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PL continued: Quantifiers and the syntax of PL

PL continued: Quantifiers and the syntax of PL. Quantifiers of PL Quantifier symbols Variables used with quantifiers Truth functional compounds of PL The formal syntax of PL Vocabulary Quantifier of PL Atomic formula of PL Recursive definition of ‘formula of PL ’.

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PL continued: Quantifiers and the syntax of PL

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  1. PL continued:Quantifiers and the syntax of PL • Quantifiers of PL • Quantifier symbols • Variables used with quantifiers • Truth functional compounds of PL • The formal syntax of PL • Vocabulary • Quantifier of PL • Atomic formula of PL • Recursive definition of ‘formula of PL’

  2. PL continued:Quantifiers and the syntax of PL • Kinds of formulas of PL • Main logical operators • Subformulas (immediate and otherwise) • Variables: bound and free • Sentences of PL

  3. PL continued:Quantifiers and the syntax of PL Quantity terms: all, each, everyone, everything, someone, something, no one, none, nothing Quantifier symbols:  and  Quantifiers: (x): A universal quantifier ‘Each x is such that…’ (x): An existential quantifier ‘There is at least one x such that…’ (or ‘there is some x such that …’)

  4. PL continued: Quantifiers and the syntax of PL UD: people in Michael’s office Lxy: x likes y m: Michael r: Rita s: Sue ‘Michael likes everyone’ ‘Each person is such that Michael likes them’ (x) Lmx Note: the variable ‘x’ appears twice: as part of the quantifier and as part of the expression that follows it.

  5. PL continued: Quantifiers and the syntax of PL ‘Michael likes someone’ (x) Lmx ‘Everyone likes Michael’ (x) Lxm Michael doesn’t like anyone. ~(x) Lmx Michael doesn’t like some people. (x) ~Lmx

  6. PL continued: Quantifiers and the syntax of PL Variables used in quantifiers and in sentences of PL: w, x, y and z They serve as placeholders for individual variables in the specification of predicates, such as Lxy: x likes y They can be replaced by constants, such as ‘m’ in Lmm: (‘Michael likes himself’) And they serve as placeholders for terms such as ‘thing’ in ‘something’, ‘one’ in ‘someone’ ‘body’ in ‘somebody’

  7. PL continued: Quantifiers and the syntax of PL We can use any of the 4 variables in specifying predicates (for example: Ey: y is easygoing) And any of the four variables in quantifiers and expressions that include quantifiers: (y), (w), and (z) are all quantifiers Finally, we can use (z) Lmz to symbolize ‘Michael likes everyone’ Even if our symbolization key includes: Lxy: x likes y

  8. So far, we have considered sentences that include only one quantifier, one predicate, and, in some cases, a tilde. But we can easily form truth-functional compounds of such sentences: UD: people in Michael’s office Ex: x is easygoing Lxy: x likes y m: Michael r: Rita ‘Either everyone is easygoing or no one is’ (x) Ex v (x) ~Ex OR (x) Ex v ~(x) Ex BUT NOT AS: (x) Ex v ~(x) Ex

  9. And because of what we’ve said about variables, we can also symbolize: ‘Either everyone is easygoing or no one is’ as: (x) Ex v (w) ~Ew (y) Ey v ~(z) Ez We can symbolize ‘If Michael is easygoing, everyone is’ as Em  (x) Ex and ‘Michael likes everyone but Rita doesn’t’ as (x) Lmx & ~(y) Lry

  10. Again, using the symbolization key, we can symbolize ‘If anyone is easygoing, Michael is’ as (x) Ex  Em and ‘Rita is easygoing if and only if everyone is’ as Er  (x) Ex

  11. Why we don’t actually need both existential and universal quantifiers. Any sentence of the form: ‘Everything is this or that’, say ‘Everyone likes Michael’, which can be symbolized as (x) Lxm Can be rephrased as: ‘There is nothing that is not this or that’ (‘There isn’t anyone who doesn’t like Michael’) ~(x) ~Lxm

  12. And any sentence of the form ‘Something is this or that’ (such as ‘Michael likes someone’), which can be symoblized as (x) Lmx can be paraphrased as: ‘It is not the case that Michael doesn’t like everyone’ or ‘It is not the case that Michael likes no one’ and symbolized as: ~(x) ~Lmx

  13. The syntax of PL Vocabulary: Sentence letters of PL: Capital Roman letters, A through Z, with or without subscripts (just the sentence letters of SL) Predicates of PL: Capital letters, A through Z, with or without subscripts followed by one or more variables: Ax, Axy, Axyz… Bx, Bwz, Bwzx.. Individual constants of PL: lowercase Roman letters, a through v, without or without subscripts Individual variables of PL: the lowercase Roman letters, ‘w’ through ‘z’ with or without subscripts. Truth-functional connectives: ~ & v   Quantifier symbols:   Punctuation: ( ) [ ]

  14. The syntax of PL 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 PL: a sequence of not necessarily distinct elements of the vocabulary of PL. Examples: (((a  bba) A  Fx (x) Txx (x) (y) Fxy

  15. 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.

  16. The formal syntax of PL Quantifier 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)

  17. Recursive definition of a formula of PL • Every atomic formula of PL is a formula of PL. • If P is a formula of PL, so is ~P. • If P is a formula of PL, so are (P & Q), (P v Q), (P  Q), and (P  Q). • 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. • Nothing else is a formula of PL. Step 4 is to rule out expressions such as: (x) (x) Lx

  18. 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

  19. 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.

  20. 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.

  21. Immediate Formula Subformula MLO Type Gadz Gadz none atomic ~Gadz Gadz ~ truth-functional (z) Gadz Gadz (z) quantified ~ (z) Gadz (z) Gadz ~ truth-functional Scope 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.

  22. 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.

  23. Sentences of PL Which of the following are sentences of PL? • (x) Bx  ~(x) ~Bx • (x) Bx  ~Bx • (x) (Bx  ~Bx) • (x) Fxa • (x) Fya • G & (y) Cyy • ~(Bxa & Bax) v (x) (Gx) • ~~ Dbc  (x) Dbx 1, 3, 4, 6, and 8

  24. Sentences of PL Substitution instances We use P(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 instanceof 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.

  25. Sentences of PL Homework: 7.5E Exercises 1-3 7.6E Exercises 1 and 2 Read section 7.7

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