Semantics of the minimal logic of quantum mechanics

The question what is the logic of the atomic world, belongs to the empirical science. It can be solved only by ways of hypotheses framing and testing. Birkhoff and von Neumann [1] put forward a hypothesis that the propositions about quantum obser? vations constitute a modular ortholattice. But at the present time the truth of the mo? dularity condition seems very unlikely [2]. In connection with this fact we shall limit ourselves to considering a "minimal55 quantum logic, namely we accept that the pro? positions of a quantum logic form an ortholattice. In this paper we propose the se? mantics of a quantum logic for which this logic is complete. The alphabet of a quantum logic consists of the signs of negation ~~|, conjunction &, alternative V, a non-empty set 6 of propositional variables x,y,z,... and parentheses. The rules of forming are as follows: if A is a propositional variable, then A is a formula (atomic); if X and Y are formulas, then ~| X, (X & Y) and (X v Y) are formulas. We shall speak about the set of formulas CW? or about the algebra of formulas ?itf = . In the sequel we will use the following metalinguistic signs: ~, -, v, =>, o, V, 3 (negation, conjunction, alternative,...). For a given relation R, by T* we shall denote any A such that TRA is true. Thus VT*9(r*) will mean "for all A such that TRA, 9(A)" and 3r*9(r*) will mean "there is a A such that TRA and 9(A)53. Definition 1. By a semantic model of the algebra of formulas C~W we mean an ordered triple 9JI = , where G is a non-empty set, R is a relation between elements of G, satisfying the following conditions: for any r, F* e G RO. TRr (reflexivity), ri. 3r** vr*** (TRr***")