Antilogism
Antilogism
If in the syllogism in Barbara the conclusion is replaced by its contradictory there is obtained the following set of three (formulas representing) propositions,
M(x) ?x P(x),
S(x) ?x M(x),
S(x) ?x ~P(x),
from any two of which the negation of the third may be inferred. Such an inconsistent triad of propositions is called an antilogism.
From the principle of the antilogism, together with obversion, simple conversion of E and I, and the fact that in the pairs, A and O, E and I, each proposition of the pair is equivalent to the negation of the other, all of the traditional valid moods of the syllogism may be derived except those which require a third (existential) premiss (see logic, formal, 4, 5). With the further aid of subalternation the remaining valid moods may be derived.
This extension of the traditional reductions of the syllogistic moods is due to Christine Ladd Franklin. She, however, stated the matter within the algebra of classes (see logic, formal, 7), taking the three terms of the syllogism as classes. From this point of view the three propositions of an antilogism appear as follows
m n -p = ?, s n -p ? ?.
— A.C.
A contradiction in terms, concepts, or propositions forming an inconsistent triad (Mrs. Ladd-Franklin), a set of three propositions such that if any two are true the third must be false; thus any two will strictly imply the contradictory of the third. An antilogism may be obtained from any strictly valid Aristotelian syllogism by contradicting the conclusion, q.v. Antilogism. — C.A.B.