Transcript Predicate logic, motivation

For Monday, read chapter 5, section 3. As
nongraded H.W., do exercise set I on p. 165.
Graded H.W. #6 is due Monday at the
beginning of class.
Exam #3 is next Friday. It will consist of
proofs, plus symbolizations in predicate logic.
Predicate logic, motivation
Simone is a philosopher.
Simone is female.
Therefore, there exists at least one female
philosopher.
Valid or Invalid?
Clearly valid.
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. You’ll get
nowhere.
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.
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 ($z) serve as existential
quantifiers. They mean “there exists at least
one thing of which the following is true.”
There’s another operator, the universal
quantifier, which you’ll read about for
Monday.
Quantifiers are like logical operators in that
they determine truth conditions for the
statements they apply to.
To do so, they work together with attached
individual variables: small-case x, y, and z,
which function like pronouns.
‘($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
Ps
Fs
\ ($x)(Px & Fx)
(Dictionary: P_: _ is a philosopher; F_: _ is female;
s: Simone)
Even if we don’t yet have a way of proving this
argument is valid, we can see the reasoning. Use
&I and generalize (if Simone is a female
philosopher, then there has to exist at least one
female philosopher).
Scope and Binding in predicate logic
Scope: A quantifier’s scope is calculated in
the same way as the scope of a tilde: look
directly to the right of the quantifier and
--if there is a predicate letter, the quantifier
applies only to the atomic formula of which
that predicate letter is a part.
--if there is a tilde, the quantifier applies to
the tilde and to whatever the tilde applies to
--if there is a parenthesis (or bracket), the
quantifier applies to everything in that pair of
parentheses (or brackets)
A variable is bound if and only if it is within
the scope of a quantifier that contains a
matching small-case letter. If a variable is
unbound, it is free.
A statement that contains at least one free
variable (but is otherwise well-formed) is an
open sentence. These count as formulae, but
they don’t have truth-conditions.
When symbolizing in predicate logic, the
result should never be an open sentence
(i.e., no free variables allowed when
translating).
($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