N‐Categories in Logic

This paper deals with a reformulations of HALMOS’ monadic and polyadii logics in terms of preorder categories. This is not a work on categorial logic but rather on algebraic ,logic treated in the language of categories. Our justification for presenting such a reformulation is based, firstly, on the capability of category theory to unify notions and, secondly, on the fact that the introduction of isomorphic objects allows us to construct the propositional and predicate calculi, without having to refer to LINDENBAUM’S algebras. The ideas developped in this paper reflect some well-known notions and results of BOOLE’S and HALMOS’ algebras. The difference with the standard presentation is that these notions and results appear here as consequences of a very simple construction, that of N-category. In order not to make the paper uniecessarily long, some proofs are only sketched or simply omitted.