David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Studia Logica 61 (2):237-280 (1998)
In this work we develop goal-directed deduction methods for the implicational fragment of several modal logics. We give sound and complete procedures for strict implication of K, T, K4, S4, K5, K45, KB, KTB, S5, G and for some intuitionistic variants. In order to achieve a uniform and concise presentation, we first develop our methods in the framework of Labelled Deductive Systems [Gabbay 96]. The proof systems we present are strongly analytical and satisfy a basic property of cut admissibility. We then show that for most of the systems under consideration the labelling mechanism can be avoided by choosing an appropriate way of structuring theories. One peculiar feature of our proof systems is the use of restart rules which allow to re-ask the original goal of a deduction. In case of K, K4, S4 and G, we can eliminate such a rule, without loosing completeness. In all the other cases, by dropping such a rule, we get an intuitionistic variant of each system. The present results are part of a larger project of a goal directed proof theory for non-classical logics; the purpose of this project is to show that most implicational logics stem from slight variations of a unique deduction method, and from different ways of structuring theories. Moreover, the proof systems we present follow the logic programming style of deduction and seem promising for proof search [Gabbay and Reyle 84, Miller et al. 91].
|Keywords||Philosophy Logic Mathematical Logic and Foundations Computational Linguistics|
|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
No references found.
Citations of this work BETA
No citations found.
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.
Takahito Aoto (1999). Uniqueness of Normal Proofs in Implicational Intuitionistic Logic. Journal of Logic, Language and Information 8 (2):217-242.
Stéphane Demri & Dov Gabbay (2000). On Modal Logics Characterized by Models with Relative Accessibility Relations: Part II. Studia Logica 66 (3):349-384.
Ryo Kashima & Norihiro Kamide (1999). Substructural Implicational Logics Including the Relevant Logic E. Studia Logica 63 (2):181-212.
Sara Negri (2011). Proof Analysis: A Contribution to Hilbert's Last Problem. Cambridge University Press.
Luca Viganò (2000). Labelled Non-Classical Logics. Kluwer Academic Publishers.
M. W. Bunder (1982). Deduction Theorems for Weak Implicational Logics. Studia Logica 41 (2-3):95 - 108.
Dov M. Gabbay (2000). Goal-Directed Proof Theory. Kluwer Academic.
A. Avron (2000). Implicational F-Structures and Implicational Relevance Logics. Journal of Symbolic Logic 65 (2):788-802.
Added to index2009-01-28
Total downloads11 ( #156,525 of 1,679,387 )
Recent downloads (6 months)1 ( #183,003 of 1,679,387 )
How can I increase my downloads?