Critique of Putnam's quantum logic

John Bell, Michael Hallett

This paper treats some of the issues raised by Putnam's discussion of, and claims for, quantum logic, specifically: that its proposal is a response to experimental difficulties; that it is a reasonable replacement for classical logic because its connectives retain their classical meanings, and because it can be derived as a logic of tests. We argue that the first claim is wrong(1), and that while conjunction and disjunction can be considered to retain their classical meanings, negation cruciallydoes not.The argument is conductedviaa thorough analysis of how the meet, join and complementation operations are defined in the relevant logical structures, respectively Boolean- and ortholattices(3). Since Putnam wishes to reinstate a realist interpretation of quantum mechanics, we ask how quantum logic can be a logic of realism. We show that it certainly cannot be a logic ofbivalencerealism (i.e., of truth and falsity), although itisconsistent with some form ofontologicalrealism(4). Finally, we show that while a reasonable explication of the idealized notion of test yields interesting mathematical structure, it by no means yields the rich ortholattice structure which Putnam (following Finkelstein) seeks.

Leopoldo Rom�n, B. Rumbos

Jon Michael Dunn

It has been argued, most notably by Putnam (1969, 1974) that socalled quantum logic (which we shall often refer to as. orthomodular logic (OML)) is the one true logic,-as has been shown by quantum mechanical experiments. Putnam also has strong universalist tendencies, which would require him to use the same logic in all domains of reasoning, including mathematics, and it has been urged, recently (as one horn of a dilemma) by Hellman (1981), that there might be some problems in working out the classical mathematics of the Hilbertspace foundations of.quantum mechanics in a quantum logic framework. In this paper we show (given one natural framework)(§3) first that if the first-order Peano arithmetic is formulated with quantum logic that it has the same theorems as the classical first-order Peano arithmetic. Distribution (for first-order arithmetical formulas) is a theorem not of quantum logic but rather of arithmetic.

Matthew Ingleby

Thomas A. Brody

Michael R. Gardner

Putnam and Finkelstein have proposed the abandonment of distributivity in the logic of quantum theory. This change results from defining the connectives, not truth-functionally, but in terms of a certain empirical ordering of propositions. Putnam has argued that the use of this ordering (“implication”) to govern proofs resolves certain paradoxes. But his resolutions are faulty; and in any case, the paradoxes may be resolved with no changes in logic. There is therefore no reason to regard the partially ordered set of propositions as a logic—i.e. as embodying a criterion for soundness of proofs. Its role in quantum theory ought to be understood in an entirely different way.