

They serve as placeholders for individual variables in They serve as placeholders for individual variables in the specification of predicates, such asthe specification of predicates, such as

Lxy: x likes yLxy: x likes y

They can be replaced by constants, such as ‘m’ inThey can be replaced by constants, such as ‘m’ in

Lmm: (‘Michael likes himself’)Lmm: (‘Michael likes himself’)And they serve as placeholders for terms such as And they serve as placeholders for terms such as

‘thing’ in ‘something’,‘thing’ in ‘something’,‘‘one’ in ‘someone’one’ in ‘someone’‘‘body’ in ‘somebody’body’ in ‘somebody’

PL continued: Quantifiers and the syntax of PLPL continued: Quantifiers and the syntax of PLWe can use any of the 4 variables in specifying We can use any of the 4 variables in specifying

predicates (for example: Ey: y is easygoing)predicates (for example: Ey: y is easygoing)And any of the four variables in quantifiers and And any of the four variables in quantifiers and

expressions that include quantifiers:expressions that include quantifiers:((y), (y), (w), and (w), and (z) are all quantifiersz) are all quantifiers

Finally, we can useFinally, we can use((z) Lmz z) Lmz to symbolize ‘Michael likes everyone’to symbolize ‘Michael likes everyone’Even if our symbolization key includes:Even if our symbolization key includes:Lxy: x likes yLxy: x likes y

So far, we have considered sentences that include So far, we have considered sentences that include only one quantifier, one predicate, and, in some only one quantifier, one predicate, and, in some cases, a tilde.cases, a tilde.

But we can easily form truth-functional compounds of But we can easily form truth-functional compounds of such sentences:such sentences:

UD: people in Michael’s officeUD: people in Michael’s officeEx: x is easygoingEx: x is easygoingLxy: x likes yLxy: x likes ym: Michaelm: Michaelr: Ritar: Rita‘‘Either everyone is easygoing or no one is’Either everyone is easygoing or no one is’((x) Ex v (x) Ex v (x) ~Ex x) ~Ex OROR ((x) Ex v ~(x) Ex v ~(x) Ex x) Ex BUT NOT ASBUT NOT AS::((x) Ex v ~(x) Ex v ~(x) Exx) Ex

And because of what we’ve said about variables, we And because of what we’ve said about variables, we can also symbolize:can also symbolize:

‘‘Either everyone is easygoing or no one is’Either everyone is easygoing or no one is’as:as:

((x) Ex v (x) Ex v (w) ~Ew w) ~Ew ((y) Ey v ~(y) Ey v ~(z) Ezz) Ez

We can symbolize ‘If Michael is easygoing, everyone We can symbolize ‘If Michael is easygoing, everyone is’is’asasEm Em ( (x) Exx) Exand ‘Michael likes everyone but Rita doesn’t’and ‘Michael likes everyone but Rita doesn’t’asas((x) Lmx & ~(x) Lmx & ~(y) Lryy) Lry

Again, using the symbolization key, we can symbolizeAgain, using the symbolization key, we can symbolize

‘‘If anyone is easygoing, Michael is’If anyone is easygoing, Michael is’asas

((x) Ex x) Ex Em Em

andand

‘‘Rita is easygoing if and only if everyone is’Rita is easygoing if and only if everyone is’

asas

Er Er ( (x) Exx) Ex

Why we don’t actually Why we don’t actually needneed both existential both existential and universal quantifiers.and universal quantifiers.

Any sentence of the form:Any sentence of the form:‘‘Everything is this or that’, say ‘Everyone likes Everything is this or that’, say ‘Everyone likes

Michael’, which can be symbolized as Michael’, which can be symbolized as ((x) Lxmx) Lxm

Can be rephrased as:Can be rephrased as:‘‘There is nothing that is not this or that’ (‘There There is nothing that is not this or that’ (‘There

isn’t anyone who doesn’t like Michael’)isn’t anyone who doesn’t like Michael’)~(~(x) ~Lxmx) ~Lxm

And any sentence of the form ‘Something is this or And any sentence of the form ‘Something is this or that’ (such as ‘Michael likes someone’), which can that’ (such as ‘Michael likes someone’), which can be symoblized asbe symoblized as

((x) Lmxx) Lmx can be paraphrased as:can be paraphrased as:

‘‘It is not the case that Michael doesn’t like everyone’ It is not the case that Michael doesn’t like everyone’ or ‘It is not the case that Michael likes no one’or ‘It is not the case that Michael likes no one’

and symbolized as:and symbolized as:

~(~(x) ~Lmxx) ~Lmx

The syntax of The syntax of PLPL

Vocabulary:Vocabulary:Sentence letters of PLSentence letters of PL: Capital Roman letters, A : Capital Roman letters, A

through Z, with or without subscripts (just the through Z, with or without subscripts (just the sentence letters of SL)sentence letters of SL)

Predicates of PLPredicates of PL: Capital letters, A through Z, with or : Capital letters, A through Z, with or without subscripts followed by one or more without subscripts followed by one or more variables: Ax, Axy, Axyz… Bx, Bwz, Bwzx..variables: Ax, Axy, Axyz… Bx, Bwz, Bwzx..

Individual constants of PLIndividual constants of PL: lowercase Roman letters, : lowercase Roman letters, a through v, without or without subscriptsa through v, without or without subscripts

Individual variables of PLIndividual variables of PL: the lowercase Roman : the lowercase Roman letters, ‘w’ through ‘z’ with or without subscripts.letters, ‘w’ through ‘z’ with or without subscripts.

Truth-functional connectivesTruth-functional connectives: ~ & v : ~ & v Quantifier symbols: Quantifier symbols: Punctuation: Punctuation: ( ) [ ]( ) [ ]

The syntax of The syntax of PLPL

PL contains expressions.PL contains formulas. Not all expressions of PL are

formulas.PL contains sentences. Not all formulas of PL are

sentences.An expression of An expression of PLPL: : a sequence of not necessarily

distinct elements of the vocabulary of PL.Examples:(((a bba)A Fx(x) Txx(x) (y) Fxy

The formal syntax of PL

We use the bold letters P, Q, and R as meta variables ranging over expressions of PL.

We use bold ‘a’ as a meta variable to range over individual constants of PL.

We use bold ‘x’ as a meta variable to range over individual variables of PL.

The formal syntax of PL

Quantifier of PLQuantifier of PL: : An expression of PL of the form (x) or (x).

A quantifier contains a variable: (y) and (y) contain the variable ‘y’ and are ‘y’ quantifiers; (x) and (x) contain the variable ‘x’ and are ‘x’ quantifiers… and so forth for ‘w’ and ‘z’ variables and quantifiers.

Atomic formulas of PL: Every expression of PL that is either a sentence letter of PL or an n-place predicate of PL followed by n individual terms of PL. (E.g., ‘B’, ‘Fab’, and ‘Gxx’ are atomic formulas)

Recursive definition of a formula of PL

1. Every atomic formula of PL is a formula of PL.2. If P is a formula of PL, so is ~P.3. If P is a formula of PL, so are (P & Q), (P v Q),

(P Q), and (P Q).4. If P is a formula of PL that contains at least

one occurrence of x and no x-quantifier, then(x) and (x) are both formulas of PL.

5. Nothing else is a formula of PL.Step 4 is to rule out expressions such as:

(x) (x) Lx

Logical operators of PL

Logical operator of PL: An expression of PL that is either a quantifier or a truth functional connective.

Every formula of PL is either atomic, quantified, or a truth-functional compound.

It is atomic if it contains no logical operations. It is quantified if its main logical operator is a quantifier. It is truth-functional if its main logical operator is a truth-

functional connective.

Cabz No logical operator~Cabz & Gwx & is the main logical operator(y) (Gy Fya) (y) is the main logical operator(x) Gx v Cabz v is the main logical operator

Logical operators of PL

Main logical operators and subformulas:Defined by cases (see p. 300 in text):1. If P is an atomic formula of PL, then P contains no logical

operator, and hence no main logical operator, and P is the only subformula of P.

2. If P is a formula of PL of the form ~Q, then the tilde that precedes Q is the main logical operator of P and Q is the immediate subformula of P.

3. If P is a formula of PL of the form Q & R, Q v R, Q R, or Q R, then the binary connective between Q and R is the main logical operator of P, and Q and R are the immediate subformulas of P.

Logical operators of PL

Main logical operators and subformulas:Defined by cases (see p. 300 in text):4. If P is a formula of PL of the form (x) Q or (x)

Q, then the quantifier that occurs before Q is the main logical operator of P and Q is the immediate subformula of P.

5. If P is a formula of PL, then every subformula (immediate or not) of a subformula of P is a subformula of P and P is a subformula of itself.

ImmediateFormula Subformula MLO TypeGadz Gadz none atomic~Gadz Gadz ~ truth-functional(z) GadzGadz (z) quantified~ (z) Gadz (z) Gadz ~ truth-

functionalScope of a quantifier: The scope of a quantifier in a

formula P of SL is the subformula Q of which that quantifier is the main connective.

(z) Gadz : the scope of (z) is all of (z) Gadz .(z) Gadz & Em: the scope of (z) is (z) Gadz.(z) (Gadz & Ez): the scope of (z) is the entire

sentence.

Bound variable: An occurrence of a variable x in a formula of PL that is within the scope of an x-quantifier.

Free variable: An occurrence of a variable x in a formula of PL that is not bound.

Sentence of PL: A formula P of PL is a sentence of PL if and only if no occurrence of a variable in P is free.

(Gx) (x) Fx is not a sentence of PL.(x) Gx (x) Fx is a sentence of PL.(x) (Gx Fx)A formula of PL that is not a sentence of PL is called an

open sentence of PL.

Sentences of PL

Which of the following are sentences of PL?1. (x) Bx ~(x) ~Bx2. (x) Bx ~Bx3. (x) (Bx ~Bx)4. (x) Fxa5. (x) Fya6. G & (y) Cyy7. ~(Bxa & Bax) v (x) (Gx)8. ~~ Dbc (x) Dbx

1, 3, 4, 6, and 8

Sentences of PL

Substitution instancesWe useP(a/x)to specify the formula of PL that is like P except that it

contains the individual constant a wherever P contains the individual variable x.

So if P is ‘(x) Fx’, then P (a/x) is ‘Fa’Substitution instance of P: If P is a sentence of PL of

the form (x) Q or (x) Q, and a is an individual constant, then Q (a/x) is a substitution instance of P. The constant a is the instantiating constant.

Sentences of PL

Homework:7.5E Exercises 1-3

7.6E Exercises 1 and 2

Read section 7.7