Contraposition

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In logic, contraposition is a form of immediate inference in which from a given proposition another is inferred having for its subject the contradictory of the original predicate, and in some cases involving a change of quality (affirmation or negation).

Traditional logic

In traditional logic the process of contraposition is a schema comprised of several steps of inference involving categorial propositions. A categorical proposition contains a subject and predicate where the existential impact of the copula implies the proposition as unconditional in contrast to the conditional form of hypothetical or materially implicative propositions, which are compounds of other propositions, e.g. If P, then Q, where P and Q are both propositions.

Conversion by contraposition is the simultaneous interchange and negation of the subject and predicate, and is valid only for the type "A" and type "O" propositions of Aristotelian logic. This is considered full contraposition. Since in the process of contraposition the obverse can be obtained in all four types of traditional propositions, yielding propositions with the contradictory of the original predicate, contraposition is first obtained by converting the obvert of the original proposition. Thus, partial contraposition can be obtained conditionally in an "E" type proposition with a change in quantity. Because nothing is said in the definition of contraposition with regard to the predicate of the inferred proposition, it can be either the original subject, or its contradictory, resulting in two contrapositives which are the obverts of one another in the "A", "O", and "E" type propositions.

By example: from an original, 'A' type categorical proposition,

"All residents are voters",

which presupposes that all classes have members and the existential import presumed in the form of categorical propositions, one can derive first by obversion the 'E' type proposition,

"No residents are non-voters".

The contrapositive of the original proposition is then derived by conversion to another 'E' type proposition,

"No non-voters are residents".

The process is completed by further obversion resulting in the 'A' type proposition that is the obverted contrapositive of the original proposition,

"All non-voters are non-residents".

The schema of contraposition:

Original Proposition	Obversion		Contraposition	Obverted Contraposition
(A) All S is P	(E) No S is non-P	↔	(E) No non-P is S	(A) All non-P is non-S
(E) No S is P	(A) All S is non-P	→	(I) Some non-P is S	(O) Some non-P is not non-S
(I) Some S is P	(O) Some S is not non-P		None	None
(O) Some S is not P	(I) Some S is non-P	↔	(I) Some non-P is S	(O) Some non-P is not non-S

Notice that contraposition is a valid form of immediate inference only when applied to "A" and "O" propositions. It is not valid for "I" propositions, where the obverse is an "O" proposition which has no converse. The contraposition of the "E" proposition is valid only with limitations (per accidens). This is because the obverse of the "E" proposition is an "A" proposition which cannot be validly converted except by limitation, that is, contraposition plus a change in the quantity of the proposition from universal to particular.

Also, notice that contraposition is a method of inference which may require the use of other rules of inference. The contrapositive is the product of the method of contraposition, with different outcomes depending upon whether the contraposition is full, or partial. The successive applications of conversion and obversion within the process of contraposition may be given by a variety of names.

The process of the logical equivalence of a statement and its contrapositive as defined in this section is not one of the axioms of propositional calculus. In traditional logic there is more than one contrapositive inferred from each original statement. In regard to the "A" proposition this is circumvented in the symbolism of modern logic by the rule of transposition, or the law of contraposition. In its technical usage within the field of philosophic logic, the term "contraposition" may be limited by logicians (e.g. Irving Copi, Susan Stebbing) to traditional logic and categorical propositions. In this sense the use the term "contraposition" is not applicable to hypothetical propositions or material implications.

Modern logic

In modern logic the law of contraposition is a tautology and its proof is the same as the rule of transposition as given in propositional calculus. The forms of hypothetical and disjunctive propositions are better suited in the conditionalizations of modern logic.

References

• Aristotle. Organon.
• Blumberg, Albert E. "Logic, Modern". Encyclopedia of Philosophy, Vol.5. Macmillan, 1967
• Copi, Irving. Introduction to Logic. MacMillan, 1953.
• Prior, A.N. "Logic, Traditional". Encyclopedia of Philosophy, Vol.5. Macmillan, 1967
• Stebbing, Susan. A Modern Introduction to Logic. 1931.

See also

• Transposition (logic)
• Conversion (logic)
• Obversion
• Organon
• Propositional calculus.

fr:Propriété contraposée