Up A Level
"And"
"And" of An "Or"
Contrapositive
"For All"
"If and Only If"
"If..., Then..."
"Not"
"Not" of An "And"
"Not" of An "If...Then"
"Not" of An "Or"
"Or"
"Or" of An "And"
Short Tautologies
"There exists"

# "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.

Note that "not[if (not(Q)), then (not(P))]" means the same as "(not(Q)) and [not(not(P))]" {see "Not" Applied To An "If...Then" Sentence}. However "not(not(P))" has the same meaning as "P" {see not}. 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 and}. Finally "P and (not(Q))" has the same meaning as "not(if P, then Q)". {See "Not" Applied To An "If...Then" Sentence.} 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 not}:
"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!