PHIL12A
Section answers, 9 February 2011
Julian Jonker
1
How much do you know?
1.
I have constructed a world in
Tarski’s World
using objects named
a
through
f
, but I’m not going
to show it to you. Now consider the sentences below, and decide whether you can determine their truth
value even though you can’t see my world. You should be able to explain why or why not.
(a)
d=d
We know that we can use the
=Intro
to introduce an identity at any point. Of course there must exist
such an object in the world, but I have told you that objects
a
through
f
exist. So
d=d
must be true.
(b)
BackOf(b,a)
∨
BackOf(a,b)
You cannot determine the truth value of this sentence without looking at the world. After all, if
a
and
b
are in the same row, then neither
BackOf(b,a)
nor
BackOf(a,b)
will be true.
(c)
LeftOf(e,f)
∧
RightOf(f,e)
LeftOf(e,f)
and RightOf(f,e) are equivalent, and if
e
and
f
are in the same column, then these will
not be true.
(d)
b=a
∨
a
6
=
b
b=a
is the same as
a=b
, and
a
6
=
b
is its negation, so this sentence says that either
a
and
b
are identical
objects, or they are not. One of the must be true, so the whole sentence is true, regardless of the world.
(e)
Large(a)
∨¬
Large(a)
Either
a
is large, or it is not. So this sentence must be true, regardless of the world.
2.
(Based on Ex 3.14) Is
¬
(Small(a)
∨
Small(b))
a logical consequence of
¬
Small(a)
∨
Small(b)
? If
it is, write an informal proof. If it is not, describe a counterexample.
The following counterexample shows that
¬
(Small(a)
∨
Small(b))
is not a logical consequence of
¬
Small(a)
∨
Small(b)
Let
a
be a large object, and
b
be a small object. Then
a
is not small, so
¬
Small(a)
is true and therefore
¬
Small(a)
∨
Small(b)
is true.
1
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However,
¬
(Small(a)
∨
Small(b))
is false, since
(Small(a)
∨
Small(b))
is true, since
Small(b)
is true.
3.
Is
¬
(Small(a)
∨
Small(b))
a logical consequence of
¬
Small(a)
∧¬
Small(b)
?
If it is, write an
informal proof. If it is not, describe a counterexample.
Yes. Suppose
¬
Small(a)
∧¬
Small(b)
is true. Then both conjuncts must be true, that is
¬
Small(a)
is
true and
¬
Small(b)
is true. But then
Small(a)
is false and so is
Small(b)
. This just means that
a
is
either medium or large, and
b
is either medium or large.
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 Spring '08
 FITELSON
 Logic, Logical connective, sentential connectives, equivalent formula

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