Converse (logic) - Categorical Converse

Categorical Converse

In traditional logic, the process of going from All S is P to its converse All P is 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." The "exposita" is more usually called the "convertend." In its simple form, conversion is valid only for E and I propositions:

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 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.

In first-order predicate calculus, All S is P can be represented as . It is therefore clear that the categorical converse is closely related to the implicational converse, and that S and P cannot be swapped in All S is P.