Classical Logic and Quantum Logic with Multiple and Common Lattice Models

Mladen Pavičić

doi:10.1155/2016/6830685

Abstract

We consider a proper propositional quantum logic and show that it has multiple disjoint lattice models, only one of which is an orthomodular lattice (algebra) underlying Hilbert (quantum) space. We give an equivalent proof for the classical logic which turns out to have disjoint distributive and nondistributive ortholattices. In particular, we prove that both classical logic and quantum logic are sound and complete with respect to each of these lattices. We also show that there is one common nonorthomodular lattice that is a model of both quantum and classical logic. In technical terms, that enables us to run the same classical logic on both a digital (standard, two-subset, 0-1-bit) computer and a nondigital (say, a six-subset) computer (with appropriate chips and circuits). With quantum logic, the same six-element common lattice can serve us as a benchmark for an efficient evaluation of equations of bigger lattice models or theorems of the logic.

Highlights

Is Logic Empirical?In his seminal paper “Is Logic Empirical?” [1], Putnam argues that logic we make use of to handle the statements and propositions of the theories we employ to describe the world around us is uniquely determined by it
We show that an axiomatic logic is wider than its relational logic variety in the sense of having many possible models and distributive ortholattice (Boolean algebra) for the classical logic and orthomodular lattice for the quantum logic
According to Definitions 16, 20, 21, and 19, of WOML∗, WOML1∗, WOML2∗, and WDL∗, respectively, these lattices denote set-theoretical differences and that is going to play a crucial role in our proof of completeness in Section 4.2 in contrast to [27] where we considered only WOML without excluding the orthomodular equation

Summary

Introduction

In his seminal paper “Is Logic Empirical?” [1], Putnam argues that logic we make use of to handle the statements and propositions of the theories we employ to describe the world around us is uniquely determined by it. We shall consider a classical and a quantum logic defined as a set of axioms whose Lindenbaum-Tarski algebras of equivalence classes of expressions from appropriate lattices correspond to the models of the logic. We shall make use of the PM classical logical system—Whitehead and Russell’s Principia Mathematica axiomatization in Hilbert and Ackermann’s presentation [24] in the schemata form and of Kalmbach’s axiomatic quantum logic [25, 26] (slightly modified by Pavicicand Megill [27, 28]—original Kalmbach axioms A1, A11, and A15 are dropped because they were proven redundant in [29]), as typical examples of axiomatic logic It is well-known that there are many interpretations of the classical logic, for example, two-valued, general Boolean algebra (distributive ortholattice) and set-valued ones [30, Ch. 8, 9].

Kinds of Logic

Lattices

Soundness and Completeness

Discussion