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THE LOGIC BOOK, 4
TH
EDITION: CHAPTER EIGHT
Predicate Logic (PL): Semantics
•
In SL, every sentence is either TruthFunctional or
Atomic. To find out the truthvalue of a truth
functional sentence, all we need is the truthvalues of
the atomic sentences.
•
In PL, however, many of the atomic sentences are
complex expressions composed of predicates and
constants, and such expressions cannot be assigned
set truthvalues. Why?
•
The truthconditions of sentences in PL depend on
the
interpretation
of that sentence, and such
interpretations depend on the
universe of discourse
.
Consider
:
UD:
Living Things
UD:
Living Things
Mx:
x is a mammal
Mx:
x is a reptile
w:
Willy the Whale
w:
Willy the Whale
Mw
Mw
1
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View Full Document Sentences without Quantifiers.
(1) Use the interpretation to determine the truthvalue of each atomic
sentence.
(2)
Proceed to determine the truth value of the sentence on the basis of
your knowledge of the connectives, just like we did in
SL
.
(Pb
∨
Pa)
⊃
Lcb
UD:
unrestricted
Px:
x is a politician.
Lxy:
x lives next to y
a: Jean Chretien
b: Celine Dion
c: Queen Elizabeth
If either Dion or Chretien is a politician
then Queen Elizabeth lives next to Dion.
Pa:
true
Pb:
false
Lcb: false
(Pb
∨
Pa)
⊃
Lcb:
false
UD:
Positive integers
Px:
x is prime
Lxy:
x is less than y
a:
four
b:
three
c:
one
If three or four is prime, then
one is less than three.
Pa: false
Pb:
true
Lcb: true
(Pb
∨
Pa)
⊃
Lcb: true
2
Sentences with Quantifiers
A universally quantified sentence is true if and only if it is true for
every
single member of the universe of discourse.
An existentially quantified sentence is true if and only if it is true for
at
least one
member of the universe of discourse.
(1) Use the interpretation to determine the truthvalue of any
unquantified atomic sentences.
(2) Use the interpretation, the results of (1), and your knowledge of
the connectives to determine the truth value or partial truth value
of any quantified sentential components.
(3) Use the interpretation, the results of (1) and the results of (2) to
determine the truthvalue of the sentence.
On the interpretation below, is the following true or false?
(
2200
x)(Ax
≡
~ Bx)
UD: positive integers
Px:
x is evenly divisible by 2
Bx:
x is an odd number
3
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View Full Document •
A universally quantified sentence is false if there is at
least one member of the UD for which the condition
specified after the quantifier does not hold.
•
A universally quantified sentence is true if there is no
member of the UD for which the condition specified
after the quantifier does not hold.
•
An existentially quantified sentence is false if it is
not the case that there is at least one member of the
UD for which the condition specified after the
quantifier does hold.
•
An existentially quantified sentence is true if it is the
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This note was uploaded on 09/28/2009 for the course PHL 245 taught by Professor Bangu during the Winter '05 term at University of Toronto Toronto.
 Winter '05
 bangu

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