Journal of Symbolic Logic

Monoid based semantics for linear formulas

W. P. R. Mitchell and H. Simmons

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Each Girard quantale (i.e., commutative quantale with a selected dualizing element) provides a support for a semantics for linear propositional formulas (but not for linear derivations). Several constructions of Girard quantales are known. We give two more constructions, one using an arbitrary partially ordered monoid and one using a partially ordered group (both commutative). In both cases the semantics can be controlled be a relation between pairs of elements of the support and formulas. This gives us a neat way of handling duality.

Article information

J. Symbolic Logic, Volume 67, Issue 2 (2002), 505-527.

First available in Project Euclid: 18 September 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03F52: Linear logic and other substructural logics [See also 03B47]
Secondary: 03G30: Categorical logic, topoi [See also 18B25, 18C05, 18C10]


Mitchell, W. P. R.; Simmons, H. Monoid based semantics for linear formulas. J. Symbolic Logic 67 (2002), no. 2, 505--527. doi:10.2178/jsl/1190150094.

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