ENGRI 126
SOLUTIONS TO HOMEWORK I
Spring 2007
1.
The idea is that a situation is paradoxical if it leads inevitably to a logical contradic-
tion of the form “
A
implies not
A
.” If there’s
some
set of circumstances that leads to no
such contradiction, then it’s not a paradox.
(a) Certainly, I could not possibly be telling the truth, since in that case the indicated
statement would be false, meaning that I am not telling the truth after all. I.e., I
am telling the truth implies I am not telling the truth. However, I could be lying.
In other words, it is possible that I do tell the truth sometimes, but not when I
say this sentence to you. In other words, if I am lying when I speak the indicated
sentence, there’s no logical contradiction.
(b) Could Lisa be telling the truth? If so, Bart must be lying, which means that Lisa
is
not
telling the truth after all. So Lisa could not be telling the truth. Could
she be lying? If so, we conclude that Bart must tell the truth sometimes. Might
he be telling the truth now? If so, Lisa could not be lying as we assumed, so he
must actually be lying now. If he is lying now, then Lisa is lying now — which is
exactly what we assumed to start this chain of reasoning. Conclusion: there’s no
logical contradiction if Lisa is lying and Bart is also lying. Hence the situation is
not a paradox.
(c) Could Lisa be telling the truth? If so, what Bart says in response must be a lie,
which means that Lisa is
not
telling the truth after all. So Lisa could not be
telling the truth. Could Lisa be lying? If so, we conclude that what Bart says
in response must be true. This implies Lisa is
not
lying as we assumed. Hence,
we have a paradox. Regardless of whether Lisa is lying or telling the truth (and
these two possibilities exhaust the set of possible circumstances), we get a logical
contradiction. (Note: I did not intend for you to worry about the time-delay
feature of this example. Compare part (e).)
(d) If (A) is true, then (B) must be false, meaning that (A) must be false — a
contradiction. If (A) is false, then (B) is true, meaning that (A) is true — a
contradiction. Thus we have a paradox. This is exactly like (c) when you think
about it.
(e) I don’t have a good answer for this one. I was just hoping to have you think about
what’s going on in situations with linked declarative sentences that are uttered
at diﬀerent times. The ﬁeld of temporal logic, which is relevant to theoretical
computer science among other things, addresses situations of this kind.
One
reasonable way to approach the question is to assert that the sentence I utter
before I go to bed, despite its declarative nature, does not have a truth value
when I utter it because it refers to an event in the future. Only after that event
takes place can we hope to (attempt to) assign a truth value to that ﬁrst sentence.
But at that point, we end up sweating over questions like, “Is what I said last