Converse (logic)

In logic, the converse of a categorical or implicational statement is the result of reversing its two parts. For the implication P → Q, the converse is Q → P. For the categorical proposition All S is P, the converse is All P is S. In neither case does the converse necessarily follow from the original statement. The categorical converse of a statement is contrasted with the contrapositive and the obverse.

Let S be a statement of the form P implies Q (P → Q). Then the converse of S is the statement Q implies P (Q → P). In general, the verity of S says nothing about the verity of its converse, unless the antecedent P and the consequent Q are logically equivalent.

For example, consider the true statement "If I am a human, then I am mortal." The converse of that statement is "If I am mortal, then I am a human," which is not necessarily true.

On the other hand, the converse of a statement with mutually inclusive terms remains true, given the truth of the original proposition. Thus, the statement "If I am a bachelor, then I am an unmarried man" is logically equivalent to "If I am an unmarried man, then I am a bachelor."

A truth table makes it clear that S and the converse of S are not logically equivalent unless both terms imply each other:

Going from a statement to its converse is the fallacy of affirming the consequent. However, if the statement S and its converse are equivalent (i.e., if P is true if and only if Q is also true), then affirming the consequent will be valid.

In mathematics, the converse of a theorem of the form P → Q will be Q → P. The converse may or may not be true. If true, the proof may be difficult. For example, the Four-vertex theorem was proved in 1912, but its converse only in 1998.

In practice, when determining the converse of a mathematical theorem, aspects of the antecedent may be taken as establishing context. That is, the converse of Given P, if Q then R will be Given P, if R then Q. For example, the Pythagorean theorem can be stated as:

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