# Conversion (logic)

In traditional logic conversion is a form of immediate inference in which from a given categorical proposition another proposition is inferred which has as its subject the predicate of the original proposition, and has as its predicate the subject of the original proposition, with the quality of the proposition remaining unchanged. The immediately inferred proposition is termed the converse of the original proposition.

The process of conversion results in an equivalent proposition only in type "E" and type "I" propositions. In the "E" type proposition both the subject term and the predicate term remain distributed in conversion, and in the "I" type proposition both the subject term and the predicate term remain undistributed in conversion.

For example, in the "E" type proposition No S is P conversion yields No P is S. Both of the terms remain distributed, that is, their class membership is exhausted. It can be expressed grammatically in the statements:

No Romans are philosophers and No philosophers are Romans.

In the "I" type proposition Some S is P conversion yields Some P is S. Both of the terms remain undistributed, that is, their class membership is not exhausted. It can be expressed grammatically in the statements:

Some Greeks are philosophers and Some philosophers are Greek.

In an "A" type proposition, conversion of "All S is P" to "All P is S" may violate the rules of distribution, for example:

All popes are saints and All saints are popes. In an "A" type proposition the subject term is distributed (exhausted) and the predicate undistributed. Conversion distributes the predicate of the original proposition as the subject in the inferred proposition. Contrast this with two other converted propositions:

(1) All isosceles triangles have their base angles equal, and All triangles with their base angles equal are isosceles.
(2) All equilateral triangles have three equal angles and All triangles with three equal angles are equilateral.

Conversion (1) violates the rules of distribution and is invalid, whereas conversion (2) does not, and is valid. Thus, logicians allow for conversion of the "A" type proposition with limitations, or per accidens. For example, from All isosceles triangles have their base angles equal one can infer the "I" type proposition Some triangles with their base angles equal are isosceles. The notion of limitation, or conversion per accidens, requires a change in the quantity of the proposition from universal to particular in instances where the rules of distribution may be violated.

Conversion of the "O" type proposition Some S is not P is not possible, in every instance violating the rules of distribution.

The schema of conversion is:

Original Proposition Converse
(A) All S is P (I) Some P is S
(E) No S is P (E) No P is S
(I) Some S is P (I) Some P is S
(O) Some S is not P None