﻿ Contrapositive

# "If P, then Q" IS EQUIVALENT TO "If (not(Q)), then (not(P))".

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!