Conversion (logic)

Conversion is a concept in traditional logic referring to a "type of immediate inference in which from a given proposition, another proposition is inferred which has as its subject the predicate of the original proposition and as its predicate the subject of the original proposition (the quality of the proposition being retained)". The immediately inferred proposition is termed the converse of the original proposition. Conversion has distinctive applications in philosophical logic and mathematical logic. This article concerns its philosophical application distinct from the other traditional inference processes of contraposition and obversion where equivocation varies with different proposition types.

For simple implications, the converse of $P\rightarrow Q$ is $Q\rightarrow P$.

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 Some triangles with three equal angles are equilateral.

Conversion (1) violates the rules of distribution and is invalid as a logical inference (yet both statements are true), whereas conversion (2) obeys the rules and is valid (yet if read informally, the second statement could be misleading). Thus, logicians allow for conversion of the "A" type proposition with limitations ("some" rather than "all"), or per accidens. 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 would otherwise 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 (convertend)		Converse
(A) All S is P		None
(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

Bibliography

• Aristotle. Organon.
• Brody, Bobuch A. "Glossary of Logical Terms". Encyclopedia of Philosophy. Macmillan, 1973.
• Copi, Irving. Introduction to Logic. MacMillan, 1953.
• Copi, Irving. Symbolic Logic. MacMillan, 1979, fifth edition.
• Stebbing, Susan. A Modern Introduction to Logic. Cromwell Company, 1931.

Footnotes

See also

• Aristotle
• Contraposition
• Inference
• Obversion

• Relational oppositeness
• Syllogism
• Term logic
• Transposition (logic)