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 rale (regularity), and we consider any subsetW 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 operationμ on an IL-algebra. Then we introduce a relational semantics based on pretopologies with an additional binary relationr 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 onr (in the spirit of correspondence theory), for any of its extensions axiomatized by any subsetW 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.
Similar content being viewed by others
References
Battilotti, G. and Sambin, G.: 1993, ‘Presentation of quantales and locales by means of pretopologies’,Preprint, Dip. di Matematica Pura e Appl., Univ. di Padova.
Benthem van, J.: 1985,Modal logic and classical logic, Bibliopolis, Napoli.
Benthem van, J.: 1986, ‘Correspondence theory’, pp. 167–247 inHandbook of Philosophical Logic, Vol II: Extensions of Classical Logic, D. Gabbay and F. Guenthner, eds., Dordrecht: Reidel.
Girard, J. Y.: 1987, ‘Linear logic’,Theoretical Computer Science 50, 1–102.
Ono, H.: 1992, ‘Algebraic aspects of logics without structurales rules’,Contemporary Mathematics 131, (Part 3), 601–621.
Ono, H.: 1993, ‘Semantics for Substructural Logics’, pp. 259–291 inSubstructurals Logics, P. Schroeder-Heister and K. Došen, eds, Clarendon Press, Oxford.
Sambin, G.: 1992, ‘Pretopologies and completeness proofs’,Preprint, Dip. di Matematica Pura e Appl., Univ. di Padova.
Sambin, G.: 1993, ‘The semantics of pretopologies’, pp. 293–307 inSubstructurals Logics, P. Schroeder-Heister and K. Došen, eds, Clarendon Press, Oxford.
Sambin, G., Valentini, S. and Virgili, P.: 1992, ‘Constructive domain theory as a branch of intuitionistic pointfree topology’,Preprint, Dip. di Matematica Pura e Appl., Univ. di Padova.
Troelstra, A. S.: 1992, ‘Lectures on linear logic’,CSLI Lectures Notes 29, Center for the Study of Language and Information, Stanford.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bucalo, A. Modalities in linear logic weaker than the exponential “of course”: Algebraic and relational semantics. J Logic Lang Inf 3, 211–232 (1994). https://doi.org/10.1007/BF01053246
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01053246