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"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! Your comments and questions are welcome. Please email them here. |