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Quantales and (noncommutative) linear logic

Published online by Cambridge University Press:  12 March 2014

David N. Yetter
David N. Yetter*
Affiliation:
Department of Mathematics and Statistics, McGill University, Montréal, P. Q. H3A 2K6, Canada
*
Department of Mathematics, Ohio State University at Mansfield, 1680 University Drive, Mansfield, Ohio 44906
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Extract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 55 , Issue 1 , March 1990 , pp. 41 - 64
Copyright
Copyright © Association for Symbolic Logic 1990

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References

REFERENCES

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