No new reading for Wednesday. Keep No new reading for Wednesday. Keep working on chapter 5, section 3.working on chapter 5, section 3.

Exam #3 is next MondayExam #3 is next Monday

Predicate logic, motivationPredicate logic, motivation

Simone is a philosopher.Simone is a philosopher.

Simone is female.Simone is female.

Therefore, there exists at least one female Therefore, there exists at least one female philosopher.philosopher.

Valid or Invalid?Valid or Invalid?

Clearly valid.Clearly valid.

Statement logic gives the wrong answer. The Statement logic gives the wrong answer. The only form it recognizes is an invalid one.only form it recognizes is an invalid one.

P, F P, F O O

Try a truth-table. The argument is easily Try a truth-table. The argument is easily proven invalid.proven invalid.

Try a natural deduction proof. YouTry a natural deduction proof. You’’ll get ll get nowhere.nowhere.

The validity of the argument depends partly The validity of the argument depends partly on the attribution of properties (category on the attribution of properties (category membership) to an individual. membership) to an individual.

We need separate symbols for subjects and We need separate symbols for subjects and predicates. We also need some way of predicates. We also need some way of expressing the indefinite idea that there expressing the indefinite idea that there exists at least one thing with certain exists at least one thing with certain properties.properties.

Predicate logic (a.k.a. the predicate calculus) Predicate logic (a.k.a. the predicate calculus) to the rescue.to the rescue.

Predicate logic:Predicate logic: Three elements Three elements

I. Small-case I. Small-case ‘‘aa’’ through through ‘‘ss’’ serve as serve as individual constantsindividual constants. They refer to specific . They refer to specific persons, places, or things.persons, places, or things.

For example, For example, ‘‘ss’’ can stand for Simone. can stand for Simone.

II. Capital letters II. Capital letters ‘‘AA’’ through through ‘‘ZZ’’ abbreviate abbreviate predicatespredicates. .

A predicate is an atomic statement with the A predicate is an atomic statement with the subject deleted. subject deleted.

‘‘Kermit is greenKermit is green’’ is a simple statement. is a simple statement.

‘‘___ is green___ is green’’ is a predicate, symbolized by is a predicate, symbolized by ‘‘GxGx’’ or or ‘‘G_G_’’..

How do we symbolize How do we symbolize ‘‘Simone is a Simone is a philosopherphilosopher’’??

Using Using ‘‘P_P_’’ for for ‘‘___ is a philosopher___ is a philosopher’’, we get , we get

PsPs

Note that the individual constant comes after Note that the individual constant comes after the predicate, even though the individual the predicate, even though the individual constant corresponds to the subject of the constant corresponds to the subject of the sentence.sentence.

III. Quantifiers and variables:III. Quantifiers and variables:

((x), (x), (y), and (y), and (z) serve as z) serve as existential existential quantifiersquantifiers. They mean . They mean ““there exists at least there exists at least one thing of which the following is true.one thing of which the following is true.””

ThereThere’’s another operator, the universal s another operator, the universal quantifier, more about which soon.quantifier, more about which soon.

Quantifiers are like logical operators in that Quantifiers are like logical operators in that they determine truth conditions for the they determine truth conditions for the statements they apply to. statements they apply to.

To do so, they work together with attached To do so, they work together with attached individual variablesindividual variables: small-case x, y, and z, : small-case x, y, and z, which function like pronouns.which function like pronouns.

‘‘((x)(Px & Fx)x)(Px & Fx)’’ says, says, ““it is true of at least one it is true of at least one thing that it is a philosopher and it is female.thing that it is a philosopher and it is female.””

Our original argument becomesOur original argument becomes

PsPsFsFs ((x)(Px & Fx)x)(Px & Fx)

(Dictionary: P_: _ is a philosopher; F_: _ is female; (Dictionary: P_: _ is a philosopher; F_: _ is female; s: Simone)s: Simone)

Even if we donEven if we don’’t yet have a way of proving this t yet have a way of proving this argument is valid, we can see the reasoning. Use argument is valid, we can see the reasoning. Use &I and generalize (if Simone is a female &I and generalize (if Simone is a female philosopher, then there has to exist at least one philosopher, then there has to exist at least one female philosopher). female philosopher).

Scope and Binding in predicate logicScope and Binding in predicate logic

Scope: A quantifierScope: A quantifier’’s scope is calculated in s scope is calculated in the same way as the scope of a tilde: look the same way as the scope of a tilde: look directly to the right of the quantifier anddirectly to the right of the quantifier and

--if there is a predicate letter, the quantifier --if there is a predicate letter, the quantifier applies only to the atomic formula of which applies only to the atomic formula of which that predicate letter is a part.that predicate letter is a part.

--if there is a tilde, the quantifier applies to --if there is a tilde, the quantifier applies to the tilde and to whatever the tilde applies tothe tilde and to whatever the tilde applies to

--if there is a parenthesis (or bracket), the --if there is a parenthesis (or bracket), the quantifier applies to everything in that pair of quantifier applies to everything in that pair of parentheses (or brackets)parentheses (or brackets)

A variable is A variable is boundbound if and only if it is within if and only if it is within the scope of a quantifier that contains a the scope of a quantifier that contains a matching small-case letter. If a variable is matching small-case letter. If a variable is unbound, it is unbound, it is freefree..

A statement that contains at least one free A statement that contains at least one free variable (but is otherwise well-formed) is an variable (but is otherwise well-formed) is an open sentence. These count as formulae, but open sentence. These count as formulae, but they donthey don’’t have truth-conditions.t have truth-conditions.

When symbolizing in predicate logic, the When symbolizing in predicate logic, the result should never be an open sentence result should never be an open sentence (i.e., no free variables allowed when (i.e., no free variables allowed when translating).translating).

((x)(Px & Fx) & Axx)(Px & Fx) & Ax

In this formula, the last x is free. The scope In this formula, the last x is free. The scope of the existential quantifier extends only to of the existential quantifier extends only to the closed parenthesis.the closed parenthesis.

The final The final ‘‘xx’’ is like a pronoun with no referent. is like a pronoun with no referent. The statement is incomplete; it does not The statement is incomplete; it does not have definite truth-conditions.have definite truth-conditions.

Statements with Individual Statements with Individual Constants and No QuantifiersConstants 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 158Problems on p 158