Question

Basically I am having trouble reasoning out the truth table for conditional proposition / P implies Q / If P then Q / etc.

From my books and quick research on google no one seems to explain on what reasoning the definition was defined on, they all just basically give you the truth table and say accept it. I am capable of doing that, but I just totally fail to see how the 4 combined possibilities represent some coherent notion or idea.

Answer 1

The answer is: it’s set up this way to make the rest of the math work out easier.

I assume the weirdness you find in the definition is that if you have P -> Q, and P is False, then you find it strange you don’t have to handle the case. If you continue going through your mathematical curriculum, you will find this actually lines up with the idea that from a contradiction you can prove anything. The statement “If P, then Q” basically means “If P is true, then it must be the case that Q is true, but if not, then it doesn’t matter what I do.” You might find it more natural to ocasionally say “P must be true, and then Q must also be true,” but this corresponds to P /\ Q.

In some basic sense, however, it is just taken for granted, it seems to correspond to what you’d think at a high level as being implication, but there are sixteen possible logical relations (for binary connectives..). If you crank the logic, things work mechanically, it’s occasionally best not to question it, as sometimes you really define truth before the high level intuition, not the other way around.