NEGATION APPLIED TO A CONDITIONAL

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))".

The Negation of a Conditional is equivalent to a Conjunction of the antecedent with the Negation of the consequent
P Q if P, then Q not(if P, then Q) not(Q) P and (not(Q))
T T T F F F
T F F T T T
F T T F F F
F F T F T F



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.