Converse (logic)

In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements.

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.

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.

Converse implication is logically equivalent to the disjunction of

[3] In practice, when determining the converse of a mathematical theorem, aspects of the antecedent may be taken as establishing context.

For example, the Pythagorean theorem can be stated as: Given a triangle with sides of length

, if the angle opposite the side of length

The converse, which also appears in Euclid's Elements (Book I, Proposition 48), can be stated as: Given a triangle with sides of length

, then the angle opposite the side of length

In traditional logic, the process of switching the subject term with the predicate term is called conversion.

In the words of Asa Mahan: "The original proposition is called the exposita; when converted, it is denominated the converse.

In its simple form, conversion is valid only for E and I propositions:[6] 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.

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.

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.

In first-order predicate calculus, All S are P can be represented as