First-Order Classical Modal Logic

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40].We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics.

[1]  Moshe Y. Vardi.  On Epistemic Logic and Logical Omniscience , 1986, TARK.

[2]  James W. Garson,et al.  Quantification in Modal Logic , 1984 .

[3]  Ruth C. Barcan.  The identity of individuals in a strict functional calculus of second order , 1947, Journal of Symbolic Logic.

[4]  Angelo Gilio,et al.  Probabilistic Reasoning Under Coherence in System P , 2002, Annals of Mathematics and Artificial Intelligence.

[5]  Timothy Williamson,et al.  Bare Possibilia , 1998 .

[6]  Horacio L. Arló-Costa.  First Order Extensions of Classical Systems of Modal Logic; The role of the Barcan schemas , 2002, Stud Logica.

[7]  D. Scott,et al.  Assigning Probabilities to Logical Formulas , 1966 .

[8]  Dov M. Gabbay,et al.  Investigations in modal and tense logics with applications to problems in philosophy and linguistics , 1976 .

[9]  Isaac Levi,et al.  For the Sake of the Argument - Ramsey Test Conditionals, Inductive Inference and Nonmonotonic Reasoning , 2007 .

[10]  Torben Braüner,et al.  First-order modal logic , 2007, Handbook of Modal Logic.

[11]  Pierpaolo Battigalli,et al.  Recent results on belief, knowledge and the epistemic foundations of game theory , 1999 .

[12]  Henry E. Kyburg,et al.  Probabilistic inference and non-monotonic inference , 1988, UAI.

[13]  M. de Rijke,et al.  Modal Logic , 2001, Cambridge Tracts in Theoretical Computer Science.

[14]  Krister Segerberg,et al.  An essay in classical modal logic , 1971 .

[15]  Martin Gerson.  The Inadequacy of the Neighbourhood Semantics for Modal Logic , 1975, J. Symb. Log..

[16]  Lawrence S. Moss,et al.  Vicious circles - on the mathematics of non-wellfounded phenomena , 1996, CSLI lecture notes series.

[17]  Marc Pauly,et al.  A Modal Logic for Coalitional Power in Games , 2002, J. Log. Comput..

[18]  Bas C. van Fraassen,et al.  Fine-grained opinion, probability, and the logic of full belief , 1995, J. Philos. Log..

[19]  J. Hintikka,et al.  Aspects of Inductive Logic. , 1968 .

[20]  Henry E. Kyburg,et al.  The Logic of Risky Knowledge , 2002, WoLLIC.

[21]  L. Dubins.  Finitely Additive Conditional Probabilities, Conglomerability and Disintegrations , 1975 .

[22]  B. M. Hill,et al.  Theory of Probability , 1990 .

[23]  Ronald Fagin,et al.  Reasoning about knowledge , 1995 .

[24]  J.F.A.K. van Benthem,et al.  Two Simple Incomplete Modal Logics , 2008 .

[25]  Dana S. Scott,et al.  Advice on Modal Logic , 1970 .

[26]  Bernard Linsky,et al.  IN DEFENSE OF THE SIMPLEST QUANTIFIED MODAL LOGIC , 1994 .

[27]  Robert Goldblatt,et al.  A General Semantics for Quantified Modal Logic , 2006, Advances in Modal Logic.

[28]  Giovanna Corsi,et al.  A unified completeness theorem for quantified modal logics , 2002, Journal of Symbolic Logic.

[29]  Marcus Kracht,et al.  Normal monomodal logics can simulate all others , 1999, Journal of Symbolic Logic.

[30]  Arnim von Stechow,et al.  Semantik: Ein Internationales Handbuch Der Zeitgenössischen Forschung , 1991 .

[31]  Ruth C. Barcan.  A Functional calculus of first order based on strict implication , 1946, Journal of Symbolic Logic.

[32]  W. V. Quine.  Review: Ruth C. Barcan, A Functional Calculus of First Order Based on Strict Implication , 1946 .

[33]  James W. Garson.  Unifying Quantified Modal Logic , 2005, J. Philos. Log..

[34]  Horacio Arl.  Qualitative and Probabilistic Models of Full Belief , 2000 .

[35]  L. M. M.-T..  Theory of Probability , 1929, Nature.

[36]  Ruth Barcan Marcus.  Possibilia and Possible Worlds , 1986 .

[37]  László Dezsö,et al.  Universal Grammar , 1981, Certainty in Action.

[38]  B. V. Fraassen.  Fine-grained opinion, probability, and the logic of full belief , 1995, Journal of Philosophical Logic.

[39]  J. Harsanyi.  Games with Incomplete Information Played by “Bayesian” Players Part II. Bayesian Equilibrium Points , 1968 .

[40]  Ruth Barcan Marcus,et al.  Modalities and intensional languages , 1961, Synthese.

[41]  Max J. Cresswell,et al.  A New Introduction to Modal Logic , 1998 .

[42]  Thomas A. Henzinger,et al.  Alternating-time temporal logic , 1999 .

[43]  Daniel Lehmann,et al.  What does a Conditional Knowledge Base Entail? , 1989, Artif. Intell..

[44]  Helle Hvid Hansen,et al.  Monotonic modal logics , 2003 .

[45]  Silvio Ghilardi,et al.  Incompleteness results in Kripke semantics , 1991, Journal of Symbolic Logic.

[46]  Rohit Parikh,et al.  Conditional Probability and Defeasible Inference , 2005, J. Philos. Log..

[47]  Horacio Arló Costa.  Trade-Offs between Inductive Power and Logical Omniscience in Modeling Context , 2001 .

[48]  Neal R. Norrick,et al.  Phraseologie / Phraseology: Ein internationales Handbuch der zeitgenössischen Forschung , 2007 .

[49]  Dov M. Gabbay,et al.  Handbook of Philosophical Logic , 2002 .

[50]  D. V. Gokhale,et al.  Theory of Probability, Vol. I , 1975 .

[51]  Horacio Arló-Costa,et al.  Qualitative and Probabilistic Models of Full Belief , 2000 .

[52]  K. Lambert.  Philosophical Problems in Logic , 1970 .

[53]  Marc Snir,et al.  Probabilities over rich languages, testing and randomness , 1982, Journal of Symbolic Logic.

[54]  Horacio L. Arló-Costa.  Non-Adjunctive Inference and Classical Modalities , 2005, J. Philos. Log..

[55]  Joseph Y. Halpern.  INTRANSITIVITY AND VAGUENESS , 2004, The Review of Symbolic Logic.

[56]  R. Parikh.  The logic of games and its applications , 1985 .