An axiom system for the modular logic

Abstract

G. Birkhoff and J. v. Neumann in their paper [4] concerning logic of the quantum mechanics come to the conclusion that not all laws of the Boolean algebra of logic apply to the statements of the quantum mechanics. Linear subspaces play an important part in the quantum mechanics. This fact suggested the authors the thought to con? sider the lattice of all linear subspaces of some linear space as a model of the sentential calculus of the quantum mechanics, analogously to the lattice of all subsets of a given set which can be considered as a model of the Boolean logic. Birkhof fand v. Neumann de? fine the modular logic as a free, modular and orthocomplementary lattice 3J? = freely generated by a set P. Elements belonging to P are called sentential variables, elements of M are called sentences, "v" and "a" are binary operations called alternative and conjunction respectively. " ' " is an unary operation called negation. One of the simplest models of the modular logic is the lattice of all linear subspaces of an ^-dimensional projective space. These subspaces are considered as sentences, the meet and linear union of two linear subspaces are considered as the conjunction and alternative of sentences. The orthogonal complementation of given subspace to the ^-dimensional space is referred to as negation. By a meet of two linear subspaces we understand their common part and by their linear union ? the smallest subspace containing both of them. In the modular logic the distributive law is abandoned. It is replaced by a weaker law called modular identity: (a a b) v (a a c) = a a (b v (a a c)). The modular logic must necessarily be an orthocomplementary lattice as in modular non-distributive lattices the complementarity relation is not a one-to-one relation, whereas the ortho complementarity is a relation of this kind [5]. It follows from the definition of the modular logic that there exist two fixed pro? positions denoted by 0 and 1. 0 is the false proposition, 1 is the true proposition. No implication operation is defined in the Birkhoff? v. Neumann modular lo? gic, however, the authors define an implication relation of partial order type. In the Boolean algebra "a implies b" if and only if "a v b = b". The implication is in the Boolean algebra uniquely defined as the operation a! ^ b (see [5]). This operation has in the modular logic few properties of the ordinary implication, e.g. a! v b = 1 does not imply the implication relation a v b = b.