PHIL12A
Section answers, 11 April 2011
Julian Jonker
1
How much do you know?
1.
Translate the following pairs of sentences, and show by a chain of equivalences that they are equivalent:
(a)
i.
It is not the case that all P’s are Q’s.
¬∀
x(P(x)
→
Q(x))
ii.
Some P’s are not Q’s.
∃
x(P(x)
∧ ¬
Q(x))
Proof of equivalence:
¬∀
x(P(x)
→
Q(x))
⇔
¬∀
x
¬
(P(x)
∧ ¬
Q(x))
equivalence for
→
⇔
∃
x
¬¬
(P(x)
∧ ¬
Q(x))
De Morgan
⇔
∃
x(P(x)
∧ ¬
Q(x))
Double negation
(b)
i.
It is not the case that some P’s are Q’s.
¬∃
x(P(x)
∧
Q(x))
ii.
No P’s are Q’s.
∀
x(P(x)
→ ¬
Q(x))
Proof of equivalence:
¬∃
x(P(x)
∧
Q(x))
⇔
∀
x
¬
(P(x)
∧
Q(x))
De Morgan
⇔
∀
x(
¬
P(x)
∨ ¬
Q(x))
De Morgan
⇔
∀
x(P(x)
→ ¬
Q(x))
equivalence for
→
2.
Which of the following sentences are logical truths? For each sentence, you should be able to explain why
it is a logical truth, or produce a counterexample that shows that it is not.
(a)
(Ex 10.24)
(
∀
xCube(x)
∨ ∀
xDodec(x))
↔ ∀
x(Cube(x)
∨
Dodec(x))
The
←
direction of implication doesn’t hold. Suppose half of the objects in the world are cubes and half
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 Spring '08
 FITELSON
 Logic, logical truth, wff Tet

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