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Logics and quantum gravity. (English) Zbl 0818.03031

The author claims here that gravitation theory (because it takes ‘the entire universe into account’) cannot allow an observer/observed distinction, and so cannot use ‘quantum logic’. This claim is unfortunately not further developed but serves as the starting point for investigating a non-Boolean but distributive ‘quantum logic’.

The bulk of this paper, section 4, is devoted to investigating the structure of such a logic, and the author adopts a relatively pseudocomplemented system in which distribution holds but both Excluded Middle and Double Negation fail, apparently an intuitionist logic represented by a Heyting algebra. The paper also includes discussion of modal extensions of the system, and of Kripke semantics developed using a non-symmetric accessibility relation. The ‘essential theorem’ of the paper is apparently that the intuitionist logic is an ‘extension of quantum logics’. This seems a peculiar result given that quantum logics are by definition here non-distributive, and the author himself has commented that to lose both distributivity and complementation is uninteresting. Discussion of ‘quantum nets’, topoi based on fuzzy sets, and Yang-Mills fields follows.

This paper is very wide-ranging and here lies its greatest weakness. Certainly it is at times confused and lacking in rigour. For example we are told “in a general Heyting algebra the list of logical connectives $\wedge$, $\vee$, $⇒$, $⇔$, $¬$, $\forall$, $\exists$, is not redundant as in the case of classical logic...”. Such claims are not followed up by proper clarification and there is throughout a confusion of algebraic and logical terms. In fact many foundational terms are undefined – we are told for instance how to define a formal language with a set of rules including an alphabet and rules of formation, yet no such definitions are provided for the formal languages used here. However, the paper has many interesting excursions, including discussion of a wide range of mathematical models (e.g. from graph theory, topology) for his systems. It also includes a very extensive bibliography.

MSC:
 03G12 Quantum logic 06D20 Heyting algebras 81P10 Logical foundations of quantum mechanics; quantum logic 03B20 Subsystems of classical logic
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