# 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.