First-order modal logic

Publisher Summary First-order modal logics are modal logics in which the underlying propositional logic is replaced by a first-order predicate logic. They pose some of the most difficult mathematical challenges. This chapter surveys basic first-order modal logics and examines recent attempts to find a general mathematical setting in which to analyze them. A number of logics that make use of constant domain, increasing domain, and varying domain semantics is discussed, and a first-order intensional logic and a first-order version of hybrid logic is presented. One criterion for selecting these logics is the availability of sound and complete proof procedures for them, typically axiom systems and/or tableau systems. The first-order modal logics are compared to fragments of sorted first-order logic through appropriate versions of the standard translation. Both positive and negative results concerning fragment decidability, Kripke completeness, and axiomatizability are reviewed. Modal hyperdoctrines are introduced as a unifying tool for analyzing the alternative semantics. These alternative semantics range from specific semantics for non-classical logics, to interpretations in well-established mathematical framework. The relationship between topological semantics and D. Lewis's counterpart semantics is investigated and an axiomatization is presented.

[1]  Daniel Gallin,et al.  Intensional and Higher-Order Modal Logic , 1975 .

[2]  Dmitrij P. Skvortsov.  On the Predicate Logic of Linear Kripke Frames and some of its Extensions , 2005, Stud Logica.

[3]  Shin'ichi Yokota.  Axiomatization of the First-Order Intermediate Logics of Bounded Kripkean Heights I , 1989, Math. Log. Q..

[4]  Melvin Fitting,et al.  First-order intensional logic , 2004, Ann. Pure Appl. Log..

[5]  Erich Grädel,et al.  Invited Talk: Decision procedures for guarded logics , 1999, CADE.

[6]  Maarten Marx,et al.  Repairing the interpolation theorem in quantified modal logic , 2003, Ann. Pure Appl. Log..

[7]  Jerzy Tiuryn,et al.  Dynamic logic , 2001, SIGA.

[8]  Gennaro Chierchia,et al.  Meaning and Grammar: An Introduction to Semantics , 1990 .

[9]  Thierry Lucas,et al.  Formal Systems for Topos-theoretic Modalities , 1989 .

[10]  Roderic A. Girle,et al.  Modal Logics and Philosophy , 2000 .

[11]  Torben Braüner,et al.  Natural Deduction for First-Order Hybrid Logic , 2005, J. Log. Lang. Inf..

[12]  R. Sikorski,et al.  The mathematics of metamathematics , 1963 .

[13]  Melvin Fitting,et al.  Basic modal logic , 1993 .

[14]  Valentin B. Shehtman,et al.  Semantics of Non-Classical First Order Predicate Logics , 1990 .

[15]  Melvin Fitting,et al.  A Modal Herbrand Theorem , 1996, Fundam. Informaticae.

[16]  D.H.J. de Jongh,et al.  The logic of the provability , 1998 .

[17]  Michael Fara,et al.  Counterparts and Actuality , 2005 .

[18]  Hiroakira Ono,et al.  A Study of intermediate Predicate logics , 1972 .

[19]  Ettore Casari,et al.  Logic and the Foundations of Mathematics , 1981 .

[20]  S MacLane.  Applications of categorical algebra , 1985 .

[21]  Gonzalo E. Reyes,et al.  Topos-theoretic approaches to modality , 1991 .

[22]  Maarten Marx,et al.  Constructive interpolation in hybrid logic , 2003, Journal of Symbolic Logic.

[23]  Patrick Blackburn,et al.  Internalizing labelled deduction , 2000, J. Log. Comput..

[24]  Dov M. Gabbay,et al.  Undecidability of modal and intermediate first-order logics with two individual variables , 1993, Journal of Symbolic Logic.

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

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

[27]  S. Niefield.  CHANGE OF BASE FOR RELATIONAL VARIABLE SETS , 2004 .

[28]  S. Ghilardi,et al.  Modal and tense predicate logic: Models in presheaves and categorical conceptualization , 1988 .

[29]  Harold T. Hodes.  Some theorems on the expressive limitations of modal languages , 1984, J. Philos. Log..

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

[31]  Marcus Kracht,et al.  The Semantics of Modal Predicate Logic I. Counterpart-Frames , 2000, Advances in Modal Logic.

[32]  Kit Fine.  Model theory for modal logic part I—The De re/de dicto distinction , 1978, J. Philos. Log..

[33]  Maarten Marx,et al.  Tolerance Logic , 2001, J. Log. Lang. Inf..

[34]  Dmitrij P. Skvortsov,et al.  On the predicate logics of finite Kripke frames , 1995, Stud Logica.

[35]  Robert L. Berger.  The undecidability of the domino problem , 1966 .

[36]  Dov M. Gabbay.  Decidability of Some Intuitionistic Predicate Theories , 1972, J. Symb. Log..

[37]  Hiroakira Ono.  On finite linear intermediate predicate logics , 1988, Stud Logica.

[38]  Torben Braüner,et al.  Natural Deduction for Hybrid Logic , 2004, J. Log. Comput..

[39]  Allen Hazen.  Counterpart-Theoretic Semantics for Modal Logic , 1979 .

[40]  Melvin Fitting,et al.  A Modal Logic Analog of Smullyan's Fundamental Theorem , 1973 .

[41]  Johan van Benthem,et al.  Beyond Accessibility: Functional Models for Modal Logic , 1993 .

[42]  Peter Øhrstrøm,et al.  A. N. Prior's rediscovery of tense logic , 1993 .

[43]  Alan Robinson,et al.  The Inverse Method , 2001, Handbook of Automated Reasoning.

[44]  Silvio Ghilardi.  Presheaf semantics and independence results for some non-classical first-order logics , 1989, Arch. Math. Log..

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

[46]  Torben Braüner,et al.  Axioms for classical, intuitionistic, and paraconsistent hybrid logic , 2006, J. Log. Lang. Inf..

[47]  Saul A. Kripke.  The Undecidability of Monadic Modal Quantification Theory , 1962 .

[48]  Giovanna Corsi.  Quantified Modal Logics of Positive Rational Numbers and Some Related Systems , 1993, Notre Dame J. Formal Log..

[49]  Kit Fine,et al.  Model theory for modal logic—Part III existence and predication , 1981, J. Philos. Log..

[50]  I. Moerdijk,et al.  Sheaves in geometry and logic: a first introduction to topos theory , 1992 .

[51]  Matthew Davidson,et al.  Essays In The Metaphysics Of Modality , 2003 .

[52]  Alvin Plantinga,et al.  Actualism and possible worlds , 2008 .

[53]  Silvio Ghilardi.  Quantified extensions of canonical propositional intermediate logics , 1992, Stud Logica.

[54]  Gonzalo E. Reyes,et al.  Completeness Results for Intuitionistic and Modal Logic in a Categorical Setting , 1995, Ann. Pure Appl. Log..

[55]  R. Montague.  On the Nature of Certain Philosophical Entities , 1969 .

[56]  Richard L. Mendelsohn,et al.  First-Order Modal Logic , 1998 .

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

[58]  Kit Fine.  Model theory for modal logic—part II the elimination of De re modality , 1978, J. Philos. Log..

[59]  Maarten Marx,et al.  Hybrid logics: characterization, interpolation and complexity , 2001, Journal of Symbolic Logic.

[60]  J.F.A.K. van Benthem,et al.  Modal logic and classical logic , 1983 .

[61]  Dmitrij P. Skvortsov.  On Intermediate Predicate Logics of some Finite Kripke Frames, I. Levelwise Uniform Trees , 2004, Stud Logica.

[62]  Richmond H. Thomason,et al.  Modality and reference , 1968 .

[63]  M. Fitting.  Herbrand’s Theorem for a Modal Logic , 1999 .

[64]  Zane Parks.  Investigations into quantified modal logic-I , 1976 .

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

[66]  Dmitrij P. Skvortsov.  The superintuitionistic predicate logic of finite Kripke frames is not recursively axiomatizable , 2005, J. Symb. Log..

[67]  David Lewis.  Counterpart Theory and Quantified Modal Logic , 1968 .

[68]  Frank Wolter,et al.  Decidable fragments of first-order modal logics , 2001, Journal of Symbolic Logic.

[69]  W. Quine.  The ways of paradox, and other essays , 1966 .

[70]  M. de Rijke,et al.  Diamonds and Defaults , 1993 .

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

[72]  Maarten Marx,et al.  Tableaux for Quantified Hybrid Logic , 2002, TABLEAUX.

[73]  Jaakko Hintikka,et al.  Time And Modality , 1958 .

[74]  Johan van Benthem,et al.  Modal Languages and Bounded Fragments of Predicate Logic , 1998, J. Philos. Log..

[75]  Dmitrij P. Skvortsov.  Not Every "Tabular" Predicate Logic is Finitely Axiomatizable , 1997, Stud Logica.

[76]  Leopold Löwenheim.  Über Möglichkeiten im Relativkalkül , 1915 .

[77]  Kit Fine.  The permutation principle in quantificational logic , 1983, J. Philos. Log..

[78]  Krister Segerberg,et al.  Two-dimensional modal logic , 1973, J. Philos. Log..

[79]  Hiroyuki Shirasu.  Duality in Superintuitionistic and Modal Predicate Logics , 1996, Advances in Modal Logic.

[80]  Tatsuya Shimura.  Kripke completeness of some intermediate predicate logics with the axiom of constant domain and a variant of canonical formulas , 1993, Stud Logica.

[81]  Richmond H. Thomason,et al.  Abstraction in First‐Order Modal Logic1 , 2008 .

[82]  Silvio Ghilardi.  Substitution, Quantifiers and Identity in Modal Logic , 2001 .

[83]  Peter Øhrstrøm,et al.  A.N. Prior's logic , 2006, Logic and the Modalities in the Twentieth Century.

[84]  M. Fitting.  Types, Tableaus, and Gödel's God , 2002 .

[85]  Frank Wolter,et al.  First-order Expressivity for S5-models: Modal vs. Two-sorted Languages , 2001, J. Philos. Log..

[86]  Allen Hazen,et al.  Expressive completeness in modal language , 1976, J. Philos. Log..

[87]  Graeme Forbes,et al.  The metaphysics of modality , 1985 .

[88]  Sergei N. Artëmov,et al.  Finite Kripke models and predicate logics of provability , 1990, Journal of Symbolic Logic.

[89]  Kit Fine,et al.  Failures of the interpolation lemma in quantified modal logic , 1979, Journal of Symbolic Logic.

[90]  R. Montague.  Formal philosophy; selected papers of Richard Montague , 1974 .

[91]  Patrick Blackburn,et al.  Arthur Prior and Hybrid Logic , 2006, Synthese.

[92]  N. Cocchiarella.  Philosophical Perspectives on Quantification in Tense and Modal Logic , 2002 .

[93]  Dov M. Gabbay,et al.  Handbook of logic in artificial intelligence and logic programming (vol. 1) , 1993 .

[94]  Thomas Jager.  An actualistic semantics for quantified modal logic , 1982, Notre Dame J. Formal Log..

[95]  Tatsuya Shimura,et al.  Kripke Incompleteness of Predicate Extensions of the Modal Logics Axiomatized by a Canonical Formula for a Frame with a Nontrivial Cluster , 2000, Stud Logica.

[96]  D. Lewis.  Counterparts of Persons and Their Bodies , 1971 .

[97]  Milton Karl Munitz.  Identity and individuation , 1971 .

[98]  Gonzalo E. Reyes.  A Topos-Theoretic Approach to Reference and Modality , 1991, Notre Dame J. Formal Log..

[99]  Michael Mortimer,et al.  On languages with two variables , 1975, Math. Log. Q..

[100]  A. Prior.  Papers On Time And Tense , 1968 .

[101]  D. P. Skvortsov,et al.  Maximal Kripke-Type Semantics for Modal and Superintuitionistic Predicate Logics , 1993, Ann. Pure Appl. Log..