Answers

Question #1

It is not the case that if Tom is a philosopher then Tom is clever.

We might translate the above statement as "~(P→Q)". This WFF of course entails P, and it also entails ~Q. In other words, according to the translation the speaker is in fact saying that Tom is a philosopher and that Tom is not clever. However, in ordinary conversation a person who utters the English statement above might intend no such thing. He might simply be saying that even if Tom is a philosopher, it does not follow that he must be clever. Such a speaker might not in fact know whether Tom is a philosopher, nor might he know whether Tom is clever.

So this example shows that the usual truth-table associated with "if __ then ___" does not capture its correct meaning (given that negation is the correct translation for "it is not the case that").

Question #2

If "P" and "$P" are logically equivalent, then they always have the same truth value. So the truth-table is the following one :

P	$P
T	T
F	F

Question #3

We only know that when "P" is true, "$P" must be true. We CANNOT conclude that when "P" is F, "$P" is F. Even when "$P" is T when "P" is F, "P" still entails "$P". In other words,

P	$P
T	T
F	?

Question #4

Suppose "#" has the same truth-table as "→". Then if "P" is true and "Q" is false, "(P∨Q)" will then be true, and "(P#Q)" will be false. So "(P∨Q)" does not entail "(P#Q)".

Since it is given that "(P∨Q)" entails "(P#Q)", "#" cannot have the same truth-table as "→".