The purpose of this paper is to give an axiom system for quantum logic. Here quantum logic is considered to have the structure of an orthomodular lattice. Some authors assume that it has the structure of an orthomodular poset. In finding this axiom system the implication algebra given in Finch [1] has been very useful. Finch shows there that this algebra can be produced from an orthomodular lattice and vice versa. Definition. An orthocomplementation N on a poset (partially ordered set) whose partial ordering is denoted by ≤ and which has least and greatest elements 0 and 1 is a unary operation satisfying the following: (1) the greatest lower bound of a and Na exists and is 0, (2) a ≤ b implies Nb ≤ Na, (3) NNa = a. Definition. An orthomodular lattice is a lattice with meet ∧, join ∨, least and greatest elements 0 and 1 and an orthocomplementation N satisfying where a ≤ b means a ∧ b = a, as usual. Definition. A Finch implication algebra is a poset with a partial ordering ≤, least and greatest elements 0 and 1 which is orthocomplemented by N. In addition, it has a binary operation → satisfying the following: An orthomodular lattice gives a Finch implication algebra by defining → by A Finch implication algebra can be changed into an orthomodular lattice by defining the meet ∧ and join ∨ by The orthocomplementation is unchanged in both cases.
The purpose of this paper is to give an axiom system for quantum logic. Here quantum logic is considered to have the structure of an orthomodular lattice. Some authors assume that it has the structure of an orthomodular poset. In finding this axiom system the implication algebra given in Finch [1] has been very useful. Finch shows there that this algebra can be produced from an orthomodular lattice and vice versa. Definition. An orthocomplementation N on a poset (partially ordered set) whose partial ordering is denoted by ≤ and which has least and greatest elements 0 and 1 is a unary operation satisfying the following: (1) the greatest lower bound of a and Na exists and is 0, (2) a ≤ b implies Nb ≤ Na, (3) NNa = a. Definition. An orthomodular lattice is a lattice with meet ∧, join ∨, least and greatest elements 0 and 1 and an orthocomplementation N satisfying where a ≤ b means a ∧ b = a, as usual. Definition. A Finch implication algebra is a poset with a partial ordering ≤, least and greatest elements 0 and 1 which is orthocomplemented by N. In addition, it has a binary operation → satisfying the following: An orthomodular lattice gives a Finch implication algebra by defining → by A Finch implication algebra can be changed into an orthomodular lattice by defining the meet ∧ and join ∨ by The orthocomplementation is unchanged in both cases.
A weak logical structure is defined as a set of boolean propositional logics in which one can define common operations of negation and implication. The set union of the boolean components of a weak logical structure is a logic of propositions which is an orthocomplemented poset, where orthocomplementation is interpreted as negation and the partial order as implication. It is shown that if one can define on this logic an operation of logical conjunction which has certain plausible properties, then the logic has the structure of an orthomodular lattice. Conversely, if the logic is an orthomodular lattice then the conjunction operation may be defined on it.
When a physicist performs a quantic measurement, new information about the system at hand is gathered. This article studies the logical properties of how this new information is combined with previous information. It presents Quantum Logic as a propositional logic under two connectives: negation and the and then operation that combines old and new information. The and then connective is neither commutative nor associative. Many properties of this logic are exhibited, and some small elegant subset is shown to imply all the properties considered. No independence or completeness result is claimed. Classical physical systems are exactly characterized by the commutativity, the associativity, or the monotonicity of the and then connective. Entailment is defined in this logic and can be proved to be a partial order. In orthomodular lattices, the operation proposed by Finch in [3] satisfies all the properties studied in this article. All properties satisfied by Finch's; operation in modular lattices are valid in Quantum Logic. It is not known whether all properties of Quantum Logic are satisfied by Finch's; operation in modular lattices. Non-commutative, non-associative algebraic structures generalizing Boolean algebras are defined, ideals are characterized and a homomorphism theorem is proved.
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.
Abstract States of quantum systems correspond to vectors in a Hilbert space and observations to closed subspaces. Hence, this logic corresponds to the algebra of closed subspaces of a Hilbert space. This can be considered as a complete lattice with orthocomplementation, but it is not distributive. It satisfies a weaker condition, the so-called orthomodularity. Later on, it was recognized that joins in this structure need not exist provided the subspaces are not orthogonal. Hence, the resulting structure need not be a lattice but a so-called orthomodular poset, more generally an orthoposet only. For orthoposets, we introduce a binary relation Δ and a binary operator d(x,y) that are generalizations of the binary relation C and the commutator c(x,y), respectively, known for orthomodular lattices. We characterize orthomodular posets among orthogonal posets. Moreover, we describe connections between the relations Δ and ↔ (the latter was introduced by P. Pták and S. Pulmannová) and the operator d(x,y). In addition, we investigate certain orthomodular posets of subsets of a finite set. In particular, we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of Δ-blocks with respect to Boolean subalgebras and distributive subposets they include.
In the present article, we explore a new approach for the study of orthomodular lattices, where we replace the problematic conjunction by a binary operator, called the Sasaki projection. We present a characterization of orthomodular lattices based on the use of an algebraic version of the Sasaki projection operator (together with orthocomplementation) rather than on the conjunction. We then define of a new logic, which we call Sasaki Orthologic, which is closely related to quantum logic, and provide a rule-based definition of this logic.
Sentences Associated with a Quantum System and Their Truth Values In a sense this chapter is marginal to the content of this volume; the reader can skip it without prejudice to the comprehension of chapters to follow. Quantum logic is a discipline that branched off from the 1936 paper of Birkhoff and von Neumann and has the orthomodular-poset structures encountered in previous chapters as basic mathematical carriers: that is why we think it worthwhile to devote some pages to the subject. But the main interests quantum logic calls into play are in the territory of logicians and philosophers: that is why we (who are neither logicians nor philosophers) shall confine ourselves to a very simplified introduction to the subject. Roughly, the starting question is whether the propositions of a quantum system can be associated with, or can be interpreted as, sentences of a language (or propositional calculus) and which rules this language inherits from the ordered structure of propositions. In raising this question one has in mind the fact that when the physical system is classical its propositions form a Boolean algebra, and Boolean algebras are the algebraic models of the calculus of classical logic. Thus the question above can also be phrased as follows: when a Boolean algebra is relaxed into an orthomodular nondistributive lattice, which logic is it the model of? “Quantum logic” is the name that designates the answer, but there are several views about the content of this name. Here we sketch one approach to a quantum logic.