The Ladd-Franklin “antilogism” provides a very convenient procedure for testing syllogistic arguments, especially when they are expressed algebraically. It is well known, however, that the antilogism test indicates the invalidity of certain moods which the traditional logic regards as valid. These are the moods in which a particular conclusion is inferred from two universal premises. It is the aim of this paper to show that the antilogism can be generalized in such a way as to cover these moods, and also to be applicable to certain forms of “immediate inference” and “sorites,” as these designations have been traditionally employed.
We shall find that when generalized in this fashion there are two principal kinds of antilogisms: (1) those which include an inequation of the form ab≠0, and (2) those which include an inequation of the form a≠0.
Definition. An antilogism is a group of propositions so constructed that the contradictory of any one of them is implied by the logical product of the others.