POLYADIC MODAL LOGICS WITH APPLICATIONS IN NORMATIVE REASONING by Kam Sing Leung B.Arch., University of Hong Kong, 1985 B.A., Simon Fraser University, 1999 M.A., Simon Fraser University, 2003 a Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Philosophy c Kam Sing Leung 2008 SIMON FRASER UNIVERSITY Summer 2008 All rights reserved. This work may not be reproduced in whole or in part, by photocopy or other means, without the permission of the author. APPROVAL Name: Kam Sing Leung Degree: Doctor of Philosophy Title of Dissertation: Polyadic Modal Logics with Applications in Normative Rea- soning Examining Committee: Martin Hahn Chair Ray Jennings, Professor, Department of Philosophy Senior Supervisor Jeff Pelletier, Professor, Department of Philosophy Supervisor Audrey Yap, Assistant Professor, Department of Philosophy, University of Victoria Internal Examiner Lou Goble, Professor, Department of Philosophy, Willamette University External Examiner Date Approved: ii Abstract The study of modal logic often starts with that of unary operators applied to sentences, de- noting some notions of necessity or possibility. However, we adopt a more general approach in this dissertation. We begin with object languages that possess multi-ary modal oper- ators, and interpret them in relational semantics, neighbourhood semantics and algebraic semantics. Some topics on this subject have been investigated by logicians for some time, and we present a survey of their results. But there remain areas to be explored, and we examine them in order to gain more knowledge of our territory. More specifically, we pro- pose polyadic modal axioms that correspond to seriality, reflexivity, symmetry, transitivity and euclideanness of multi-ary relations, and prove soundness and completeness of normal systems based on these axioms. We also put forward polyadic classical systems determined by classes of neighbourhood frames of finite types such as superset-closed frames, quasi- filtroids and filtroids. Equivalences between categories of modal algebras and categories of relational frames and neighbourhood frames are demonstrated. Furthermore some of the systems studied in this dissertation are shown to be translationally equivalent. While the first part of our study is purely formal, we take a different route in the sec- ond part. The multi-ary modal operators, previously interpreted in classes of mathematical structures, are given meanings in ordinary discourse. We read them as modalities in nor- mative thinking, for instance, as the \ought" when we say \you ought to visit your parents, or at least call them if you cannot visit them". A series of polyadic modal logics, called systems of deontic residuation, are proposed. They represent real-life situations involving, for example, normative conflicts and contrary-to-duty obligations better than traditional deontic logics based on unary modal operators do. Keywords: polyadic modal logic; relational semantics; neighbourhood semantics; transla- tional equivalence; deontic logic; deontic residuation iii In memory of my brother Kam Yuen iv Acknowledgments I am particularly indebted to Ray Jennings for his constant support during my study at Simon Fraser University. Without his guidance, this dissertation would have been much impoverished. I am also grateful to Lou Goble, Jeff Pelletier and Audrey Yap for their valuable comments, which have improved the content of this dissertation. Any remaining deficiencies are solely my responsibility. Versions of Chapter 10 (co-authored with Ray Jennings) were presented at the Annual Meeting of the Society for Exact Philosophy (2007) and at Dynamic Logic Montr´eal2007. Thanks to the participants of these two conferences who commented on the material pre- sented. v Contents Approval ii Abstract iii Dedication iv Acknowledgments v Contents vi List of Figures x 0 Introduction 1 1 Modal Languages and Set-Theoretic Semantics 4 1.1 Object languages . 4 1.2 Semantics for propositional languages . 8 1.3 Relational semantics for modal languages . 10 1.4 Neighbourhood semantics for modal languages . 13 1.5 Hybrids of relational and neighbourhood semantics . 15 2 From PL to Normal Modal Systems 17 2.1 Logics: syntax and semantics . 17 2.2 Propositional Logic and its extensions . 21 2.3 Normal monadic systems . 23 2.4 Normal polyadic systems . 24 2.5 Determination for Kn ............................... 27 vi 3 Normal Systems from Kn to S5n 32 3.1 The normal monadic systems P, D, T, B, S4 and S5 . 33 3.2 The normal polyadic systems Pn,Dn,Tn,Bn, S4n and S5n . 34 3.3 Classes of frames for Pn,Dn,Tn,Bn, S4n and S5n . 40 3.4 Determination for Pn,Dn,Tn,Bn, S4n and S5n . 47 3.5 First-order relational properties . 51 4 Maximal Normal Systems 57 4.1 The systems Trivn and Vern ............................ 57 4.2 Classes of frames and determination for Trivn and Vern . 60 4.3 Maximality . 63 5 Classical Systems of Modal Logic 68 5.1 Classical monadic systems . 68 5.2 Classical polyadic systems . 70 5.3 Properties of neighbourhood functions . 73 5.4 Classes of frames for classical systems . 79 5.5 General neighbourhood frames and completeness . 83 6 Modal Algebras, General Relational Frames 87 6.1 Modal algebras and normal modal algebras . 88 6.2 General relational frames . 91 6.3 Transformation of DRF to NMA .......................... 95 6.4 Transformation of NMA to DRF .......................... 99 6.5 Dual equivalence between DRF and NMA . 106 7 Modal Algebras, General Neighbourhood Frames 114 7.1 General neighbourhood frames . 115 7.2 Transformation of DNF to MA . 118 7.3 Transformation of MA to DNF . 120 7.4 Dual equivalence between DNF and MA . 123 8 Translation in Modal Logic 131 8.1 Translation and translational equivalence . 131 8.2 Monadic fragments of polyadic systems . 132 vii 8.3 Non-normal systems and normal systems . 138 8.4 Polyadic systems and monadic systems . 143 9 Formal Representation of Deontic Reasoning 149 9.1 Modern deontic logic . 150 9.2 Problem set for deontic logic . 154 9.3 After SDL: new approaches to deontic logic . 164 10 The Logic of Deontic Residuation 169 10.1 From Anderson's sanction to deontic residuation . 170 10.2 Normal deontic systems . 174 10.3 Classes of frames for Dn, DRn and DR!n . 180 10.4 Determination for Dn, DRn and DR!n . 183 10.5 Embedding of DR!n in D1 .............................185 10.6 The logic of deontic residuation: an interpretation . 188 A Algebraic Systems and Boolean Algebras 192 A.1 Algebras . 192 A.2 Operations on algebras . 195 A.3 Equational classes . 196 A.4 Algebraic semantics for propositional languages . 198 A.5 Algebraic semantics for modal languages . 202 A.6 Lindenbaum-Tarski algebras . 204 B Basic Category Theory 206 B.1 Categories . 206 B.2 Functors . 209 B.3 Natural transformations . 211 B.4 Equivalence of categories . 212 B.5 Contravariance and opposites . 213 C Contemporary Deontic Logics 216 C.1 van Fraassen's deontic logics . 216 C.2 Goble's deontic logics . 218 C.3 Horty's deontic logics . 222 viii C.4 Nute's defeasible deontic logic . 225 C.5 Makinson and van der Torre's Input/Output Logics . 227 Bibliography 230 ix List of Figures 3.1 Symmetry at the i-th place . 43 3.2 Transitivity at the i-th place . 43 3.3 Euclideanness at the i-th place . 43 3.4 ySymmetry at the i-th place . 48 3.5 yTransitivity at the i-th place . 48 3.6 yEuclideanness at the i-th place . 48 3.7 yTransitivity at the i-th place: an alternative picture . 49 3.8 yEuclideanness at the i-th place: an alternative picture . 49 3.9 Inclusions among normal systems . 51 9.1 The traditional square of opposition . 152 9.2 The modal square of opposition . 152 9.3 The deontic square of opposition . 152 10.1 Semita at the i-th place . 177 10.2 Strong semita at the i-th place . 179 x Chapter 0 Introduction Modern modal logic since C.I. Lewis's A Survey of Symbolic Logic (1918) has been pre- dominantly monadic in character: it posits unary modal operators, usually labelled by their intended meanings such as necessity, possibility, impossibility and contingency. The advent of relational semantics in the 1950's has not changed the dominance of the monadic lan- guage in the study of modal logic. In spite of some exceptions, most contemporary systems of modal logic (for example, the weakest normal modal system K, and its extensions such as KT and KD) deal with a single unary operator, but there are no reasons, other than simplicity, to insist on this self-imposed constraint on the arity of modal operators. It may be argued, as we do here, that a more general approach is preferred, both from a formal and from an interpretational point of view. Mathematicians have long been working on functions and relations of finite ranks (i.e. functions taking n arguments and relations consisting of n-tuples, for some finite number n). Logic, as the formal study of reasoning, has had a close relationship with mathematics. (We note here that Boole, in the 19th-century, conceived propositional logic as an algebra of propositions.) Thus viewed, there are good reasons to develop a general theory of modal logic, in which unary operators are merely special cases of multi-ary operators that can take finitely many sentences as their operands. In fact such an approach has already been suggested by J´onssonand Tarski's paper \Boolean Algebras with Operators. Part I" (1951), which shows that every Boolean algebra supplemented with finitary functions satisfying the conditions of normality and additivity can be represented as a subalgebra of the complex algebra of a relational structure. Effectively their work provides a multi-ary relational semantics for the polyadic modal language although its implications for modal logic had not 1 CHAPTER 0.