4.4 CONVERSION, OBVERSION AND CONTRAPOSITION

The operations of conversion, obversion and contraposition are applied to categorical propositions to yield new categorical propositions.

Conversion consists of simply switching the subject term with the predicate term while leaving the quality and quantity of the proposition unaltered.

The result of applying conversion to a categorical proposition is called the converse of the proposition.

Thus, for example, the converse of “All dogs are mammals” is “All mammals are dogs.”

The converses of E and I propositions are logically equivalent to them (that is, they necessarily have the same truth values.)

The converse of A and O propositions are not, in general, logically equivalent to them. Similarly, if we form an argument whose premise is a categorical proposition and whose conclusion is the converse of it, then the argument is valid if the premise is an E and I proposition and, in general, is invalid if the premise is an A or an O proposition.

Obversion consists of both (1) changing the quality of the proposition (leaving the quantity the same), and (2) complementing the predicate term.

To complement the predicate term, one typically attaches the prefix “non-“ to it.

The result of obversion is called obverse of the proposition to which it is applied.

The obverse of any categorical proposition, A, E, I or O, is logically equivalent to it.

Any argument whose premise is a categorical proposition and whose conclusion is its obverse is a valid argument.

Contraposition consist of both (1) switching the subject with the predicate term (while leaving the quality and quantity of the proposition unaltered), and (2) complementing both terms.

Thus, for example, the contrapositive of “All dogs are mammals” is “All non-mammals are non-dogs.”

The result of contraposition is called contrapositive of the proposition to which it is applied.

The contrapositive of A propositions and O propositions are logically equivalent to the originals, while the contrapositives of E and I propositions are not, in general, logically equivalent to the originals.

If we form an argument whose premises is a categorical proposition and whose conclusion is the contrapositive of it, then the argument is valid if the premise is an A or an O proposition and, in general is invalid if the premise is an E or an I proposition. Conversion, obversion and contraposition may be used in sequence to prove certain arguments valid.

There are two basic points to remember:

The first is that doubly complementing a term (e.g. non non-P) yields the equivalent of the term that is doubly complemented.

The second is that the operations of conversion and contraposition must be correctly applied: conversion must be applied only to E and I propositions and contraposition only to A and O propositions.

Here is an example of using the operations to show that “All A are non-B; therefore, no B are A” is a valid argument:

All A are non-B. All non-non-B are non-A.  Contraposition of an A proposition All B are non-A.  Replacing “non-non-B” with “B.” No B are non-non-A  Obversion No B are A.  Replacing “non-non-A” with “A.”