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Zeitschrift für Naturforschung A

Quasi-Implicative Lattices and the Logic of Quantum Mechanics

Gary M. Hardegree

Published Nov 1, 1975• 10 citations
Publisher

Abstract

Mittelstaedt has defined the class of quasi-implicative lattices and shown that an ortholattice (orthocomplemented lattice) is quasi-implicative exactly if it is orthomodular (quasi-modular). He has also shown that the quasi-implication operation is uniquely determined by the quasi-implicative conditions. One of Mittelstaedt's conditions, however, seems to lack immediate intuitive motivation. Consequently, this paper seeks to provide a number of reformulations of the quasi-implicative conditions which are more intuitively plausible. Three sets of conditions are examined, and it is shown that each set of conditions is both necessary and sufficient to ensure that an ortholattice is orthomodular, and each set of conditions uniquely specifies the implication operation to be Mittelstaedt's quasi-implication. Various properties of the quasi-implication are then investigated. In particular, it is shown that the quasi-implication fails to satisfy a number of laws associated with the classical material conditional. Various weakenings of these laws, satisfied by the quasi-implication, are also discussed

DOI
10.1515/zna-1975-1102

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Mittelstaedt has defined the class of quasi-implicative lattices and shown that an ortholattice (orthocomplemented lattice) is quasi-implicative exactly if it is orthomodular (quasi-modular). He has also shown that the quasi-implication operation is uniquely determined by the quasi-implicative conditions. One of Mittelstaedt's conditions, however, seems to lack immediate intuitive motivation. Consequently, this paper seeks to provide a number of reformulations of the quasi-implicative conditions which are more intuitively plausible. Three sets of conditions are examined, and it is shown that each set of conditions is both necessary and sufficient to ensure that an ortholattice is orthomodular, and each set of conditions uniquely specifies the implication operation to be Mittelstaedt's quasi-implication. Various properties of the quasi-implication are then investigated. In particular, it is shown that the quasi-implication fails to satisfy a number of laws associated with the classical material conditional. Various weakenings of these laws, satisfied by the quasi-implication, are also discussed
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.
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.
Quasi-set theory was proposed as a mathematical context to investigate collections of indistinguishable objects. After presenting an outline of this theory, we define an algebra that has most of the standard properties of an orthocomplete orthomodular lattice, which is the lattice of the closed subspaces of a Hilbert space. We call the mathematical structure so obtained I\mathfrak{I}I-lattice. After discussing, in a preliminary form, some aspects of such a structure, we indicate the next problem of axiomatizing the corresponding logic, that is, a logic which has I\mathfrak{I}I-lattices as its algebraic models. We suggest that the intuitions that the `logic of quantum mechanics' would be not classical logic (with its Boolean algebra), is consonant with the idea of considering indistinguishability right from the start, that is, as a primitive concept. In other words, indiscernibility seems to lead `directly' to I\mathfrak{I}I-lattices. In the first sections, we present the main motivations and a `classical' situation which mirrors that one we focus on the last part of the paper. This paper is our first study of the algebraic structure of indiscernibility within quasi-set theory.
It has been shown by Birkhoff and v. Neumann (1936) and by Jauch and Piron (1963,1964,1968) that the subspaces of Hilbert space constitute an orthocomplemented quasi-modular lattice Lq, if one considers between two subspaces (elements) a, b the relation a⊆b and the operations a∩b, a∪b, a⊥. Furthermore, since the subspaces can be interpreted as quantum mechanical propositions, and since the operations ∩,∪ ,⊥ have some similarity with the logical operations ⋀ (and), ⋁ (or) and ⌝ (not), the question has been raised already by Birkhoff and v. Neumann, whether the lattice of subspaces of Hilbert space can be interpreted as a propositional calculus, sometimes called quantum logic. There are many kinds of lattices which can be interpreted as a propositional or logical calculus. A Boolean lattice LB of propositions corresponds to the calculus of classical logic and an implicative (Birkhoff, 1961) lattice Li has as a model the calculus of effective (intuitionistic) logic.
This article employs a descriptive-analytical method with an algebraic and axiomatic approach to address quantum logic. Through several thought experiments, it is demonstrated that the distributive law for propositions does not hold in quantum mechanics. Therefore, the logic governing this domain of science is not classical logic. In fact, the algebra corresponding to the logic of quantum propositions is an orthomodular lattice, which is why the distributive law is not necessarily valid for such propositions. In quantum logic, various definitions are offered for the implication operator. Since none of these operators satisfy the deduction theorem, a suitable implication operator with this property must be defined. This objective is achieved by defining a unified quantum logic. By constructing a Lindenbaum-Tarski algebra for this unified quantum logic, it can be shown that it is indeed a quantum logic. Dishkant presented an embedding of quantum logic into the Br-modal logic system. This provides an appropriate semantics for quantum logic. A suitable model for this logic is the closed subspaces of a Hilbert space.
Abstract Standard quantum logics are typically defined over the orthomodular lattice associated with Hilbert space, yet there is no consensus on how this structure should be interpreted logically or ontologically. This paper offers a unified perspective on these issues by combining an epistemological three-layer model for scientific theories with quantum set theory. We argue that quasi-set theory is best understood not as an alternative semantics for quantum logics, but as an ontological framework operating at a different epistemological level: the ontological one. From this standpoint, the distinctive non-classical features of quantum logic are explained by representational assumptions about identity and individuality, rather than by positing any direct revisions of the logical principles themselves.