Contrapositive and converse

Two further statements can be built out of any "if p, then q" statement: its contrapositive and its converse. They are easy to confuse but behave very differently.

Contrapositive. The contrapositive of "if p, then q" is "if $\sim q$, then $\sim p$." Crucially, a statement and its contrapositive always carry the same meaning — one is true exactly when the other is true. For example, the contrapositive of "If the physical environment changes, then the biological environment changes" is "If the biological environment does not change, then the physical environment does not change" — different words, same claim.

Converse. The converse of "if p, then q" is "if q, then p" — the two halves swapped. Unlike the contrapositive, the converse is generally a different statement from the original and need not share its truth value. For instance, the converse of "If a number is divisible by 10, it is divisible by 5" (true) is "If a number is divisible by 5, then it is divisible by 10" (false — 15 is divisible by 5 but not 10). So checking a converse is a genuinely separate task from checking the original implication. …