Definability and Commonsense Reasoning

Abstract The definition of concepts is a central problem in commonsense reasoning. Many themes in nonmonotonic reasoning concern implicit and explicit definability. Implicit definability in nonmonotonic logic is always relative to the context—the current theory of the world. We show that fixed point equations provide a generalization of explicit definability, which correctly captures the relativized context. Theories expressed within this logical framework provide implicit definitions of concepts. Moreover, it is possible to derive these fixed points entirely within the logic.

[1]  V. Lifschitz.  Formal theories of action , 1987 .

[2]  Hector J. Levesque,et al.  All I Know: A Study in Autoepistemic Logic , 1990, Artif. Intell..

[3]  Claudio Bernardi.  The fixed-point theorem for diagonalizable algebras , 1975 .

[4]  J. McCarthy.  Circumscription|a Form of Nonmonotonic Reasoning , 1979 .

[5]  Kurt Konolige,et al.  On the Relation Between Default and Autoepistemic Logic , 1987, Artif. Intell..

[6]  E. Rosch,et al.  Categorization of Natural Objects , 1981 .

[7]  Donald Perlis,et al.  Languages With Self-Reference I: Foundations , 1985, Artif. Intell..

[8]  Robert Stalnaker,et al.  A Note on Non-Monotonic Modal Logic , 1993, Artif. Intell..

[9]  M. Fitting.  Proof Methods for Modal and Intuitionistic Logics , 1983 .

[10]  Dov M. Gabbay,et al.  A Structural Property on Modal Frames Characterizing Default Logic , 1996, Log. J. IGPL.

[11]  Victor W. Marek,et al.  Modal nonmonotonic logics: ranges, characterization, computation , 1991, JACM.

[12]  Raymond M. Smullyan.  Languages in Which Self Reference is Possible , 1957, J. Symb. Log..

[13]  Matthew L. Ginsberg,et al.  Readings in Nonmonotonic Reasoning , 1987, AAAI 1987.

[14]  Victor W. Marek,et al.  Nonmonotonic logic - context-dependent reasoning , 1997, Artificial intelligence.

[15]  Grigori Schwarz.  In Search of a "True" Logic of Knowledge: The Nonmonotonic Perspective , 1995, Artif. Intell..

[16]  Raymond Reiter,et al.  Circumscription Implies Predicate Completion (Sometimes) , 1982, AAAI.

[17]  Raymond Reiter,et al.  A Logic for Default Reasoning , 1987, Artif. Intell..

[18]  Miroslaw Truszczynski,et al.  Minimal Knowledge Problem: A New Approach , 1994, Artif. Intell..

[19]  Miroslaw Truszczynski,et al.  Modal Interpretations of Default Logic , 1991, IJCAI.

[20]  Robert C. Moore.  Semantical Considerations on Nonmonotonic Logic , 1985, IJCAI.

[21]  Fiora Pirri,et al.  Intuitionistic Autoepistemic Logic , 1997, Stud Logica.

[22]  Drew McDermott,et al.  Nonmonotonic Logic II: Nonmonotonic Modal Theories , 1982, JACM.

[23]  Drew McDermott,et al.  Non-Monotonic Logic I , 1987, Artif. Intell..

[24]  R. Goldblatt.  Logics of Time and Computation , 1987 .

[25]  Miroslaw Truszczynski,et al.  Modal Logic S4F and The Minimal Knowledge Paradigm , 1992, TARK.

[26]  John McCarthy,et al.  Circumscription - A Form of Non-Monotonic Reasoning , 1980, Artif. Intell..

[27]  Evert W. Beth,et al.  On Padoa’s Method in the Theory of Definition , 1953 .

[28]  Krzysztof R. Apt,et al.  Logics and Models of Concurrent Systems , 1989, NATO ASI Series.

[29]  J. Doyle.  Some theories of reasoned assumptions : an essay in rational psychology , 1983 .

[30]  Craig Smorynski,et al.  Quantified modal logic and self-reference , 1987, Notre Dame J. Formal Log..

[31]  Miroslaw Truszczynzki.  Modal interpretations of default logic , 1991, IJCAI 1991.

[32]  Peter Gärdenfors,et al.  A Geometric Model of Concept Formation , 1992 .

[33]  Grigoris Antoniou,et al.  Nonmonotonic reasoning , 1997 .

[34]  Franco Montagna,et al.  The predicate modal logic of provability , 1984, Notre Dame J. Formal Log..

[35]  Donald Perlis,et al.  Languages with Self-Reference II: Knowledge, Belief, and Modality , 1988, Artif. Intell..

[36]  Jon Doyle,et al.  A model for deliberation, action, and introspection , 1981, SGAR.

[37]  J. Monk,et al.  Unprovability of Consistency , 1976 .

[38]  R. Smullyan.  Diagonalization and Self-Reference , 1994 .

[39]  Vladimir Lifschitz,et al.  Computing Circumscription , 1985, IJCAI.

[40]  Fangzhen Lin,et al.  State Constraints Revisited , 1994, J. Log. Comput..

[41]  G. Boolos,et al.  Self-Reference and Modal Logic , 1985 .

[42]  R. Solovay.  Provability interpretations of modal logic , 1976 .

[43]  Grigori Schwarz,et al.  Minimal model semantics for nonmonotonic modal logics , 1992, [1992] Proceedings of the Seventh Annual IEEE Symposium on Logic in Computer Science.

[44]  Robert C. Moore.  Possible-World Semantics for Autoepistemic Logic , 1987, NMR.

[45]  Fangzhen Lin,et al.  Epistemic Semantics for Fixed-Points Non-Monotonic Logics , 1990, TARK.

[46]  Grigori F. Shvarts.  Autoepistemic Modal Logics , 1990, TARK.

[47]  Setsuo Ohsuga,et al.  Information Modelling and Knowledge Bases , 1990 .