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Modus Ponens in Physics

Kate Clements, Fay Dowker B, Petros Wallden

Published Jan 30, 2012• 1 citation
OpenAlex

Abstract

It is proved that the classical rules of inference about physical events are equivalent to three axioms: (i) something happens, (ii) modus ponens and (iii) if an event doesn't happen then its complement does. The multiplicative scheme for Generalised Measure Theory retains the first two axioms and replaces the third with a principle of maximal detail.

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Binary orthologic with modus ponens is either orthomodular or distributive

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We show that binary orthologic becomes either quantum or classical logic when nothing but modus ponens rule is added to it, depending on the kind of the operation of implication used. We also show that in the usual approach the rule characterizes neither quantum nor classical logic. The dierence turns out to stem from the chosen valuation on a model of a logic. Thus algebraic mappings of axioms of standard quantum logics would fail to yield an orthomodular lattice if a unary|as opposed to binary|valuation were used. Instead, non-orthomodular nontrivial varieties of orthologic are obtained. We also discuss the computational eciency of the binary quantum logic and stress its importance for quantum computation and related algorithms.

Helvetica Physica Acta (Switzerland)•Dec 1, 1998•15 citations
It is proved that the classical rules of inference about physical events are equivalent to three axioms: (i) something happens, (ii) modus ponens and (iii) if an event doesn't happen then its complement does. The multiplicative scheme for Generalised Measure Theory retains the first two axioms and replaces the third with a principle of maximal detail.
Classical logic does not apply in quantum theory. However one would like to be able to reason according to some standard rules for quantum systems as well (whether they are an electron or the universe itself). We examine what exactly is needed in order to be able to construct deductive arguments and in particular to maintain the "Modus Ponens" which is the basic deductive rule of inference. It turns out that this requirement restricts the possible theories and in the context of the coevent interpretation it results uniquely to the "multiplicative-scheme".
The process of inference reflects the structure of propositions with assigned truth values, either true or false. Modus ponens is a fundamental form of inference that involves affirming the antecedent to affirm the consequent. Inspired by the quantum computer, the superposition of true and false is used for the parallel processing. In this work, we propose a quantum version of modus ponens. Additionally, we introduce two generations of quantum modus ponens: quantum modus ponens inference chain and multidimensional quantum modus ponens. Finally, a simple implementation of quantum modus ponens on the OriginQ quantum computing cloud platform is demonstrated.
A physical theory, in the narrow sense of the word, is a logical structure based on assumptions and definitions which permits one to predict the outcome of a maximum number of different experiments on the basis of a minimum number of postulates. One usually desires the postulates to be as self-evident as possible; one demands simplicity, beauty, and even elegance. A physical theory seems to have only a finite lifetime. Newtonian mechanics was replaced by relativistic mechanics, thermodynamics by statistical mechanics, classical by quantum mechanics. There is a logical order in theoretical physics. Indeed, such an order is necessary if the endeavor of theoretical physics is to be meaningful and is to succeed in the direction of more and more general axiomatic deductive systems. The relationship between classical and quantum mechanics is especially interesting. The chapter also presents an overview of the key concepts discussed in this book.
We show that binary orthologic becomes either quantum or classical logic when nothing but modus ponens rule is added to it, depending on the kind of the operation of implication used. We also show that in the usual approach the rule characterizes neither quantum nor classical logic. The dierence turns out to stem from the chosen valuation on a model of a logic. Thus algebraic mappings of axioms of standard quantum logics would fail to yield an orthomodular lattice if a unary|as opposed to binary|valuation were used. Instead, non-orthomodular nontrivial varieties of orthologic are obtained. We also discuss the computational eciency of the binary quantum logic and stress its importance for quantum computation and related algorithms.