The systems of modal calculus of names. II

Abstract

1. One can distinguish two branches of modal logic: modal calculus of propositions and modal calculus of names. In the language of modal calculus of propositions there appear propositional variables q,p,r,..., logical constants of propositional calculus ~5 ->5 A5 v5 = which are proposition formative functors of propositional arguments and modal terms M (it is possible that) and L (it is necessary that) ? proposition formative functors of propositional arguments, either. Usually one of the modal functors M or L is primitive and the other one can be defined. In the language of modal calculus of names there appear nominal variables S,P,X, Y,..., x,y, z,..., logical constants of propositional calculus, constants of nominal calculus (e.g.: a, i, e, o, -, +5 ') which are proposition formative or name formative functors of either propositional or nominal arguments. Historically the modal logic reaches back to Aristotle's times. Aristotle had introduced neither the term "modal proposition" nor any term equivalent to it, the term was formed later2. The chapters 3, 8?12 of First Analytics and the chapters 12?13 of De Interpretatione are devoted to the propositions called modal ones today. As Luka? siewicz mentioned one could say "Aristotle's modal logic of terms (i.e. of names)" and "Aristotle's modal logic of propositions"3. The first system of modal logic was Aristotle's modal syllogistic, that is modal calculus of names4. Aristotle formulated some laws of modal logic of propositions as well, namely (0.1) Mp = ~ L ~ p (0.2) Lp = ~ M ~ p (0.3) ~ p-*M~ p (0.4) (/>->?)(Lp -> I?) (0.5) (p -> q) -> {Mp -> Mq)