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

1

• No new reading for Wednesday. Keep working on
  chapter 5, section 3.
• Exam 3 is next Monday

2

• Predicate logic, motivation
• Simone is a philosopher.
• Simone is female.
• Therefore, there exists at least one female
  philosopher.
• Valid or Invalid?
• Clearly valid.

3

• Statement logic gives the wrong answer. The only
  form it recognizes is an invalid one.
• P, F \ O
• Try a truth-table. The argument is easily proven
  invalid.
• Try a natural deduction proof. Youll get nowhere.

4

• The validity of the argument depends partly on
  the attribution of properties (category
  membership) to an individual.
• We need separate symbols for subjects and
  predicates. We also need some way of expressing
  the indefinite idea that there exists at least
  one thing with certain properties.
• Predicate logic (a.k.a. the predicate calculus)
  to the rescue.

5

• 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.

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• 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_.

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• 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.

8

• 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.
• Theres another operator, the universal
  quantifier, more about which soon.

9

• 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.

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• Our original argument becomes
• Ps
• Fs
• \ (x)(Px Fx)
• (Dictionary P_ _ is a philosopher F_ _ is
  female s Simone)
• Even if we dont 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).

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• Scope and Binding in predicate logic
• Scope A quantifiers 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.

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• --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.

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• A statement that contains at least one free
  variable (but is otherwise well-formed) is an
  open sentence. These count as formulae, but they
  dont have truth-conditions.
• When symbolizing in predicate logic, the result
  should never be an open sentence (i.e., no free
  variables allowed when translating).

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• (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.

15
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
  shes not American. (Dictionary P_ _ is a
  philosopher F_ _ is female A_ _is American
  s Simone)
• (Ps Fs) As

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• Problems on p 158