The sentence "If (not(Q)), then (not(P))" is known as the
**contrapositive** of "If P, then Q". The reader is invited
to construct a truth table that shows that these two sentences have the
same meaning. What is given below is another approach to this principle.

P | Q | if P, then Q | not(Q) | not(P) | if (not(Q)), then (not(P)) |
---|---|---|---|---|---|

T | T | T | F | F | T |

T | F | F | T | F | F |

F | T | T | F | T | T |

F | F | T | T | T | T |

Note that "not[if (not(Q)), then (not(P))]" means the same as "(not(Q)) and [not(not(P))]" {see Negation of a Conditional}. However "not(not(P))" has the same meaning as "P" {see negation}. So,

"Not[if (not(Q)), then (not(P))]" has the same meaning as "(not(Q)) and P".However, "and" is commutative, so both of these have the same meaning as "P and (not(Q))" {see conjunction}. Finally "P and (not(Q))" has the same meaning as "not(if P, then Q)". {See Negation of a Conditional.} Thus,

"Not[if (not(Q)), then (not(P))]" has the same meaning as "not(if P, then Q)".Apply "not" to both of these sentences, and then "cancel" the double "not"'s {see negation}:

"If (not(Q)), then (not(P))" has the same meaning as "if P, then Q".On second thought, maybe a truth table would have been easier to present!