Exam #3 is Friday. It will consist of proofs, plus symbolizations in predicate logic.

Predicate logic: Three elements I. Small-case ‘a’ through ‘s’ serve as individual constants. They refer to specific persons, places, or things. For example, ‘s’ can stand for Simone.

II. Capital letters ‘A’ through ‘Z’ abbreviate predicates. A predicate is an atomic statement with the subject deleted. ‘Kermit is green’ is a simple statement. ‘___ is green’ is a predicate, symbolized by ‘Gx’ or ‘G_’.

How do we symbolize ‘Simone is a philosopher’? Using ‘P_’ for ‘___ is a philosopher’, we get Ps Note that the individual constant comes after the predicate, even though the individual constant corresponds to the subject of the sentence.

III. Quantifiers and variables: (  x), (  y), and (  z) serve as existential quantifiers. They mean “there exists at least one thing of which the following is true.” (  x), (  y), and (  z) serve as universal quantifiers. They mean “the following statement true for everything in the universe.”

Quantifiers are like logical operators in that they determine truth conditions for the statements they apply to. To do so, they work together with attached individual variables: small-case x, y, and z, which function like pronouns. ‘(  x)(Px & Fx)’ says, “it is true of at least one thing that it is a philosopher and it is female.”

Our original argument becomes PsFs  (  x)(Px & Fx) (Dictionary: P_: _ is a philosopher; F_: _ is female; s: Simone) Even if we don’t yet have a way of proving this argument is valid, we can see the reasoning. Use &I and generalize (if Simone is a female philosopher, then there has to exist at least one female philosopher).

Scope and Binding in predicate logic Scope: A quantifier’s scope is calculated in the same way as the scope of a tilde: look directly to the right of the quantifier and --if there is a predicate letter, the quantifier applies only to the atomic formula of which that predicate letter is a part.

--if there is a tilde, the quantifier applies to the tilde and to whatever the tilde applies to --if there is a parenthesis (or bracket), the quantifier applies to everything in that pair of parentheses (or brackets) A variable is bound if and only if it is within the scope of a quantifier that contains a matching small-case letter. If a variable is unbound, it is free.

A statement that contains at least one free variable (but is otherwise well-formed) is an open sentence. These count as formulae, but they don’t have truth-conditions. When symbolizing in predicate logic, the result should never be an open sentence (i.e., no free variables allowed when translating).

(  x)(Px & Fx) & Ax In this formula, the last x is free. The scope of the existential quantifier extends only to the closed parenthesis. The final ‘x’ is like a pronoun with no referent. The statement is incomplete; it does not have definite truth-conditions.

Statements with Individual Constants and No Quantifiers Many of our symbolizations have no quantifiers, simply because there is no quantity term in (and no corresponding idea expressed by) the English sentence being symbolized. Example: Simone is a female philosopher, but she’s not American. (Dictionary: P_: _ is a philosopher; F_: _ is female; A_: _is American; s: Simone) (Ps & Fs) & ~ As

Problems on p 158, 165