Deductive systems and categories

Some years ago, G. D. Findlay and the author observed a connection between proofs in certain deductive systems and canonical mappings in certain categories [6, see also 8, Appendix II]. It is the object of the present paper to give a more formal expression to this connection. Roughly speaking, the situation is this: While deductive systems may be viewed as free algebras on the category of pre-ordered sets, they may also be used to construct free algebras on the category of all categories in some universe. The crucial step in the argument is the construction of a category whose objects are the terms of a given deductive system and whose maps are equivalence classes of proofs. An equivalence relation is manufactured ad hoc to serve this purpose. However, in the pilot project under investigation, it turns out that two proofs are equivalent if and only if they have the same "scope", in the sense that any method of generalizing one proof (without changing the rules of inference) leads to a generalization of the other. The present investigation is not carried out in all generality. Instead, we take as a pilot project a particular deductive system, which has been called "syntactic calculus" elsewhere [8, 9]. This system possesses a very simple decision procedure ~ la Gentzen, which has the pedagogical advantage of being free from so-called "structural" rules of inference [7]. As has already been pointed out [9], this decision procedure may be lifted to the level of categories, where it corresponds to Bourbaki's method of multilinear mappings. The author plans to pursue this subject further, both by investigating standard constructions ( = triples = monads) on the category of categories in general, and by looking at special deductive systems, in particular at the propositional calculus and its relation to Curry's theory of combinators (see [3, Chapter 9]).