DOCSLIB.ORG

Polyadic Modal Logics with Applications in Normative Reasoning

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Polyadic Modal Logics with Applications in Normative Reasoning 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.
Load more

Recommended publications
  • The Modal Logic of Potential Infinity, with an Application to Free Choice
    The Modal Logic of Potential Infinity, With an Application to Free Choice Sequences Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Ethan Brauer, B.A. ∼6 6 Graduate Program in Philosophy The Ohio State University 2020 Dissertation Committee: Professor Stewart Shapiro, Co-adviser Professor Neil Tennant, Co-adviser Professor Chris Miller Professor Chris Pincock c Ethan Brauer, 2020 Abstract This dissertation is a study of potential infinity in mathematics and its contrast with actual infinity. Roughly, an actual infinity is a completed infinite totality. By contrast, a collection is potentially infinite when it is possible to expand it beyond any finite limit, despite not being a completed, actual infinite totality. The concept of potential infinity thus involves a notion of possibility. On this basis, recent progress has been made in giving an account of potential infinity using the resources of modal logic. Part I of this dissertation studies what the right modal logic is for reasoning about potential infinity. I begin Part I by rehearsing an argument|which is due to Linnebo and which I partially endorse|that the right modal logic is S4.2. Under this assumption, Linnebo has shown that a natural translation of non-modal first-order logic into modal first- order logic is sound and faithful. I argue that for the philosophical purposes at stake, the modal logic in question should be free and extend Linnebo's result to this setting. I then identify a limitation to the argument for S4.2 being the right modal logic for potential infinity.
    [Show full text]
  • Branching Time
    24.244 Modal Logic, Fall 2009 Prof. Robert Stalnaker Lecture Notes 16: Branching Time Ordinary tense logic is a 'multi-modal logic', in which there are two (pairs of) modal operators: P/H and F/G. In the semantics, the accessibility relations, for past and future tenses, are interdefinable, so in effect there is only one accessibility relation in the semantics. However, P/H cannot be defined in terms of F/G. The expressive power of the language does not match the semantics in this case. But this is just a minimal multi-modal theory. The frame is a pair consisting of an accessibility relation and a set of points, representing moments of time. The two modal operators are defined on this one set. In the branching time theory, we have two different W's, in a way. We put together the basic tense logic, where the points are moments of time, with the modal logic, where they are possible worlds. But they're not independent of one another. In particular, unlike abstract modal semantics, where possible worlds are primitives, the possible worlds in the branching time theory are defined entities, and the accessibility relation is defined with respect to the structure of the possible worlds. In the pure, abstract theory, where possible worlds are primitives, to the extent that there is structure, the structure is defined in terms of the relation on these worlds. So the worlds themselves are points, the structure of the overall frame is given by the accessibility relation. In the branching time theory, the possible worlds are possible histories, which have a structure defined by a more basic frame, and the accessibility relation is defined in terms of that structure.
    [Show full text]
  • 29 Alethic Modal Logics and Semantics
    29 Alethic Modal Logics and Semantics GERHARD SCHURZ 1 Introduction The first axiomatic development of modal logic was untertaken by C. I. Lewis in 1912. Being anticipated by H. McCall in 1880, Lewis tried to cure logic from the ‘paradoxes’ of extensional (i.e. truthfunctional) implication … (cf. Hughes and Cresswell 1968: 215). He introduced the stronger notion of strict implication <, which can be defined with help of a necessity operator ᮀ (for ‘it is neessary that:’) as follows: A < B iff ᮀ(A … B); in words, A strictly implies B iff A necessarily implies B (A, B, . for arbitrary sen- tences). The new primitive sentential operator ᮀ is intensional (non-truthfunctional): the truth value of A does not determine the truth-value of ᮀA. To demonstrate this it suf- fices to find two particular sentences p, q which agree in their truth value without that ᮀp and ᮀq agree in their truth-value. For example, it p = ‘the sun is identical with itself,’ and q = ‘the sun has nine planets,’ then p and q are both true, ᮀp is true, but ᮀq is false. The dual of the necessity-operator is the possibility operator ‡ (for ‘it is possible that:’) defined as follows: ‡A iff ÿᮀÿA; in words, A is possible iff A’s negation is not necessary. Alternatively, one might introduce ‡ as new primitive operator (this was Lewis’ choice in 1918) and define ᮀA as ÿ‡ÿA and A < B as ÿ‡(A ŸÿB). Lewis’ work cumulated in Lewis and Langford (1932), where the five axiomatic systems S1–S5 were introduced.
    [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]
  • 31 Deontic, Epistemic, and Temporal Modal Logics
    31 Deontic, Epistemic, and Temporal Modal Logics RISTO HILPINEN 1 Modal Concepts Modal logic is the logic of modal concepts and modal statements. Modal concepts (modalities) include the concepts of necessity, possibility, and related concepts. Modalities can be interpreted in different ways: for example, the possibility of a propo- sition or a state of affairs can be taken to mean that it is not ruled out by what is known (an epistemic interpretation) or believed (a doxastic interpretation), or that it is not ruled out by the accepted legal or moral requirements (a deontic interpretation), or that it has not always been or will not always be false (a temporal interpretation). These interpre- tations are sometimes contrasted with alethic modalities, which are thought to express the ways (‘modes’) in which a proposition can be true or false. For example, logical pos- sibility and physical (real or substantive) possibility are alethic modalities. The basic modal concepts are represented in systems of modal logic as propositional operators; thus they are regarded as syntactically analogous to the concept of negation and other propositional connectives. The main difference between modal operators and other connectives is that the former are not truth-functional; the truth-value (truth or falsity) of a modal sentence is not determined by the truth-values of its subsentences. The concept of possibility (‘it is possible that’ or ‘possibly’) is usually symbolized by ‡ and the concept of necessity (‘it is necessary that’ or ‘necessarily’) by ᮀ; thus the modal formula ‡p represents the sentence form ‘it is possible that p’ or ‘possibly p,’ and ᮀp should be read ‘it is necessary that p.’ Modal operators can be defined in terms of each other: ‘it is possible that p’ means the same as ‘it is not necessary that not-p’; thus ‡p can be regarded as an abbreviation of ÿᮀÿp, where ÿ is the sign of negation, and ᮀp is logically equivalent to ÿ‡ÿp.
    [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]
  • Modal Logic Programming Revisited
    Modal logic programming revisited Linh Anh Nguyen University of Warsaw Institute of Informatics ul. Banacha 2, 02-097 Warsaw (Poland) [email protected] ABSTRACT. We present optimizations for the modal logic programming system MProlog, in- cluding the standard form for resolution cycles, optimized sets of rules used as meta-clauses, optimizations for the version of MProlog without existential modal operators, as well as iter- ative deepening search and tabulation. Our SLD-resolution calculi for MProlog in a number of modal logics are still strongly complete when resolution cycles are in the standard form and optimized sets of rules are used. We also show that the labelling technique used in our direct ap- proach is relatively better than the Skolemization technique used in the translation approaches for modal logic programming. KEYWORDS: modal logic, logic programming, MProlog. DOI:10.3166/JANCL.19.167–181 c 2009 Lavoisier, Paris 1. Introduction Modal logic programming is the field that extends classical logic programming to deal with modalities. As modal logics can be used, among others, to reason about knowledge and belief, developing a good formalism for modal logic programs and an efficient computational procedure for it is desirable. The first work on extending logic programming with modal logic is (Fariñas del Cerro, 1986) on the implemented system Molog (Fariñas del Cerro et al., 1986). With Molog, the user can fix a modal logic and define or choose the rules to deal with modal operators. Molog can be viewed as a framework which can be instantiated with particular modal logics. As an extension of Molog, the Toulouse Inference Machine (Balbiani et al., 1991) together with an abstract machine model called TARSKI for implementation (Balbiani et al., 1992) makes it possible for a user to select clauses which cannot exactly unify with the current goal, but just resemble it in some way.
    [Show full text]
  • The Principle of Analyticity of Logic a Philosophical and Formal Perspective
    Scuola Normale Superiore Classe di Lettere e Filosofia Corso di Perfezionamento in Filosofia The Principle of Analyticity of Logic A Philosophical and Formal Perspective Thesis Advisors: Candidate: Prof. Marcello D'Agostino Costanza Larese Prof. Massimo Mugnai July 2019 2 Contents Introduction 4 Summary 8 I Historical and philosophical reconstruction 12 1 Kant and the foundations of the analytic-synthetic distinction 14 1.1 The Kantian analytic-synthetic distinction . 14 1.1.1 Criteria for analyticity . 16 1.1.2 Containment . 19 1.1.3 Clarification, identity and contradiction . 26 1.1.4 Syntheticity and intuition . 33 1.2 Kant's pure general logic . 35 1.2.1 The topics of pure general logic . 37 1.2.2 The conception of pure general logic . 41 1.3 The relationship between analyticity and logic . 46 1.3.1 Logic as an instrument . 46 1.3.2 The status of logic . 47 2 Bolzano and the syntheticity of logic 54 2.1 Bolzano's analytic-synthetic distinction . 54 2.1.1 Preliminary notions: the method of substitution, validity, derivability . 54 2.1.2 Bolzano's conception of analyticity . 57 2.1.3 The notion of logical analyticity . 62 2.1.4 Language independence and synonymy . 65 2.1.5 Criticisms against Kant's analysis and analytic-synthetic dis- tinction . 66 2.2 The science of logic . 70 2.2.1 Grounding, deductive sciences and synthetic a priori .... 70 3 CONTENTS 4 2.2.2 Bolzano's thesis that logic is synthetic . 74 2.3 An evaluation . 76 2.3.1 A contradiction in Bolzano's system? The pragmatics of analyticity .
    [Show full text]
  • Informal Introduction to Modal Logic
    Informal Introduction to Modal Logic G. J. Mattey 1 Motivation for Modal Logic One of the first things learned by beginning logic students is the definition of a valid argument. The standard definition of a valid argument runs along these lines: An argument from a set of premises to a conclusion is valid if and only if it is not possible for all the premises of the argument to be true and for the conclusion to be false. Alternatively, one might say that an argument is valid just in case it is that necessarily, if the premises are true, then the conclusion is true. The concept of a valid argument lies at the heart of logic. The definition of the concept of validity, in turn, depends essentially on the modal concepts of possibility and necessity. These concepts are called “modal” because they indi- cate a “way” or “mode” in which the truth-values of the premises are connected with the truth-values of the conclusion. One of the main tasks of symbolic logic is to represent the form of arguments, in such a way that their validity or invalidity can be determined using standard- ized techniques. One can use truth-tables, for example, to represent the validity or invalidity of arguments whose basic units are individual sentences. In the logics that are generally taught in introductory logic courses, the properties of validity or invalidity are not represented in the symbolic language itself. There are symbols representing the “truth-functional” operators ‘and’, ‘or’, ‘not’, etc., but there is no symbol for ‘therefore’.
    [Show full text]
  • Tense Or Temporal Logic Gives a Glimpse of a Number of Related F-Urther Topics
    TI1is introduction provi'des some motivational and historical background. Basic fonnal issues of propositional tense logic are treated in Section 2. Section 3 12 Tense or Temporal Logic gives a glimpse of a number of related f-urther topics. 1.1 Motivation and Terminology Thomas Miiller Arthur Prior, who initiated the fonnal study of the logic of lime in the 1950s, called lus logic tense logic to reflect his underlying motivation: to advance the philosophy of time by establishing formal systems that treat the tenses of nat­ \lrallanguages like English as sentence-modifying operators. While many other Chapter Overview formal approaches to temporal issues have been proposed h1 the meantime, Pri­ orean tense logic is still the natural starting point. Prior introduced four tense operatm·s that are traditionally v.rritten as follows: 1. Introduction 324 1.1 Motiv<Jtion and Terminology 325 1.2 Tense Logic and the History of Formal Modal Logic 326 P it ·was the case that (Past) 2. Basic Formal Tense Logic 327 H it has always been the case that 2.1 Priorean (Past, Future) Tense Logic 328 F it will be the case that (Future) G it is always going to be the case that 2.1.1 The Minimal Tense Logic K1 330 2.1 .2 Frame Conditions 332 2.1.3 The Semantics of the Future Operator 337 Tims, a sentence in the past tense like 'Jolm was happy' is analysed as a temporal 2.1.4 The lndexicai'Now' 342 modification of the sentence 'John is happy' via the operator 'it was the case thai:', 2.1.5 Metric Operators and At1 343 and can be formally expressed as 'P HappyO'olm)'.
    [Show full text]
  • Deontic Logic and Ethics
    Deontic Logic and Ethics Shyam Nair ∗ March 2, 2021 Abstract Though there have been productive interactions between moral philoso- phers and deontic logicians, there has also been a tradition of neglecting the insights that the fields can offer one another. The most sustained interactions between moral philosophers and deontic logicians have not been systematic but instead have been scattered across a number of dis- tinct and often unrelated topics. This chapter primarily focuses on three topics. First, we discuss the \actualism/possibilism" debate which, very roughly, concerns the relevance of what one will do at some future time to what one ought to do at present (x2). This topic is also used to intro- duce various modal deontic logics. Second we discuss the particularism debate which, very roughly, concerns whether there can be any systematic generally theory of what we ought to do (x3). This topic is also used to introduce various non-modal deontic logics. Third, we discuss collective action problems which concern the connection between the obligations of individuals and the behavior and obligations of groups of individuals (x4). This topic is also used to discuss formal systems that allow us to study the relationship between individuals and groups. The chapter also contains a general discussion of the relation between ethical theory and deontic logic (x1) and a brief consideration of other miscellaneous topics (x5). Introduction Though there have been productive interactions between moral philosophers and deontic logicians, there has also been a tradition of neglecting the insights that the fields can offer one another. The most sustained interactions between moral philosophers and deontic logicians have not been systematic but instead have been scattered across a number of distinct and often unrelated topics.
    [Show full text]
  • Modal Truths from an Analytic-Synthetic Kantian Distinction Francesca Poggiolesi
    Modal Truths from an Analytic-Synthetic Kantian Distinction Francesca Poggiolesi To cite this version: Francesca Poggiolesi. Modal Truths from an Analytic-Synthetic Kantian Distinction. A. Moktefi, L. Moretti, F. Schang, eds. Soyons Logiques / Let us be logical, 22, College Publications, pp.149-160, 2016, Cahiers de Logique et d’Epistemologie 978-1-84890-090-5. halshs-00775811 HAL Id: halshs-00775811 https://halshs.archives-ouvertes.fr/halshs-00775811 Submitted on 10 Jun 2016 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. Francesca Poggiolesi, IHPST, 13 rue du Four, 75006, Paris, France [email protected] http://sites.google.com/site/poggiolesi/ Modal Truths From an Analytic- Synthetic Kantian Distinction Abstract In the article of 1965 Are logical truths analytic? Hintikka deals with the problem of establishing whether the logical truths of first-order logic are analytic or synthetic. In order to provide an answer to this issue, Hintikka firstly distinguishes two different notions of analyticity, and then he shows that the sentences of first-order logic are analytic in one sense, but synthetic in another. This interesting result clearly illustrates the non-triviality of the question.
    [Show full text]