Abstract
In this paper, we briefly argue, following ideas set forth by Jacques Dubucs, for a radical version of anti-realism and claim that it leads to the adoption of a ‘substructural’ logic, linear logic. We further argue that, in order to avoids problems such as that of ‘omniscience’, one should develop an epistemic linear logic, which would be weak enough so that the agents could still be described as omniscient, while this would not be problematic anymore. We then examine two possible ways to develop an epistemic linear logic, and eliminate one. We conclude on some remarks about complexity. The paper contains a coding in Coq of fragments of modal linear logic and a proof of the ‘wise men’ puzzle.
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References
Abramsky, S.: 1993, ‘Computational Interpretations of Linear Logic’, Theoretical Computer Science 111, 3–57.
Avron, A.: 1988, ‘Syntax and Semantics of Linear Logic’, Theoretical Computer Science 57, 161–184
Barwise, J. and J. Perry: 1983, Situations and Attitudes, Cambridge MA, MIT Press.
Brauner, T. and V. de Paiva: 1996, ‘Cut-Elimination for Full Intuitionistic Linear Logic’, Technical Report 395, Computer Laboratory University of Cambridge and BRICKS, Denmark.
Caldwell, J.: 1998, ‘Decidability Extracted: Synthesizing Correct-by-Construction Decision Pro-decure from Constructive Proofs’, Ph.D. thesis, Cornell University.
Carbone, A. and S. Semmes: 1997, ‘Making Proofs without Modus Ponens: An Introduction to the Combinatorics and Complexity of Cut Elimination’, Bulletin of the American Mathematical Society 34, 131–159.
Coquand, T. and G. Huet: 1988, ‘The Calculus of Constructions’, Information and Computation 76, 95–120.
Cornes, C., J. Courant, J-C. Filliatre, G. Huet, P. Manoury, C. Murioz, C. Murthy, C. Parent, C. Paulin-Mohring, A. Saibi and B. Werber: 1999–2003, The Coq Proof Assistant Reference Manual, Version 7.4, Rapport Technique 177, INRIA.
Dubucs, J.: 1997, ‘Logique, effectivité et faisabilité’, Dialogue 36, 45–68.
Dubucs, J.: 1998, ‘Hintikka et la question de l'omniscience logique’, in E. Rigal (ed.), Jaakko Hintikka. Questions de logique et de phénoménologie, Paris, Vrin, pp. 141–148.
Dubucs, J.: 2002, ‘Feasibility in Logic’, Synthese 132, 213–237.
Dubucs, J. and M. Marion: 2003, ‘Radical Anti-Realism and Substructural Logics’, in A. Rojszczak, J. Cachro and G. Kurczewski (eds.), Philosophical Dimensions of Logic and Science. Selected Contributed Papers from the 11th International Congress of Logic, Methodology, and the Philosophy of Science, Krakòw, 1999, Dordrecht, Kluwer, pp. 235–249.
Eberle, R. A.: 1974, ‘A Logic of Believing, Knowing and Inferring’, Synthese 26, 356–382.
Fagin, R., J. Y. Halpern, Y. Moses and M. Y. Vardi: 1995, Reasoning about Knowledge, Cambridge MA, MIT Press.
Felty, A.: 2002, ‘Two-Level Meta-Reasoning in Coq’, Fifteenth International Conference on Theorem Proving in Higher Order Logics, Springer-Verlag LNCS 2410.
Girard, J.-Y.: 1970, ‘Une extension de l'interprétation de Gödel a l'analyse et son application a l'elimination des coupures dans l'analyse et la théorie des types’, in J.-E. Fenstad (ed.), Proceedings of the Second Scandinavian Logic Symposium, Amsterdam, North-Holland, pp. 63–92.
Girard, J-Y.: 1987, ‘Linear Logic’, Theoretical Computer Science 50, 1–102.
Girard, J-Y.: 1995a, ‘Light Linear Logic’, in D. Leivant (ed.), Logic and Computational Complexity, Berlin, Springer, pp. 145–176.
Girard, J-Y.: 1995b, ‘Linear Logic: Its Syntax and Semantics’, in J-Y. Girard, Y. Lafont and L. Regnier (eds.), Advances in Linear Logic, Cambridge, Cambridge University Press, pp. 1–42.
Girard, J-Y.: 1998, ‘On the Meaning of Logical Rules I: Syntax vs. Semantics’, http://iml.univ-mrs.fr/ girard/Articles.html.
Girard, J-Y., A. Scedrov and P. Scott: 1992, ‘Bounded Linear Logic, A Modular Approach to Polynomial-Time Computability’, Theoretical Computer Science 97, 1–66.
Halpern, J. Y.: 1986, ‘Reasoning about Knowledge: An Overview’, in J. Y. Halpern (ed.), Theoretical Aspects of Reasoning about Knowledge. Proceedings of the 1986 Conference, San Francisco, Morgan Kaufmann, pp. 1–17.
Hintikka, J.: 1962, Knowledge and Belief, An Introduction to the Logic of Two Notions, New York, Cornell University Press.
Hintikka, J.: 1975, ‘Impossible Possible Worlds Vindicated’, Journal of Philosophical Logic 4, 475–484.
Hodas, J. S.: 1992, ‘Lolli: An Extension of Lambda Prolog with Linear Logic Context Management’, Workshop on the lambda Prolog Programming Language, Philadelphia, 159–168.
Howard, W. A.: 1980, ‘The Formulae-as-Types Notion of Construction’, in J. P. Seldin and J.R. Hindley (eds.), To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, London, Academic Press, pp. 479–490.
Kobayashi, N., T. Shimizu and A. Yonezawa: 1999, ‘Distributed Concurrent Linear Logic Programming’, Theoretical Computer Science 227, 185–220.
Konolige, K.: 1986, A Deduction Model of Belief, San Franscisco, Morgan Kaufmann.
Lescanne, P.: 2001, ‘Epistemic Logic in Higher Order Logic’, http://www.ens-lyon.fr/LIP/Pub/rr2001.html.
Levesque, H.: 1984a, ‘A Logic of Implicit and Explicit Belief’, Proceedings of the National Conference on Artificial Intelligence, AAAI-84, pp. 192–202.
Levesque, H.: 1984b, ‘A Logic of Implicit and Explicit Belief’, Fairchild Laboratory for Artificial Intelligence Research, Technical Report.
Levesque, H.: 1988, ‘Logic and the Complexity of Reasoning’, Journal of Philosophical Logic 17, 355–389.
Martin, A.: 2001, Modal and Fixpoint Linear Logic, Masters Thesis, Department of Mathematics and Statistics, University of Ottawa.
Martin-Löf, P.: 1984, Intuitionistic Type Theory, Naples, Bibliopolis.
Milner, R., J. Parrow and D. Walker: 1992, ‘A Calculus of Mobile Processes I, II’, Information and Computation 100, 1–77.
Powers, J. and C. Webster: 1999, ‘Working with Linear Logic in Coq’, 12th International Conference on Theorem Proving in Higher Order Logics.
Parikh, R.: 1987, ‘Knowledge and the Problem of Logical Omniscience’, in Z. W. Ras and M. Zemankova (eds.), Methodologies for Intelligent Systems, The Hague, Elsevier, pp. 432–439.
Parikh, R.: 1995, ‘Logical Omniscience’, in D. Leivant (ed.), Logic and Computational Complexity, Berlin, Springer, pp. 22–29.
Prawitz, D.: 1965, Natural Deduction. A Proof-Theoretical Study, Stockholm, Almqvist & Wicksell.
Rantala, V.: 1978, ‘Urn Models: A New Kind of Non-Standard Model for First-Order Logic’, in E. Saarinen (ed.), Game-Theoretical Semantics, Dordrecht, D. Reidel, pp. 347–366.
Rantala, V.: 1982, ‘Impossible Worlds Semantics and Logical Omniscience’, in I. Niiniluoto and E. Saarinen (ed.), Intensional Logic: Theory and Applications, Acta Philosophica Fennica, pp. 1–9.
Restall, G.: 2000, An Introduction to Substructural Logics, London & New York, Routledge.
Schroeder-Heister P. and K. Dosen (eds.): 1993, Substructural Logics, Oxford, Clarendon Press.
Urquhart, A.: 1995, ‘The Complexity of Propositional Proofs’, Bulletin of Symbolic Logic 1, 425–467.
Wright, C.: 1993, ‘Strict Finitism’, in C. Wright, Realism, Meaning and Truth, 2nd edn, Oxford, Blackwell, pp. 107–175.
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Marion, M., Sadrzadeh, M. (2009). Reasoning About Knowledge In Linear Logic: Modalities and Complexity. In: Rahman, S., Symons, J., Gabbay, D.M., Bendegem, J.P.v. (eds) Logic, Epistemology, and the Unity of Science. Logic, Epistemology, And The Unity Of Science, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-2808-3_17
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DOI: https://doi.org/10.1007/978-1-4020-2808-3_17
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