The logical theory which is called the calculus of (binary) relations, and which will constitute the subject of this paper, has had a strange and rather capricious line of historical development. Although some scattered remarks regarding the concept of relations are to be found already in the writings of medieval logicians, it is only within the last hundred years that this topic has become the subject of systematic investigation. The first beginnings of the contemporary theory of relations are to be found in the writings of A. De Morgan, who carried out extensive investigations in this domain in the fifties of the Nineteenth Century. De Morgan clearly realized the inadequacy of traditional logic for the expression and justification, not merely of the more intricate arguments of mathematics and the sciences, but even of simple arguments occurring in every-day life; witness his famous aphorism, that all the logic of Aristotle does not permit us, from the fact that a horse is an animal, to conclude that the head of a horse is the head of an animal. In his effort to break the bonds of traditional logic and to expand the limits of logical inquiry, he directed his attention to the general concept of relations and fully recognized its significance. Nevertheless, De Morgan cannot be regarded as the creator of the modern theory of relations, since he did not possess an adequate apparatus for treating the subject in which he was interested, and was apparently unable to create such an apparatus. His investigations on relations show a lack of clarity and rigor which perhaps accounts for the neglect into which they fell in the following years.

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.

We refer to the axioms in Quine's book, Mathematical logic, New York, 1940.

To prove the independence of *200, give xϵ α the truth value F in all cases and give (x)ϕ the same truth value as ϕ. Then clearly all formulas derivable from the other axioms besides *200 have the value T, whereas from *200 one can derive (∃x)(∃α)(xϵ α) which has the value F. This method of proving independence amounts to taking for a model a universe consisting of the single object Λ.

For *201 we prove a contingent independence. That is, we prove that if Quine's system is consistent, then *201 is independent. The line of argument is the following. Suppose *201 can be derived from the other axioms. Let us replace xϵ α by throughout all the axioms. Then what *201 becomes can be derived from what the other axioms become. However what *201 becomes will lead to a contradiction in Quine's system whereas the rules which the other axioms become are valid in Quine's system.

We now get down to technical details. Let us refer to the replacement of xϵ α by throughout an expression as an r replacement. Denote the result of performing an r replacement on ϕ by ϕr. Let Wα denote

Then

Note that if x and y are variables, then by D10,