Sequential method in quantum logic

Abstract

Since Birkhoff and von Neumann [2] a new area of logical investigation has grown up under the name of quantum logic. However it seems to me that many authors have been inclined to discuss algebraic semantics as such (mainly lattices of a certain kind) almost directly without presenting any axiomatic system, far from developing any proof theory of quantum logic. See, e.g., Gunson [9], Jauch [10], Varadarajan [15], Zeirler [16], etc. In this sense many works presented under the name of quantum logic are algebraic in essence rather than genuinely logical, though it is absurd to doubt the close relationship between algebraic and logical study on quantum mechanics.The main purpose of this paper is to alter this situation by presenting an axiomatization of quantum logic as natural and as elegant as possible, which further proof-theoretical study is to be based on.It is true that several axiomatizations of quantum logic are present now. Several authors have investigated the so-called material implication α → β ( = ¬α∨(α ∧ β)) very closely with due regard to its importance. See, e.g., Finch [5], Piziak [11], etc. Indeed material implication plays a predominant role in any axiomatization of a logic in Hilbert-style. Clark [4] has presented an axiomatization of quantum logic with negation ¬ and material implication → as primitive connectives. In this paper we do not follow this approach. First of all, this approach is greatly complicated because orthomodular lattices are only locally distributive.