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Logic and Logical Philosophy

In defense of operational quantum logic

Sonja Smets

Published Nov 23, 2003• 5 citations
Publisher

Abstract

In the literature the work of C. Piron on OQL, “the operational quantum logic of the Geneva School”, has a few times been criticised. Those criticisms were often due to misunderstandings, as has already been pointed out in [19]. In this paper we follow the line of defense in favour of OQL by replying to the criticisms formulated some time ago in [4] and [17]. In order for the reader to follow our argumentation, we briefly analyze the basic conceptual machinery of OQL.

DOI
10.12775/llp.2003.010

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In the literature the work of C. Piron on OQL, “the operational quantum logic of the Geneva School”, has a few times been criticised. Those criticisms were often due to misunderstandings, as has already been pointed out in [19]. In this paper we follow the line of defense in favour of OQL by replying to the criticisms formulated some time ago in [4] and [17]. In order for the reader to follow our argumentation, we briefly analyze the basic conceptual machinery of OQL.
We analyze G.M. Hardegree's interpretation of the Sasaki hook as a Stalnaker conditional and explain how he makes use of the basic conceptual machinery of OQL, i.e. the operational quantum logic which originated with the Geneva Approach to the foundations of physics. In particular we focus on measurements which are ideal and of the first kind, since these encode the content of the so-called Sasaki projections within the Geneva Approach. The Sasaki projections play a fundamental role when analyzing the condition under which the properties expressed by Sasaki hooks can be considered as actual. We finish with a note on how the Sasaki hook can be conceived as ``assigning causes for properties to be actual", which links the interpretation of G.M. Hardegree to what has been called ``dynamic OQL".
In this paper we show how ideas coming from two areas of research in logic can reinforce each other. The first such line of inquiry concerns the “dynamic turn” in logic and especially the formalisms inspired by Propositional Dynamic Logic (PDL); while the second line concerns research into the logical foundations of Quantum Physics, and in particular the area known as Operational Quantum Logic, as developed by Jauch and Piron (Helve Phys Acta 42:842–848, 1969 ), Piron (Foundations of Quantum Physics, 1976 ). By bringing these areas together we explain the basic ingredients of Dynamic Quantum Logic, a new direction of research in the logical foundations of physics.
This paper will take into account the Lindenbaum property in Ortho modular Quantum Logic (OQL) and Partial Classical Logic (PCL). The Lindenbaum property has an interest both from a logical and a physical point of view since it has to do with the problem of the completeness of quantum theory and with the possibi? lity of extending any semantically non-contradictory set of formulas to a semanti? cally non-contradictory complete set of formulas. The main purpose of this paper is4 to show that both OQL and PCL cannot satisfy the Lindenbaum property.