PHIL12A
Section answers, 2 February 2011
Julian Jonker
1
How much do you know?
The following arguments are given in the blocks language of Tarski’s World. Decide whether the argument is
valid. If it is, find a way to persuade others that it is. (In other words, write an informal proof.) If the argument
is not valid, think of a counterexample.
1.
(Ex 2.9)
1
LeftOf(a,b)
2
b=c
3
RightOf(c,a)
This argument is valid. We know that
a
is left of
b
, by premise 1. But
b
is identical to
c
, by premise 2. By the
indiscernibility of identicals, we know that what is true of
b
is true of
c
, so
a
is left of
c
.
Left of
and
right of
are inverses of each other, so we have that
c
is right of
a
, which is the conclusion we want.
2.
(Ex 2.12)
1
BackOf(a,b)
2
FrontOf(a,c)
3
FrontOf(b,c)
This argument is valid. We know that
a
is in back of
b
by premise 1. But
back of
and
front of
are inverses of
each other, so
b
is in front of
a
. We also know that
a
is in front of
c
by premise 2. By the transitivity of
front
of
, it follows that
b
is in front of
c
– which is the conclusion we want.
1
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3.
(Ex 2.14)
1
Between(b,a,c)
2
LeftOf(a,c)
3
LeftOf(a,b)
When I put this problem on the handout, I had the following simple proof in mind:
We know that
a
is left of
c
by premise 2. Furthermore,
b
is between
a
and
c
by premise 1. It follows
that
b
is left of
c
and right of
a
, and so
b
is right of
a
. But
left of
and
right of
are inverses of each
other, and so
a
is left of
b
, which is the conclusion we want.
However, in section, I realized that this is not really good enough. The proof is valid, but it is not obvious that
each step is convincing, since we assume that the reader is doing a lot of the work on her own when reading the
proof. In particular, how can we go from stating that
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 Spring '08
 FITELSON
 Logic, 10%, Julian Jonker

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