Abstract
We investigate the semantics of the logical systems obtained by introducing the modalities □ and ⋄ into the family of substructural implication logics (including relevant, linear and intuitionistic implication). Then, in the spirit of the LDS (Labelled Deductive Systems) methodology, we "import" this semantics into the classical proof system KE. This leads to the formulation of a uniform labelled refutation system for the new logics which is a natural extension of a system for substructural implication developed by the first two authors in a previous paper.
References
ANDERSON, A. R., and N. D. BELNAP JR, 1975, Entailment: the Logic of Relevance and Necessity, 1, Princeton University Press, Princeton.
ABRUSCI, V. M., 1991, ‘Phase semantics and sequent calculus for pure noncommutative classical linear propositional logic’, Journal of Symbolic Logic 56, 1403–1451.
ALLWEIN, G., and J. M. DUNN, 1993, ‘Kripke models for linear logic’, The Journal of Symbolic Logic 58, 514–545.
ALEXIEV, V., 1994, ‘Applications of linear logic to computation: An overview’, Bulletin of the IGPL 2, 77–107.
AMATI, G., and F. PIRRI, 1993, A uniform tableau method for intuitionistic modal logics I, Manuscript.
AVRON, A., 1988, ‘The semantics and proof theory of linear logic’, Theoretical Computer Science 57, 161–184.
BoŽiĆ, M., and K. DoŠen, 1984, ‘Models for normal intuitionistic modal logics’, Studia Logica 43, 217–245.
BRODA, K., M. D'AGOSTINO, and A. RUSSO, 1997, ‘Transformation methods in LDS’, In Hans Jürgen Ohlbach and Uwe Reyle, editors, Logic, Language and Reasoning. Essays in Honor of Dov Gabbay, Kluwer Academic Publishers, To appear.
BELNAP, N., 1982, ‘Display logic’, Journal of Philosophical Logic 11, 375–417.
BELNAP, N., 1990, ‘Linear logic displayed’, Notre Dame Journal of Formal Logic 31, 14–25.
BULL, R., and K. SEGERBERG, 1984, ‘Basic modal logic’, In Dov Gabbay and Franz Guenthner, editors, Handbook of Philosophical Logic, volume II, chapter II.I, 1–88. Kluwer Academic Publishers.
D'AGOSTINO, M., and D. M. GABBAY, 1994, ‘A generalization of analytic deduction via labelled deductive systems.Part I: Basic substructural logics’, Journal of Automated Reasoning 13, 243–281.
D'AGOSTINO, M., and M. MONDADORI, 1994, ‘The taming of the cut’, Journal of Logic and Computation 4, 285–319.
DÔsen, K., 1988, ‘Sequent systems and groupoid models I’, Studia Logica 47, 353–385.
DÔsen, K., 1989, ‘Sequent systems and groupoid models II’, Studia Logica 48, 41–65.
DÔsen, K., 1993, ‘A historical introduction to substructural logics’, In P. Schroeder Heister and Kosta Dôsen, editors, Substructural Logics, 1–31, Oxford University Press.
DUNN, J. M., 1986, ‘Relevant logic and entailment’, In D. Gabbay and F. Guenthner, editors, Handbook of Philosophical Logic, volume III, 117–224. Reidel, Dordrecht.
FITTING, M., 1983, Proof Methods for Modal and Intuitionistic Logics, Reidel, Dordrecht.
FISCHER-SERVI, G., 1977, ‘On modal logic with an intuitionistic base’, Studia Logica 36, 141–149.
GABBAY, D. M., 1994, ‘Labelled Deductive Systems, Volume 1 — Foundations’, Technical Report MPI-I–94-223, Max-Planck-Institut Für Informatik. Preliminary partial draft of a book intended for Oxford University Press.
GARSON, J., 1989, ‘Modularity and relevant logic’, Notre Dame Journal of Formal Logic 30, 207–223.
GIRARD, J. Y., 1987, ‘Linear logic’, Theor Computer Science 50, 1–102.
MACCAULL, W., 1996, ‘Relational semantics and a relational proof system for full Lambek calculus’, Technical report, Dept. Mathematics and Computing Sciences, St. Francis Xavier University.
ONO, H., 1993, ‘Semantics for substructural logics’, In Peter Schroeder-Heister, editor, Substructural Logics, 259–291. Oxford University Press.
PLOTKIN, G. D., and C. P. STIRLING, 1986, ‘A framework for intuitionistic modal logic’, In J. Y. Halpern, editor, Theoretical Aspects of Reasoning About Knowledge, 399–406.
SAMBIN, G., 1993, ‘The semantics of pretopologies’, In Peter Schroeder-Heister, editor, Substructural Logics, 293–307. Oxford University Press.
SYMPSON, A. K., 1993, The Proof Theory and Semantics of Intuitionistic Modal Logic, PhD thesis, University of Edinburgh.
URQUIIART, A., 1972, ‘Semantics for relevant logic’, The Journal of Symbolic Logic 37, 159–170.
WANSING, H., 1993, The Logic of Information Structures, Number 681 in Lecture Notes in Artificial Intelligence. Springer-Verlag, Berlin.
WANSING, H., 1994, ‘Sequent calculi for normal modal propositional logics’, Journal of Logic and Computation 4, 125–142.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
D'agostino, M., Gabbay, D.M. & Russo, A. Grafting Modalities onto Substructural Implication Systems. Studia Logica 59, 65–102 (1997). https://doi.org/10.1023/A:1004947400860
Issue Date:
DOI: https://doi.org/10.1023/A:1004947400860