Modalities in linear logic weaker than the exponential “of course”: Algebraic and relational semantics [Book Review]
David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Journal of Logic, Language and Information 3 (3):211-232 (1994)
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 operation 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.
|Keywords||linear logic modality pretopology relational semantics general frame|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
Giovanni Sambin (1995). Pretopologies and Completeness Proofs. Journal of Symbolic Logic 60 (3):861-878.
Citations of this work BETA
Michael Moortgat (1996). Multimodal Linguistic Inference. Journal of Logic, Language and Information 5 (3-4):349-385.
Norihiro Kamide (2003). Normal Modal Substructural Logics with Strong Negation. Journal of Philosophical Logic 32 (6):589-612.
Similar books and articles
D. M. Gabbay (1996). Fibred Semantics and the Weaving of Logics Part 1: Modal and Intuitionistic Logics. Journal of Symbolic Logic 61 (4):1057-1120.
Giovanni Sambin (1999). Subdirectly Irreducible Modal Algebras and Initial Frames. Studia Logica 62 (2):269-282.
Josep Maria Font & Miquel Rius (2000). An Abstract Algebraic Logic Approach to Tetravalent Modal Logics. Journal of Symbolic Logic 65 (2):481-518.
Norihiro Kamide (2002). Kripke Semantics for Modal Substructural Logics. Journal of Logic, Language and Information 11 (4):453-470.
Simone Martini & Andrea Masini (1994). A Modal View of Linear Logic. Journal of Symbolic Logic 59 (3):888-899.
Sergei P. Odintsov & E. I. Latkin (2012). BK-Lattices. Algebraic Semantics for Belnapian Modal Logics. Studia Logica 100 (1-2):319-338.
W. J. Blok (1980). The Lattice of Modal Logics: An Algebraic Investigation. Journal of Symbolic Logic 45 (2):221-236.
Horacio Arló-Costa & Eric Pacuit (2006). First-Order Classical Modal Logic. Studia Logica 84 (2):171 - 210.
Nobu-Yuki Suzuki (1999). Algebraic Kripke Sheaf Semantics for Non-Classical Predicate Logics. Studia Logica 63 (3):387-416.
W. J. Blok & J. Rebagliato (2003). Algebraic Semantics for Deductive Systems. Studia Logica 74 (1-2):153 - 180.
Added to index2009-01-28
Total downloads7 ( #291,913 of 1,725,430 )
Recent downloads (6 months)2 ( #268,739 of 1,725,430 )
How can I increase my downloads?