Converse (logic)

Implicational converse

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 truth of S says nothing about the truth 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.

However, the converse of a statement with mutually inclusive terms remains true, given the truth of the original proposition. This is equivalent to saying that the converse of a definition is true. Thus, the statement "If I am a triangle, then I am a three-sided polygon" is logically equivalent to "If I am a three-sided polygon, then I am a triangle," because the definition of "triangle" is "three-sided polygon".

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., P is true if and only if Q is also true), then affirming the consequent will be valid.

Converse implication is logically equivalent to the disjunction of ${\displaystyle P}$ and ${\displaystyle \mathrm{\neg }Q}$

In natural language, this could be rendered "not Q without P".

Converse of a theorem

In mathematics, the converse of a theorem of the form P → Q will be Q → P. The converse may or may not be true, and even if true, the proof may be difficult. For example, the four-vertex theorem was proved in 1912, but its converse was proved only in 1997.

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:

Given a triangle with sides of length ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$, if the angle opposite the side of length ${\displaystyle c}$ is a right angle, then ${\displaystyle a^2+b^2=c^2}$.

The converse, which also appears in Euclid's Elements (Book I, Proposition 48), can be stated as:

Given a triangle with sides of length ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$, if ${\displaystyle a^2+b^2=c^2}$, then the angle opposite the side of length ${\displaystyle c}$ is a right angle.

Converse of a relation

If ${\displaystyle R}$ is a binary relation with ${\displaystyle R\subseteq A\times B,}$ then the converse relation ${\displaystyle R^T=\{(b,a):(a,b)\in R\}}$ is also called the transpose.