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# Quantales and (noncommutative) linear logic

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It is the purpose of this paper to make explicit the connection between J.-Y. Girard's “linear logic” [4], and certain models for the logic of quantum mechanics, namely Mulvey's “quantales” [9]. This will be done not only in the case of commutative linear logic, but also in the case of a version of noncommutative linear logic suggested, but not fully formalized, by Girard in lectures given at McGill University in the fall of 1987 [5], and which for reasons which will become clear later we call “cyclic linear logic”.

For many of our results on quantales, we rely on the work of Niefield and Rosenthal [10].

The reader should note that by “the logic of quantum mechanics” we do not mean the lattice theoretic “quantum logics” of Birkhoff and von Neumann [1], but rather a logic involving an associative (in general noncommutative) operation “and then”. Logical validity is intended to embody empirical verification (whether a physical experiment, or running a program), and the validity of A & B (in Mulvey's notation) is to be regarded as “we have verified A, and then we have verified B”. (See M. D. Srinivas [11] for another exposition of this idea.)

This of course is precisely the view of the “multiplicative conjunction”, ⊗, in the phase semantics for Girard's linear logic [4], [5]. Indeed the quantale semantics for linear logic may be regarded as an element-free version of the phase semantics.

Corresponding author
Department of Mathematics, Ohio State University at Mansfield, 1680 University Drive, Mansfield, Ohio 44906
References
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[1]Birkhoff, G. and von Neumann, J., The logic of quantum mechanics, Annals of Mathematics, ser. 2, vol. 37 (1936), pp. 823843; reprinted in [6], Vol. I, pp. 1–26.
[2]Freyd, P. J. and Yetter, D. N., Braided compact closed categories with applications to low dimensional topology, Advances in Mathematics (to appear).
[3]Freyd, P. J. and Yetter, D. N., Coherence theorems via knot theory, preprint.
[4]Girard, J.-Y., Linear logic, Theoretical Computer Science, vol. 50 (1987), pp. 1102.
[5]Girard, J.-Y., Seminar lectures, Le Groupe Interuniversitaire en Études Catégoriques, McGill University, Montreal, 11 1987.
[6]Hooker, C. A. (editor), Logico-algebraic approach to quantum mechanics, Vols I, II, Reidel, Dordrecht, 1975, 1979.
[7]Joyal, A. and Street, R., Braided monoidal categories, preprint.
[8]Knuth, D. E. and Bendix, P. E., Simple word problems in universal algebra, Computational problems in abstract algebra (Leech, J., editor), Pergamon Press, Oxford, 1970, pp. 263297.
[9]Mulvey, C. J., &, Second topology conference (Taormina, 1984), Rendiconti del Circolo Matematico di Palermo, ser. 2, supplement no. 12 (1986), pp. 94104.
[10]Niefield, S. B. and Rosenthal, K. I., Constructing locales from quantales, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 104 (1988), pp. 215234.
[11]Srinivas, M. D., Foundations of a quantum probability theory, Journal of Mathematical Physics, vol. 16 (1975), pp. 16721685; reprinted in [6], Vol. II, pp. 227–260.
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The Journal of Symbolic Logic
• ISSN: 0022-4812
• EISSN: 1943-5886
• URL: /core/journals/journal-of-symbolic-logic
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