DOCSLIB.ORG

Mathematical Modal Logic: a View of Its Evolution

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mathematical Modal Logic: a View of Its Evolution Journal of Applied Logic 1 (2003) 309–392 www.elsevier.com/locate/jal Mathematical modal logic: A view of its evolution Robert Goldblatt Centre for Logic, Language and Computation, Victoria University, P.O. Box 600, Wellington, New Zealand Abstract This is a survey of the origins of mathematical interpretations of modal logics, and their devel- opment over the last century or so. It focuses on the interconnections between algebraic semantics using Boolean algebras with operators and relational semantics using structures often called Kripke models. It reviews the ideas of a number of people who independently contributed to the emergence of relational semantics, and compares them with the work of Kripke. It concludes with an account of several applications of modal model theory to mathematics and theoretical computer science. 2003 Elsevier B.V. All rights reserved. Keywords: Modal logic; History of logic; Boolean algebra; Kripke semantics 1. Introduction Modal logic was originally conceived as the logic of necessary and possible truths. It is now viewed more broadly as the study of many linguistic constructions that qualify the truth conditions of statements, including statements concerning knowledge, belief, tempo- ral discourse, and ethics. Most recently, modal symbolism and model theory have been put to use in computer science, to formalise reasoning about the way programs behave and to express dynamical properties of transitions between states. Over a period of three decades or so from the early 1930s there evolved two kinds of mathematical semantics for modal logic. Algebraic semantics interprets modal connectives as operators on Boolean algebras. Relational semantics uses relational structures, often called Kripke models, whose elements are thought of variously as being possible worlds, moments of time, evidential situations, or states of a computer. The two approaches are intimately related: the subsets of a relational structure form a modal algebra (Boolean algebra with operators), while conversely any modal algebra can be embedded into an E-mail address: [email protected] (R. Goldblatt). URL: http://www.mcs.vuw.ac.nz/~rob. 1570-8683/$ – see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/S1570-8683(03)00008-9 310 R. Goldblatt / Journal of Applied Logic 1 (2003) 309–392 algebra of subsets of a relational structure via extensions of Stone’s Boolean representation theory. Techniques from both kinds of semantics have been used to explore the nature of modal logic and to clarify its relationship to other formalisms, particularly first and second order monadic predicate logic. The aim of this article is to review these developments in a way that provides some insight into how the present came to be as it is. The pervading theme is the mathematics underlying modal logic, and this has at least three dimensions. To begin with there are the new mathematical ideas: when and why they were introduced, and how they interacted and evolved. Then there is the use of methods and results from other areas of mathematical logic, algebra and topology in the analysis of modal systems. Finally, there is the applica- tion of modal syntax and semantics to study notions of mathematical and computational interest. There has been some mild controversy about priorities in the origin of relational model theory, and space is devoted to this issue in Section 4. An attempt is made to record in one place a sufficiently full account of what was said and done by early contributors to allow readers to make their own assessment (although the author does give his). Despite its length, the article does not purport to give an encyclopaedic coverage of the field. For instance, there is much about temporal logic (see Gabbay et al. [77]) and logics of knowledge (see Fagin et al. [64]) that is not reported here, while the surface of modal predicate logic is barely scratched, and proof theory is not discussed at all. I have not attempted to survey the work of the present younger generation of modal logicians (see Chagrov and Zakharyaschev [34], Kracht [141], and Marx and Venema [171], for exam- ple). There has been little by way of historical review of work on intensional semantics over the last century, and no doubt there remains room for more. 2. Beginnings 2.1. What is a modality? Modal logic began with Aristotle’s analysis of statements containing the words “nec- essary” and “possible”.1 These are but two of a wide range of modal connectives,or modalities that are abundant in natural and technical languages. Briefly, a modality is any word or phrase that can be applied to a given statement S to create a new statement that makes an assertion about the mode of truth of S: about when, where or how S is true, or about the circumstances under which S may be true. Here are some examples, grouped according to the subject they are naturally associated with tense logic: henceforth, eventually, hitherto, previously, now, tomorrow, yesterday, since, until, inevitably, finally, ultimately, endlessly, it will have been, it is being . 1 For the early history of modal logic, including the work of Greek and medieval scholars, see Bochenski 1961 [20] and Kneale and Kneale 1962 [134]. The Historical Introduction to Lemmon [156] gives a brief but informative sketch. R. Goldblatt / Journal of Applied Logic 1 (2003) 309–392 311 deontic logic: it is obligatory/forbidden/permitted/unlawful that epistemic logic: it is known to X that, it is common knowledge that doxastic logic: it is believed that dynamic logic: after the program/computation/action finishes, the program enables, throughout the computation geometric logic: it is locally the case that metalogic: it is valid/satisfiable/provable/consistent that The key to understanding the relational modal semantics is that many modalities come in dual pairs, with one of the pair having an interpretation as a universal quantifier (“in all ...”)andtheotherasanexistentialquantifier(“insome...”).Thisis illustrated by the following interpretations, the first being famously attributed to Leibniz (see Section 4). necessarily in all possible worlds possibly in some possible world henceforth at all future times eventually at some future time it is valid that in all models it is satisfiable that in some model after the program finishes after all terminating executions the program enables there is a terminating execution such that It is now common to use the symbol for a modality of universal character, and ✸ for its existential dual. In systems based on classical truth-functional logic, is equivalent to ¬✸¬,and✸ to ¬¬,where¬ is the negation connective. Thus “necessarily” means “not possibly not”, “eventually” means “not henceforth not”, a statement is valid when its negation is not satisfiable, etc. Notation Rather than trying to accommodate all the different notations used for truth-functional connectives by different authors over the years, we will fix on the symbols ∧, ∨, ¬, → and ↔ for conjunction, disjunction, negation, (material) implication, and (material) equiv- alence. The symbol is used for a constant true formula, equivalent to any tautology, while ⊥ is a constant false formula, equivalent to ¬.Wealsouse and ⊥ as symbols for truth values. The standard syntax for propositional modal logic is based on a countably infinite list p0,p1,...of propositional variables, for which we typically use the letters p,q,r.Formu- las are generated from these variables by means of the above connectives and the symbols and ✸. There are of course a number of options about which of these to take as prim- itive symbols, and which to define in terms of primitives. When describing the work of different authors we will sometimes use their original symbols for modalities, such as M for possibly, L or N for necessarily, and other conventions for deontic and tense logics. The symbol n stands for a sequence ··· of n copies of , and likewise ✸n for ✸✸ ···✸ (n times). 312 R. Goldblatt / Journal of Applied Logic 1 (2003) 309–392 A systematic notation will also be employed for Boolean algebras: the symbols + , ·, − denote the operations of sum (join), product (meet), and complement in a Boolean algebra, and 0 and 1 are the greatest and least elements under the ordering given by x y iff x · y = x. The supremum (sum) and infimum (product) of a set X of elements will be denoted X and X (when they exist). 2.2. MacColl’s iterated modalities The first substantial algebraic analysis of modalised statements was carried out by Hugh MacColl, in a series of papers that appeared in Mind between 1880 and 1906 under the title Symbolical Reasoning,2 as well as in other papers and his book [163] of 1906. MacColl symbolised the conjunction of two statements a and b by their concatenation ab,useda +b for their disjunction, and wrote a : b for the statement “a implies b”, which he said could be read “if a is true, then b must be true”, or “whenever a is true, b is also true”. The equation a = b was used for the assertion that a and b are equivalent, meaning that each implies the other. Thus a = b is itself equivalent to the “compound implication” (a : b)(b : a),an observation that was rendered symbolically by the equation (a = b) = (a : b)(b : a). MacColl wrote a for the “denial” or “negative” of statement a, and stated that (a + b) is equivalent to ab . However, while a + b is a “necessary consequence” of a : b (written (a : b) : a + b ), he argued that the two formulas are not equivalent because their denials are not equivalent, claiming that the denial of a : b “only asserts the possibility of the com- bination ab ”, while the denial of a + b “asserts the certainty of the same combination”.3 Boole had written a = 1anda = 0for“a is true” and “a is false”, giving a tempo- ral reading of these as always true and always false respectively (see Boole 1854 [21], Chapter XI).
Load more

Recommended publications
  • Subverting the Classical Picture: (Co)Algebra, Constructivity and Modalities∗
    Informatik 8, FAU Erlangen-Nürnberg [email protected] Subverting the Classical Picture: (Co)algebra, Constructivity and Modalities∗ Tadeusz Litak Contents 1. Subverting the Classical Picture 2 2. Beyond Frege and Russell: Modalities Instead of Predicates 6 3. Beyond Peirce, Tarski and Kripke: Coalgebras and CPL 11 3.1. Introduction to the Coalgebraic Framework ............. 11 3.2. Coalgebraic Predicate Logic ...................... 14 4. Beyond Cantor: Effectivity, Computability and Determinacy 16 4.1. Finite vs. Classical Model Theory: The Case of CPL ........ 17 4.2. Effectivity, Choice and Determinacy in Formal Economics ..... 19 4.3. Quasiconstructivity and Modal Logic ................. 21 5. Beyond Scott, Montague and Moss: Algebras and Possibilities 23 5.1. Dual Algebras, Neighbourhood Frames and CA-baes ........ 23 5.2. A Hierachy of (In)completeness Notions for Normal Modal Logics . 24 5.3. Possibility Semantics and Complete Additivity ........... 26 5.4. GQM: One Modal Logic to Rule Them All? ............. 28 6. Beyond Boole and Lewis: Intuitionistic Modal Logic 30 6.1. Curry-Howard Correspondence, Constructive Logic and 2 ..... 31 6.2. Guarded Recursion and Its Neighbours ................ 33 6.3. KM, mHC, LC and the Topos of Trees ................ 34 6.4. Between Boole and Heyting: Negative Translations ......... 35 7. Unifying Lewis and Brouwer 37 7.1. Constructive Strict Implication .................... 37 7.2. Relevance, Resources and (G)BI .................... 40 ∗Introductory overview chapter for my cumulative habilitation thesis as required by FAU Erlangen-Nürnberg regulations (submitted in 2018, finally approved in May 2019). Available online at https://www8.cs.fau. de/ext/litak/gitpipe/tadeusz_habilitation.pdf. 1 8. Beyond Brouwer, Heyting and Kolmogorov: Nondistributivity 43 8.1.
    [Show full text]
  • Probabilistic Semantics for Modal Logic
    Probabilistic Semantics for Modal Logic By Tamar Ariela Lando A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Philosophy in the Graduate Division of the University of California, Berkeley Committee in Charge: Paolo Mancosu (Co-Chair) Barry Stroud (Co-Chair) Christos Papadimitriou Spring, 2012 Abstract Probabilistic Semantics for Modal Logic by Tamar Ariela Lando Doctor of Philosophy in Philosophy University of California, Berkeley Professor Paolo Mancosu & Professor Barry Stroud, Co-Chairs We develop a probabilistic semantics for modal logic, which was introduced in recent years by Dana Scott. This semantics is intimately related to an older, topological semantics for modal logic developed by Tarski in the 1940’s. Instead of interpreting modal languages in topological spaces, as Tarski did, we interpret them in the Lebesgue measure algebra, or algebra of measurable subsets of the real interval, [0, 1], modulo sets of measure zero. In the probabilistic semantics, each formula is assigned to some element of the algebra, and acquires a corresponding probability (or measure) value. A formula is satisfed in a model over the algebra if it is assigned to the top element in the algebra—or, equivalently, has probability 1. The dissertation focuses on questions of completeness. We show that the propo- sitional modal logic, S4, is sound and complete for the probabilistic semantics (formally, S4 is sound and complete for the Lebesgue measure algebra). We then show that we can extend this semantics to more complex, multi-modal languages. In particular, we prove that the dynamic topological logic, S4C, is sound and com- plete for the probabilistic semantics (formally, S4C is sound and complete for the Lebesgue measure algebra with O-operators).
    [Show full text]
  • Boxes and Diamonds: an Open Introduction to Modal Logic
    Boxes and Diamonds An Open Introduction to Modal Logic F19 Boxes and Diamonds The Open Logic Project Instigator Richard Zach, University of Calgary Editorial Board Aldo Antonelli,y University of California, Davis Andrew Arana, Université de Lorraine Jeremy Avigad, Carnegie Mellon University Tim Button, University College London Walter Dean, University of Warwick Gillian Russell, Dianoia Institute of Philosophy Nicole Wyatt, University of Calgary Audrey Yap, University of Victoria Contributors Samara Burns, Columbia University Dana Hägg, University of Calgary Zesen Qian, Carnegie Mellon University Boxes and Diamonds An Open Introduction to Modal Logic Remixed by Richard Zach Fall 2019 The Open Logic Project would like to acknowledge the gener- ous support of the Taylor Institute of Teaching and Learning of the University of Calgary, and the Alberta Open Educational Re- sources (ABOER) Initiative, which is made possible through an investment from the Alberta government. Cover illustrations by Matthew Leadbeater, used under a Cre- ative Commons Attribution-NonCommercial 4.0 International Li- cense. Typeset in Baskervald X and Nimbus Sans by LATEX. This version of Boxes and Diamonds is revision ed40131 (2021-07- 11), with content generated from Open Logic Text revision a36bf42 (2021-09-21). Free download at: https://bd.openlogicproject.org/ Boxes and Diamonds by Richard Zach is licensed under a Creative Commons At- tribution 4.0 International License. It is based on The Open Logic Text by the Open Logic Project, used under a Cre- ative Commons Attribution 4.0 Interna- tional License. Contents Preface xi Introduction xii I Normal Modal Logics1 1 Syntax and Semantics2 1.1 Introduction....................
    [Show full text]
  • Giorgi Japaridze
    Giorgi Japaridze Curriculum Vitae CONTACT INFORMATION Address: Computing Sciences Department, Villanova University, 800 Lancaster Avenue, Villanova, PA 19085, USA. Email: [email protected] Telephone: (1) 610 519 7332 Fax: (1) 610 519 7889 Home page: http://www.csc.villanova.edu/~japaridz/ EDUCATION 1998 Ph.D. Computer Science, University of Pennsylvania Dissertation: The Logic of Resources and Tasks 1987 Ph.D. Logic, Moscow State University Dissertation: Modal-Logical Means of Studying Provability 1983 M.S. Philosophy, Tbilisi State University (Georgia, former USSR) Thesis: The Notion of Truth in Formalized Languages LANGUAGES: Georgian, Russian, English, Chinese (Mandarin), German. EMPLOYMENT HISTORY 2008 - present Full Professor Computing Sciences Department, Villanova University, Villanova, PA, USA 2010 - 2013 Chair Professor School of Computer Science and Technology, Shandong University, Jinan, China 2004-2008 Associate Professor Computing Sciences Department, Villanova University, Villanova, PA, USA 2007 Visiting Professor School of Information Science and Technology, Institute of Artificial Intelligence, Xiamen University, Fujian, Xiamen, China 1998 -2003 Assistant Professor Computing Sciences Department, Villanova University, Villanova, PA, USA 1995-1998 Research Assistant Dept. of Computer and Information Science, University of Pennsylvania, Philadelphia, PA, USA 1993-1994 Visiting Associate Professor Philosophy Department, University of Notre Dame, Notre Dame, IN, USA 1992-1993 Postdoctoral Fellow Dept. of Mathematics and Computer Science, University of Amsterdam, Amsterdam, The Netherlands 1987-1992 Senior Researcher Institute of Philosophy, Georgian Academy of Sciences, Tbilisi, Georgia (former USSR) Main Contributions to Science Provability and interpretability logics (1985-1998) While a student, introduced polymodal provability logic GLP, and proved its arithmetical completeness. This contained a solution of an open problem on the logic of provability raised by George Boolos a decade earlier (1985-1988).
    [Show full text]
  • Propositional Computability Logic I∗
    Propositional computability logic I∗ Giorgi Japaridze† Department of Computing Sciences, Villanova University, 800 Lancaster Avenue, Villanova, PA 19085, USA. Email: [email protected] URL: http://www.csc.villanova.edu/∼ japaridz/ Abstract In the same sense as classical logic is a formal theory of truth, the recently initiated approach called computability logic is a formal theory of computability. It understands (interactive) computational prob- lems as games played by a machine against the environment, their computability as existence of a machine that always wins the game, logical operators as operations on computational problems, and validity of a logical formula as being a scheme of “always computable” problems. Computability logic has been introduced semantically, and now among its main technical goals is to axiomatize the set of valid formu- las or various natural fragments of that set. The present contribution signifies a first step towards this goal. It gives a detailed exposition of a soundness and completeness proof for the rather new type of a deductive propositional system CL1, the logical vocabulary of which contains operators for the so called parallel and choice operations, and the atoms of which represent elementary problems, i.e. predicates in the standard sense. This article is self-contained as it explains all relevant concepts. While not technically necessary, however, familiarity with the foundational paper “Introduction to computability logic” [Annals of Pure and Applied Logic 123 (2003), pp.1-99] would greatly help the reader in understanding the philosophy, underlying motivations, potential and utility of computability logic, — the context that determines the value of the present results.
    [Show full text]
  • Kripke Completeness Revisited
    Kripke completeness revisited Sara Negri Department of Philosophy, P.O. Box 9, 00014 University of Helsinki, Finland. e-mail: sara.negri@helsinki.fi Abstract The evolution of completeness proofs for modal logic with respect to the possible world semantics is studied starting from an analysis of Kripke’s original proofs from 1959 and 1963. The critical reviews by Bayart and Kaplan and the emergence of Henkin-style completeness proofs are detailed. It is shown how the use of a labelled sequent system permits a direct and uniform completeness proof for a wide variety of modal logics that is close to Kripke’s original arguments but without the drawbacks of Kripke’s or Henkin-style completeness proofs. Introduction The question about the ultimate attribution for what is commonly called Kripke semantics has been exhaustively discussed in the literature, recently in two surveys (Copeland 2002 and Goldblatt 2005) where the rˆoleof the precursors of Kripke semantics is documented in detail. All the anticipations of Kripke’s semantics have been given ample credit, to the extent that very often the neutral terminology of “relational semantics” is preferred. The following quote nicely summarizes one representative standpoint in the debate: As mathematics progresses, notions that were obscure and perplexing become clear and straightforward, sometimes even achieving the status of “obvious.” Then hindsight can make us all wise after the event. But we are separated from the past by our knowledge of the present, which may draw us into “seeing” more than was really there at the time. (Goldblatt 2005, section 4.2) We are not going to treat this issue here, nor discuss the parallel development of the related algebraic semantics for modal logic (Jonsson and Tarski 1951), but instead concentrate on one particular and crucial aspect in the history of possible worlds semantics, namely the evolution of completeness proofs for modal logic with respect to Kripke semantics.
    [Show full text]
  • Towards a Theory of Proofs of Classical Logic Lutz Straßburger
    Towards a Theory of Proofs of Classical Logic Lutz Straßburger To cite this version: Lutz Straßburger. Towards a Theory of Proofs of Classical Logic. Logic in Computer Science [cs.LO]. Université Paris-Diderot - Paris VII, 2011. tel-00772590 HAL Id: tel-00772590 https://tel.archives-ouvertes.fr/tel-00772590 Submitted on 10 Jan 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Towards a Theory of Proofs of Classical Logic Habilitation a` diriger des recherches Universit´eDenis Diderot – Paris 7 Lutz Straßburger Jury: Richard Blute (rapporteur) Pierre-Louis Curien (rapporteur) Gilles Dowek Martin Hyland (rapporteur) Delia Kesner Christian Retor´e Alex Simpson (rapporteur) Soutenance : 7 janvier 2011 Table of Contents Table of Contents iii 0 Vers une th´eorie des preuves pour la logique classique v 0.1 Cat´egoriesdespreuves . vi 0.2 Notationssyntaxiquepourlespreuves . xv 0.3 Tailledespreuves................................ xx 1 Introduction 1 1.1 CategoriesofProofs .............................. 1 1.2 SyntacticDenotationsforProofs . 3 1.3 SizeofProofs................................... 5 2 On the Algebra of Proofs in Classical Logic 7 2.1 WhatisaBooleanCategory?. 7 2.2 Star-AutonomousCategories . 9 2.3 Someremarksonmix .............................. 12 2.4 ∨-Monoids and ∧-comonoids........................... 16 2.5 Orderenrichment................................
    [Show full text]
  • Informal Proceedings of the 30Th International Workshop on Unification (UNIF 2016)
    Silvio Ghilardi and Manfred Schmidt-Schauß (Eds.) Informal Proceedings of the 30th International Workshop on Unification (UNIF 2016) June 26, 2016 Porto, Portugal 1 Preface This volume contains the papers presented at UNIF 2016: The 30th International Workshop on Unification (UNIF 2016) held on June 26, 2016 in Porto. There were 10 submissions. Each submission was reviewed by at least 3, and on the average 3.3, program committee members. The committee decided to accept 8 papers for publication in these Proceedings. The program also includes 2 invited talks. The International Workshop on Unification was initiated in 1987 as a yearly forum for researchers in unification theory and related fields to meet old and new colleagues, to present recent (even unfinished) work, and to discuss new ideas and trends. It is also a good opportunity for young researchers and researchers working in related areas to get an overview of the current state of the art in unification theory. The list of previous meetings can be found at the UNIF web page: http://www.pps.univ-paris-diderot.fr/~treinen/unif/. Typically, the topics of interest include (but are not restricted to): Unification algorithms, calculi and implementations • Equational unification and unification modulo theories • Unification in modal, temporal and description logics • Admissibility of Inference Rules • Narrowing • Matching algorithms • Constraint solving • Combination problems • Disunification • Higher-Order Unification • Type checking and reconstruction • Typed unification • Complexity issues • Query answering • Implementation techniques • Applications of unification • Antiunification/Generalization • This years UNIF is a satellite event of the first International Conference on Formal Structures for Computation and Deduction (FSCD).
    [Show full text]
  • Introduction to Modal Logics (Lecture Notes: Summer Term 2011)
    Introduction to Modal Logics (Draft) Stefan Wolfl¨ July 22, 2015 2 Contents 1 From Propositional to Modal Logic5 1.1 Propositional logic................................5 1.2 A simple modal logic...............................7 2 Modal Language, Frames, and Models 13 2.1 Relational structures............................... 13 2.2 Modal languages................................. 16 2.3 Relational models and satisfaction........................ 19 2.4 Constructing models............................... 24 2.5 Translating modal logic into first-order logic................... 27 2.6 Consequence relation and compactness...................... 29 2.7 Expressiveness.................................. 31 3 Normal Modal Logics, Frame Classes, and Definability 39 3.1 Normal modal logics............................... 39 3.2 Kripke frames and definability.......................... 41 3.3 Proving theorems................................. 45 3.4 Soundness and completeness........................... 48 3.5 Canonical frames, definability, and compactness................. 53 3.6 The modal logic S4 ................................ 56 3.7 The modal logic S5 ................................ 58 3.8 The modal logic KL ............................... 60 4 Decidability and Complexity 65 4.1 Finite Model Property............................... 65 4.2 Filtration..................................... 68 4.3 Complexity.................................... 70 5 Decision Procedures 81 5.1 A tableaux procedure for K(m) .......................... 81 5.2 Tableaux procedures for
    [Show full text]
  • Baltic International Yearbook of Cognition, Logic and Communication
    Baltic International Yearbook of Cognition, Logic and Communication Volume 8 Games, Game Theory and Game Semantics Article 5 2013 Ptarithmetic Giorgi Japaridze School of Computer Science and Technology, Shandong University; Department of Computing Sciences, Villanova University Follow this and additional works at: https://newprairiepress.org/biyclc Part of the Theory and Algorithms Commons This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License. Recommended Citation Japaridze, Giorgi (2013) "Ptarithmetic," Baltic International Yearbook of Cognition, Logic and Communication: Vol. 8. https://doi.org/10.4148/1944-3676.1074 This Proceeding of the Symposium for Cognition, Logic and Communication is brought to you for free and open access by the Conferences at New Prairie Press. It has been accepted for inclusion in Baltic International Yearbook of Cognition, Logic and Communication by an authorized administrator of New Prairie Press. For more information, please contact [email protected]. Ptarithmetic 2 The Baltic International Yearbook of and developed (Japaridze 2011, unpublished, forth.). In retrospect, Cognition, Logic and Communication however, the author finds that “Ptarithmetic” contained a number of potentially useful ideas that have not been subsequently adopted by November 2013 Volume 8: Games, Game Theory the “clarithmetics” line of research (at least not yet), and that, for this and Game Semantics reason, it would be a pity to let this material remain unpublished. pages 1-186 DOI: 10.4148/1944-3676.1074 Probably the most important of such ideas is the “Polynomial Time Induction” (PTI) rule of Ptarithmetic, with no close or distant relatives elsewhere in the literature.
    [Show full text]
  • First-Order Modal Logic, Most Decidedly, Is Not Just Propositional Modal Logic Plus Classical Quantifier Machinery
    SYNTHESE LIBRARY MELVIN FITTING AND RICHARD L. MENDELSOHN STUDIES IN EPISTEMOLOGY. Lehman College and the Graduate Cente" CUNY, New York. U.s.A. LOGIC. METHODOLOGY. AND PHILOSOPHY OF SCIENCE '0 Managing Editor: FIRST-ORDER JAAKKO HINTlKKA. Boston University MODAL LOGIC Editors: DIRK VAN DALEN. University of Utrecht. The Netherlands DONALD DAVIDSON. University of California. Berkeley THEO A.F. KUIPERS. University ofGroningen. The Netherlands PATRICK SUPPES. Stanford University. California JAN WOLENSKI. Jagiellonian University. Krak6w. Poland VOLUME 277 KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON A C.J.P. Catalogue record for this book is available from the Library of Congress. ISBN 0-7923-5335-8 (PB) ISBN 0-7923-5334-X (HB) To Marsha, Robin, and losh, with love PubHshed by Kluwer Academic Publishers, RLM P.D. Box 17,3300 AA Dordrecht, The Netherlands. To Roma, who knows what I mean MP Sold and distributed in North, Central and South America by Kluwer Academic Publishers. 101 Philip Drive, NorweII, MA 02061, U.S.A. In all other countries, sold and distributed by KJuwer Academij; Publishers, P.D. Box 322, 3300 AH Dordrecht, The Netherlands. Printed on acid-free paper An Rights Reserved © 1998 KJuwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any fonn or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner Printed in the Netherlands. CONTENTS
    [Show full text]
  • A Note on Strictly Positive Logics and Word Rewriting Systems
    A note on strictly positive logics and word rewriting systems Lev Beklemishev∗ August 21, 2018 Abstract We establish a natural translation from word rewriting systems to strictly positive polymodal logics. Thereby, the latter can be considered as a generalization of the former. As a corollary we obtain examples of undecidable strictly positive normal modal logics. The translation has its counterpart on the level of proofs: we formulate a natural deep inference proof system for strictly positive logics generalizing derivations in word rewriting systems. We also formulate some open questions related to the theory of modal companions of superintuitionistic logics that was initiated by L.L. Maksimova and V.V. Rybakov. In this note we study the fragment of polymodal logic consisting of impli- cations of the form A → B, where A and B are formulas built-up from ⊤ and propositional variables using just ∧ and the diamond modalities. We call such formulas A and B strictly positive and will often omit the word ‘strictly.’ The interest towards such weak logics independently emerged within two different disciplines: provability logic and description logic (see [8, 5, 1]). In both cases, it was observed that the strictly positive language combines sim- plicity and efficiency while retaining a substantial amount of expressive power of modal logic. Thus, strictly positive fragments of many standard modal logics arXiv:1509.00666v2 [math.LO] 18 Feb 2016 are polytime decidable. The positive fragment of the (Kripke incomplete) poly- modal provability logic GLP is both polytime decidable and complete w.r.t. a natural class of finite Kripke frames [5].
    [Show full text]