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Modalities in Linear Logic Weaker than the Exponential "Of Course": Algebraic and Relational Semantics

Anna Bucalo
Journal of Logic, Language, and Information
Vol. 3, No. 3 (1994), pp. 211-232
Published by: Springer
Stable URL: http://www.jstor.org/stable/40180048
Page Count: 22
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Modalities in Linear Logic Weaker than the Exponential "Of Course": Algebraic and Relational Semantics
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Abstract

We present a semantic study of a family of modal intuitionistic linear systems, providing various logics with both an algebraic semantics and a relational semantics, to obtain completeness results. We call modality a unary operator □ on formulas which satisfies only one rule (regularity), and we consider any subset W of a list of axioms which defines the exponential "of course" of linear logic. We define an algebraic semantics by interpreting the modality □ as a unary operation µ on an IL-algebra. Then we introduce a relational semantics based on pretopologies with an additional binary relation γ between information states. The interpretation of □ is defined in a suitable way, which differs from the traditional one in classical modal logic. We prove that such models provide a complete semantics for our minimal modal system, as well as, by requiring the suitable conditions on γ (in the spirit of correspondence theory), for any of its extensions axiomatized by any subset W as above. We also prove an embedding theorem for modal IL-algebras into complete ones and, after introducing the notion of general frame, we apply it to obtain a duality between general frames and modal IL-algebras.

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