"NOT" APPLIED TO AN "IF...THEN" SENTENCE

Let P and Q be sentences which are true or false, but neither of them is both. "Not(if P, then Q)" means the same thing as "P and (not(Q))".

Some people understand this principle as follows. They know that "if P, then Q" is false only when the promise is broken---that is, when P is true and Q is false. Q being false makes "not(Q)" true. So, both P and "not(Q)" are true. This is what is meant by "P and (not(Q))".

The truth table to the right is a different approach to the same principle. Note that columns 4 and 6 have the same truth values.