# Converse (logic)

In logic and mathematics, the **converse** of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication *P* → *Q*, the converse is *Q* → *P*. For the categorical proposition *All S are P*, the converse is *All P are S*. Either way, the truth of the converse is generally independent from that of the original statement.^{[1]}^{[2]}

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,^{[1]}^{[3]} 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. 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.

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.^{[4]}

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:

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

In traditional logic, the process of going from "All *S* are *P"* to its converse "All *P* are *S"* is called **conversion**. In the words of Asa Mahan:

"The original proposition is called the exposita; when converted, it is denominated the converse. Conversion is valid when, and only when, nothing is asserted in the converse which is not affirmed or implied in the exposita."^{[6]}

The "exposita" is more usually called the "convertend." In its simple form, conversion is valid only for **E** and **I** propositions:^{[7]}

The validity of simple conversion only for **E** and **I** propositions can be expressed by the restriction that "No term must be distributed in the converse which is not distributed in the convertend."^{[8]} For **E** propositions, both subject and predicate are distributed, while for **I** propositions, neither is.

For **A** propositions, the subject is distributed while the predicate is not, and so the inference from an **A** statement to its converse is not valid. As an example, for the **A** proposition "All cats are mammals", the converse "All mammals are cats" is obviously false. However, the weaker statement "Some mammals are cats" is true. Logicians define conversion *per accidens* to be the process of producing this weaker statement. Inference from a statement to its converse *per accidens* is generally valid. However, as with syllogisms, this switch from the universal to the particular causes problems with empty categories: "All unicorns are mammals" is often taken as true, while the converse *per accidens* "Some mammals are unicorns" is clearly false.