Consider the argument (symbolized and written in standard form, where the first three lines are the premises and the last line is the conclusion):
p
p · q
~p v ~q
q
The definition of an argument’s validity is thus: if all the premises are true, then the conclusion must be true. The argument is invalid if all the premises are true and the conclusion is false. So to determine whether the argument is valid or not, we simply look at all possible scenarios where the conclusion is false. This means we evaluate the premises and the conclusion given that q is false and p is false, and again given that q is false and p is true.
(Also, the expression ~p v ~q means “either p or q must be false”, and p · q means “both p and q must be true”.)
| p | p · q | ~p v ~q | q |
| T | F | T | F |
| F | F | T | F |
What the table tells us is that there is no possible situation where the conclusion is false that all of the premises are true. Logically, the argument is valid. So what’s the problem? Watch what happens when I expand the table to allow for the conclusion to be true as well:
| p | p · q | ~p v ~q | q |
| T | F | T | F |
| F | F | T | F |
| T | T | F | T |
| F | F | T | T |
Look at the third and fourth columns – p · q and ~p v ~q. As you can see, these two expressions always evaluate opposite of each other – that is, one will be true and the other will be false – regardless of the individual values of p and q. This is a problem, because both of these expressions are premises in the same argument. So while logically the argument is valid, it’ll never be correct.
Screw you, Spock!