Review of Becker 1930: On modal logic (1931e)

Abstract

Abstract The author’s starting point is C. I. Lewis’ “system of strict implication” (a propositional calculus with the primitive notions “and”, “not”, “impossible”), and he remarks that in this system we can generate, through iterated application of the predicates “not” and “impossible”, infinitely many irreducible modalities for a proposition. These modalities cannot even be linearly ordered according to their logical strength in the sense that, of any two affirming modalities, one will imply the other, and similarly for negating ones. (To be sure, strict proofs of these assertions are not given.) To remedy this defect of Lewis’ system, the author proposes various additional axioms and then seeks to specify a system, with as few assumptions as possible, for which a linear ordering still exists. All in all, three different kinds of the calculus of modalities emerge, which, so the author believes, can be applied when different kinds of “necessity” are considered. As far as the purely formal side is concerned, one can hardly take exception to anything here, but there remain essential gaps to be filled in, some of which the author himself points out. Above all, it is nowhere shown that the three systems set up really differ from one another and from Lewis’ system (in other words, that the additional axioms are not in fact equivalent and do not follow from Lewis’); nor, furthermore, that the six, or ten, basic modalities obtained cannot be still further reduced. In conclusion, the author discusses, from a formal as well as a phenomenological standpoint, the connections that in his opinion obtain between modal logic and the intuitionistic logic of Brouwer and Heyting. It seems doubtful, however, that the steps here taken to deal with this problem on a formal plane will lead to success.