Propositional Logic
Note 1
( P -> Q ) is True whenever P is False as well as when both P and Q are true according to the Meaning of Connectives.
This might be counterintuitive for some people and might be a little difficult to be convinced of.
What we are concerned about here is True or False of the statement ( P -> Q ).
You might also look at it this way. We are interested in whether or not the person who made this satement is lying. If the statement is False, then that person is
lying.
For example consider this sentence:
You get ten thousand dollars from me
if I win one million dollars in a lottery.
Here P is "I win one million dollars in a lottery" and Q is
"You get ten thousand dollars from me".
If I don't win the lottery (P is False), I don't have to give you ten thousand dollars (Q is False). My statement ( P -> Q ) is still true if you don't get the money from me when I don't win. I haven't lied to you.
This is what "( P -> Q ) is True when P is False" means.
Similarly for when P and Q are False.
On the other hand, if I did win the the lottery and did not give you $10,000, then I have lied to you, that is the statement "You get ten thousand dollars from me if I win one million dollars in a lottery" is not true. That is what "( P -> Q ) is False if P is True and Q is False" means.
Note 2
In "If P then Q",
P and Q are arbitrary propositions.
We are interested in only true or false
of P -> Q vis-a-vis true or false of P and Q.
Thus P and Q may be completely
unrelated sentences such as in
" If 3 > 1, then ODU is in
Norfolk, VA." This proposition is true since both "3 > 1" and "ODU is in Norfolk,
VA" are true.
As an English sentence this if-then statement is meaningless. However, as a
proposition it is legitimate.