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Published online by Cambridge University Press: 28 February 2022
One can and often does use the word ‘logic’ in a variety of metaphorical ways, speaking of “the logic of the situation”, “the logic of events”, “the logic of history”, and so on. No harm is done, so long as one realizes that these usages are metaphorical, and should not be given any deeper significance – at least without extensive discussion of the meaning of the word ‘logic’ in these phrases. In this metaphorical sense, one may speak of “the logic of quantum mechanics”, just as one may speak of “the logic of modern art”, for example.
But if we want to use the term ‘logic’ in a non-metaphorical sense – and I take it that this is the intent in all serious discussions of ‘quantum logic’–we are under some obligation to give an account of the significance to be given to the word ‘logic’ and to demonstrate that the proposed usage does not do violence to the several-thousand-year-old tradition in the field.
Research partially supported by the National Science Foundation.
1 In reading some of the articles on quantum logic, I cannot help wishing that their authors had done the same, and that one didn’t have to try to infer their positions on these basic questions about logic from their discussions of quantum theory.
2 I have elsewhere written ‘A Note on the Concept of Scientific Practice’ in For Dirk Struik (Boston Studies in the Philosophy of Science, Volume XV, D. Reidel Publ. Co.; Dordrecht, Holland and Boston, 1974).
3 I leave aside here the important ideological functions of language, which may aid in the mastery of one group of men by another, as not directly relevant to the topic of quantum logic; although oh other occasions I would be prepared to discuss the question of whether certain misinterpretations of quantum theory do not – in highly mediated fashion, of course – play a role, however modest, in certain general ideological misinterpretations of scientific practice. See note 2 for reference to a paper containing the beginnings of such a discussion.
4 Those who have followed recent discussions on the methodology used by Marx in his economic studies will recognize my debt to them. My formulation here is essentially an attempt to apply the methodological criteria outlined by Marx in his introduction to the Grundrisse. For an excellent translation of this introduction, with extensive commentary and other materials, see Terrell, Carver, ed., Karl Marx: Texts on Method (Basil Blackwell, Oxford, 1975)Google Scholar. For an excellent discussion of Marx’s views on science, with extensive quotations, see the biographical article by Robert S., Cohen, ‘Karl Marx’ to appear in The Dictionary of Scientific Biography.Google Scholar
5 ‘How Logical is Quantum Logic’ to appear in the University of Pittsburgh Series in the Philosophy of Science. Let me add here, dogmatically again in the interests of brevity, but to avoid possible misunderstandings of my position, that I do not accept the view of the sciences which sees logic as the foundation of mathematics, and mathematics as the foundation for the so-called empirical sciences. The relations between the development of the various sciences are too complex and many-sided to be fruitfully encompassed in such a linear geneology. I think the by now traditional distinction between the formal and the empirical sciences must be re-examined and replaced by a more adequate image, which will avoid the pseudo-problems that arise when we seek to link up the ‘formal’ and the ‘empirical’ - which should never have been sundered in the first place.
6 Hilary Putnam, one of the chief advocates of this analogy, gives us a nice example of a shift in the analogy when he first writes:
and then states:
” …just as it is impossible to understand the true nature of space and time as long as it is assumed that ‘space’ obeys the laws of Euclidean geometry, so it is impossible to understand the true nature of microprocesses as long as it is assumed that physical propositions obey the laws of Aristotle’s logic”.
After the.antecedent: “nature of space and time… ‘space’ obeys…”, we should have expected the consequent: “nature of microprocesses” …microprocesses “obey” - but Putnam is too good a logician for that! So we get “ …propositions obey…”. Of course . he could have restored the balance of the phrase by writing “so it is impossible to understand the true nature of propositions about microprocesses…” – but there would have gone the point of his argument! Hilary, Putnam, ‘How to Think Quantum-Logically’ Synthese 29, 55 (1974)Google Scholar.
7 “Physicists know what it means to interpret physical experience in terms of mathematical concepts; but to imagine that they might have to interpret somehow pre-existing mathematical symbols in physical terms is an idealistic construction alien to the spirit of science”. Leon, Rosenfeld , ‘Misunderstandings about the Foundations of Quantum Theory’ in R. S., Cohen and J., Stachel (eds.), Leon Rosenfeld, Selected Papers (Boston Studies in the Philosophy of Science, Volume XXI, D. Reidel Publ. Co.: Dordrecht, Holland and Boston, forthcoming).Google Scholar
8 Martin Strauss had long ago pointed this out for quantum theory. See the translations of his papers on complementarity dating from the 1930’s, in Martin, Strauss, Modem Physics and its Philosophy (D. Reidel Publ. Co., Dordrecht, Holland and Boston, 1972)Google Scholar. Peter Mittelstaedt has also emphasized this for quantum theory, with his concepts of ‘restricted availability’ versus ‘unrestricted availability’ for propositions. (Boston Studies in the Philosophy of Science, Vol. XVIII, D. Reidel, 1975). But it is just as true for properties of open systems in classical physics, as I will discuss in more detail elsewhere. (See Note 5).
9 A detailed discussion of this question will appear in Reference 5. A brief treatment will be found in my ‘Comments on ‘Logical Problems Suggested by Empirical Theories”, in Boston Studies in the Philosophy of Science, Volume XXXI, to appear shortly.
10 Those few properties which serve to characterize the type of microsystem in question, such as charge, mass, spin, etc., do indeed behave like classical properties of a closed system. We may assert: ‘The charge of the electron is e’ without further qualification. But whether the contingent properties of a microsystem, such as its position, momentum, etc., which serve to characterize its particular physical interactions, can be so asserted is the fundamental issue.
11 I will discuss these issues more fully in Reference 5. Brief discussions of various aspects will be found in Reference 2, pp. 425-430; and in my ‘The Rise and Fall of Geometrodynamics’ Boston Studies in the Philosophy of Science, Volume XX (D. Reidel Publ. Co., Dordrecht, Holland and Boston, 1974), especially pp. 40-41.
12 One may also choose to allow compound statements, referring to compatible properties of a quantum system (such as ‘The x-component of the momentum of the electron is px and the y component of its position is yO’), while forbidding compounds referring to incompatible properties (such as ‘The momentum of the electron is) and the position is x’), without attaching a truth value to such combinations apart from a specification of the macroscopic context. This approach, leading to the use of a partial Boolean algebra for quantum logic, has been advocated by Martin Strauss (See Note 7 for reference). It appears that one is even able to allow combinations of all statements about properties of microsystems, modally qualified, in such a way as to utilize a lattice for quantum logic, without thereby being committed to a non-standard assertoric logic. See Note 9 for reference to a preliminary discussion of this problem, and Note 5 for reference to a fuller discussion.
13 Perhaps I should state that I am fully aware that many interesting, and even important, studies of various mathematical structures in quantum theory, which have shed new light on the nature and implications of the theory, have been carried out under the label of ‘quantum logic’ and indeed that various attempts at the generalization of the existing theory have been similarly labelled. But these studies must stand on their own merits, and do not help to resolve the properly logical issues. It is only the labelling which is open to question here, not the contents of the package.