# The Formal Representation of Physical Quantities

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## Abstract

In earlier papers I have attempted to develop a certain perspective on the structure of physical theories, and to provide correlatively a critical account of the logic of quantum mechanics.^{1} After a brief outline of the basic framework, the present paper aims to distinguish various non-statistical and statistical aspects of the representation of physical quantities. I shall attempt to elucidate recent work in quantum logic in the following way: when certain mathematical entities are used to represent physical quantities (in a manner successful relative to certain purposes), then logical relations among statements about physical quantities can be defined in terms of these mathematical entities. The most important question about these logical relations is then: to what extent can we expect them in the context of physical theories in general, and to what extent do they reflect the peculiar features of quantum theory?

## Keywords

Probability Measure Physical Quantity Physical Theory Semantic Relation Quantum Logic## Preview

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## Notes

- 1.‘On the Extension of Beth’s Semantics of Physical Theories’,
*Philosophy of Science***37**(1970), 325–339; and ‘The Labyrinth of Quantum Logics’, this volume. For the semantic approach adopted in these papers, see my*Formal Semantics and Logic*Macmillan, New York, 1971.Google Scholar - 3.G. Birkhoff and J. von Neumann, ‘The Logic of Quantum Mechanics’,
*Annals of Mathematics***37**(1936), 823–843; especially p. 825.CrossRefGoogle Scholar - 6.Cf. J. Jauch,
*Foundations of Quantum Mechanics*Reading, Mass. Addison-Wesley, 1968, pp. 98–99;Google Scholar - 6a.and V. S. Varadarajan,
*Geometry of Quantum Theory*vol. 1, Van Nostrand, Princeton: 1968, pp. 108–111. Note that in the present context,*h(m*_{b}*({r*})) =*{**ϕ:m*_{e}*(**ϕ) = r}*i.e. the probability that*m*will be found to have value*r*in state*ϕ*equals 1 if and only if*ϕ*is an eigenstate of*m*corresponding to value*r*. There are reasons for not wishing to generalize upon this fact;Google Scholar - 6b.see D. L. Reisler,
*The Einstein Podolsky Rosen Paradox*unpublished dissertation, Yale University, 1967, especially pp. 162–164.Google Scholar - 9.P. Suppes, ‘The Probabilistic Argument for a Non-Classical Logic in Quantum Mechanics’,
*Philosophy of Science***33**(1966), 14–21.CrossRefGoogle Scholar - 10.A. Fine, ‘Logic, Probability, and Quantum Mechanics’,
*Philosophy of Science***35**(1968), 101–111.CrossRefGoogle Scholar - 12.Cf. Jauch,
*op. cit.*pp. 93–95.Google Scholar - 15.Jauch,
*op. cit.*p. 100.Google Scholar - 16.S. Kochen and E. P. Specker, ‘Logical Structures Arising in Quantum Theory’, in J. W. Addison
*et al.*(eds.),*The Theory of Models*North-Holland Publ Co., Amsterdam, Holland, 1965, pp. 177–189; especially pp. 183–184.Google Scholar - 17.The lattice is
*distributive*if (D)*a*⋀*(b*⋁ c)*≤**(a ⋀ b)*⋁*c*holds,*modular*if (D) holds when*c ≤ a*. Jauch defines*weak modularity*as: if*a ≤**b*then {*a, b*} generates a Boolean sublattice; Varadarajan defines it as: if*a ≤*-*b*and*b ≤**c*then*(a ⋀ b) ⋀ c ≤ (a ⋀ c) ⋁ b*. See Jauch,*op. cit.*p. 86,Google Scholar - 17a.and Varadarajan,
*op. cit.*pp. 107–108.Google Scholar - 17b.The credit for noticing the connection between lattices encountered in quantum logic and unions of Boolean lattices must apparently go to
*S.*Watanabe, ‘A Model of Mind-body Relation in Terms of Modular Logic’,*Synthese***13**(1961), 261–302;CrossRefGoogle Scholar - 17c.for a treatment of this topic with greater generality and rigor see P. D. Finch, ‘On the Structure of Quantum Logic’,
*Journal of Symbolic Logic***34**(1969), 275–282.CrossRefGoogle Scholar - 18.Cf. Jauch,
*op. cit.*p. 100.Google Scholar - 19.See my ‘Semantic Analysis of Quantum Logic’ in C. A. Hooker (ed.),
*Contemporary Research in the Foundations of Quantum Theory*D. Reidel Publ. Co., Dordrecht-Holland, 1973.Google Scholar