DOCSLIB.ORG

6 the Strength of Russell's Modal Logic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
6 the Strength of Russell's Modal Logic 6 The Strength of Russell’s Modal Logic I discussed Russell’s theory of modality in chapters 2–4. It is time to discuss Russell’s modal logics and their strength. First, I sketch my methodology for imputing a modal logic to Russell. Then I distinguish Russell’s modal theory from his modal logic in general. Then I define seven modal logics which may be implicitly attributed to Russell, including three alethic logics, a causal logic, an epistemic logic, and two deontic logics, and assess their strength. I save the deontic logics for last because they are the most difficult. Next, I discuss grades of modal involvement and problems of paraphrase of ordinary de re modal talk, especially problems of quantification into modal contexts. Last, I refute Thomas Magnell’s critique of my 1990 Erkenntnis paper. My methodology is this. I shall impute a modal logic to Russell if either of two tests is met: (i) it is more reasonable than not to paraphrase Russell’s thinking into the modal logic, or (ii) it is more reasonable than not to suppose that Russell would have substantially assented to the modal logic as a paraphrase of his thought. Test (i) is Russell’s own philosophy of paraphrase applied to whole theories. Paraphrase is regimentation of informal discourse into a formal notation. Some replacement of vague thoughts by a more determinate notation is permissible, if that replacement is more reasonable than not. This involves our judgment in balancing the letter and the spirit of the discourse, in assessing what Russell meant or intended. It is ultimately a matter of philosophical interpretation. Russell states test (i) in his famous “Mr Strawson on Referring” (MPD 178–79). Test (i) is met to the extent that a certain modal logic is logically implicit in Russell’s thinking. Test (ii) boils down to test (i), if we suppose that Russell would have substantially assented to formal paraphrases of his thinking which are more reasonable than not. I think that supposition is fairly safe. But no one is perfectly rational all the time, so it may be safer to say only that tests (i) and (ii) are not wholly distinct in the case of thinkers such as Russell. I think there is a sense in which test (ii) is more speculative. Namely, test (ii) calls for express speculation about what Russell would or might have said, if asked. This is good. Such speculation is involved in applying test (i), and it 61 62 Bertrand Russell on Modality and Logical Relevance should be brought out expressly. But test (i) is the more basic, because more purely rational, test. It is not always the case that philosophers would assent to what is logically implicit in their thinking; it might, for example, lead to a reductio ad absurdum of their beliefs. To that extent there is tension between tests (i) and (ii). But the tension is resolved by the very fact that test (i) is more basic. My disjunctive two-prong test is somewhat weaker than a test of finding it more reasonable than not that Russell actually did admit the modal logics I impute to him. On the other hand, my test is far stronger than a test of merely finding these modal logics to be logically consistent with everything or nearly everything he says. On either prong, my test requires finding a substantial positive basis in Russell for imputing the logics to him. In fact, it may be called a substantial positive basis test. This concludes my discussion of methodology. We are after fair philosophical game, if not game to a mere historian’s taste. I distinguish Russell’s modal theory from his modal logic in general. Russell’s full modal theory is MDL {1,2,3}, in which MDL appears as level (3). MDL defines a propositional function F(x) as possible with respect to x just in case F(x) is not always false. From the ontological perspective, MDL {1} is basic to understanding Russell’s theory of modality. But from the logical point of view, MDL {3} is only the stepping-stone to the modal logics. I proceed to the task of imputing modal logics to Russell and assessing their strength. I do not impute a modal logic to Principles of Mathematics. I do impute the full range of Leibniz’s possible worlds as logically implicit in that work. In fact, the 1903 Russell takes such worlds far more seriously than Leibniz does. For Leibniz, they are mere ideas in God’s mind. For Russell, they would be mind-independent complex entities. Such worlds could very easily be the semantic basis for a modal logic. But in Principles, Russell expressly adopts G. E. Moore’s theory of logical necessity as being degrees of implication among propositions. I conclude that there is no possible worlds modal logic in Principles. There are only the possible worlds themselves. I suppose you could call the totality of degrees of implication among propositions a modal logic. But that seems too remote from what is ordinarily meant by the term “modal logic.” Russell’s first modal logic is FG–MDL, meaning fully generalized MDL. Russell presents FG–MDL in his unpublished paper, “Necessity and Possibility” (Russell 1994a; c. 1903–5). FG–MDL defines logical truths as truths which are fully general, i.e., truths containing nothing but logical The Strength of Russell’s Modal Logic 63 constants and bound individual and predicate variables. I follow Gregory Landini in proposing these definitions for FG–MDL: ~A =Df (F1...Fm, x1...xn)AF1...Fm, x1...xn and A =Df (›F1...›Fm, ›x1...›xn)AF1...Fm, x1...xn (Landini 1993) It is easy to see how FG–MDL is based on MDL. In FG–MDL, truths are necessary (“~”) or possible (“”) with respect to all the variables they contain. Russell’s second and mature modal logic is FG–MDL* (“*” is pro- nounced “star”). FG–MDL* defines logical truths as truths which are both (i) fully general in accordance with Landini’s definitions and (ii) true in virtue of their form, i.e.. tautologous. I explained why Russell abandoned FG–MDL and adopted FG–MDL* in chapter 1. I proceed to define the modal operators, using the usual boxes and diamonds to represent “Necessarily” by ~ and “Possibly” by . The modal operators, ~ and , must be defined differently in the two modal logics. In FG–MDL, “~” means “true when fully generalized using only universal quantifiers.” In FG–MDL*, “~” means “true when fully generalized using only universal quantifiers and true in virtue of its form.” Both interpreta- tions are modally innocent. They contain no modal notions. In FG–MDL*, the operator for necessity might be informally called “Analytically,” or even “Logically.” That will not cover the whole field of necessary truths, but Russell aims to cover only logically necessary truths. Iteration of modal operators is admissible in both FG–MDL and FG–MDL*, since we are prefixing whole propositions with the operators. We can iterate operators all we want. Theory of types is not a problem, since an iterated operator is never predicated of itself, but always of a proposition. I shall now discuss the strength of Russell’s two modal logics for logical necessity, beginning with FG–MDL*.1 The distinguishing axiom of alethic S1 is that if P, then it is logically possible that P. That is, P 6 P. That is trivially true in FG–MDL*, due to considerations of logical form. The distinguishing axiom of alethic S2 is that if it is logically possible that P and Q, then it is logically possible that P. That is, (P & Q) 6 P. That is trivially true in FG–MDL* as well. 64 Bertrand Russell on Modality and Logical Relevance The distinguishing axiom of alethic S3 is that if P deductively implies Q, then the logical possibility of P deductively implies the logical possibility of Q. That is, (P 6 Q) 6 (P 6 Q). That is clearly true in FG–MDL*. Iteration of logical modal operators is syntactically possible in FG–MDL*, since they are predicated of statements. Thus we may proceed to consider S4 and S5. Iterated modal operators entirely of the same kind will be trivially collapsible in FG–MDL*, since they will all have the same conditions of application, namely appropriate kind of logical form. Thus ~~P 6 ~P, and P 6 P. The only way it could be logically necessary for P to be logically necessary is if P is logically necessary in virtue of its logical form. And the only way it could be logically possible for P to be logically possible is if P is not logically impossible in virtue of its logical form. The distinguishing axiom of alethic S4 is that if it is logically possible that P is logically possible, then P is logically possible. That is, P 6 P. That is true in FG–MDL*, as we just saw. The distinguishing axiom of alethic S5 is that if P is logically possible, then it is logically necessary that P is logically possible. That is, P 6 ~P. In FG–MDL*, if a statement is possible, then it is necessarily possible, since its possibility is in virtue of its logical form, which is an essential part of the statement’s being the statement it is, and which is timeless and unchanging. And if a statement is necessary, then it is both necessarily possible and necessarily necessary, in virtue of its tautological character. Thus FG–MDL* is closest to S5; note that Arthur Prior and Kit Fine compare standard quantificational logic to S5 (Prior 1977: chapter 1).
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]
  • DRAFT: Final Version in Journal of Philosophical Logic Paul Hovda
    Tensed mereology DRAFT: final version in Journal of Philosophical Logic Paul Hovda There are at least three main approaches to thinking about the way parthood logically interacts with time. Roughly: the eternalist perdurantist approach, on which the primary parthood relation is eternal and two-placed, and objects that persist persist by having temporal parts; the parameterist approach, on which the primary parthood relation is eternal and logically three-placed (x is part of y at t) and objects may or may not have temporal parts; and the tensed approach, on which the primary parthood relation is two-placed but (in many cases) temporary, in the sense that it may be that x is part of y, though x was not part of y. (These characterizations are brief; too brief, in fact, as our discussion will eventually show.) Orthogonally, there are different approaches to questions like Peter van Inwa- gen's \Special Composition Question" (SCQ): under what conditions do some objects compose something?1 (Let us, for the moment, work with an undefined notion of \compose;" we will get more precise later.) One central divide is be- tween those who answer the SCQ with \Under any conditions!" and those who disagree. In general, we can distinguish \plenitudinous" conceptions of compo- sition, that accept this answer, from \sparse" conceptions, that do not. (van Inwagen uses the term \universalist" where we use \plenitudinous.") A question closely related to the SCQ is: under what conditions do some objects compose more than one thing? We may distinguish “flat” conceptions of composition, on which the answer is \Under no conditions!" from others.
    [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]
  • Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics
    axioms Article Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics Lorenz Demey 1,2 1 Center for Logic and Philosophy of Science, KU Leuven, 3000 Leuven, Belgium; [email protected] 2 KU Leuven Institute for Artificial Intelligence, KU Leuven, 3000 Leuven, Belgium Abstract: Aristotelian diagrams, such as the square of opposition, are well-known in the context of normal modal logics (i.e., systems of modal logic which can be given a relational semantics in terms of Kripke models). This paper studies Aristotelian diagrams for non-normal systems of modal logic (based on neighborhood semantics, a topologically inspired generalization of relational semantics). In particular, we investigate the phenomenon of logic-sensitivity of Aristotelian diagrams. We distin- guish between four different types of logic-sensitivity, viz. with respect to (i) Aristotelian families, (ii) logical equivalence of formulas, (iii) contingency of formulas, and (iv) Boolean subfamilies of a given Aristotelian family. We provide concrete examples of Aristotelian diagrams that illustrate these four types of logic-sensitivity in the realm of normal modal logic. Next, we discuss more subtle examples of Aristotelian diagrams, which are not sensitive with respect to normal modal logics, but which nevertheless turn out to be highly logic-sensitive once we turn to non-normal systems of modal logic. Keywords: Aristotelian diagram; non-normal modal logic; square of opposition; logical geometry; neighborhood semantics; bitstring semantics MSC: 03B45; 03A05 Citation: Demey, L. Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics. Axioms 2021, 10, 128. https://doi.org/ 1. Introduction 10.3390/axioms10030128 Aristotelian diagrams, such as the so-called square of opposition, visualize a number Academic Editor: Radko Mesiar of formulas from some logical system, as well as certain logical relations holding between them.
    [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]
  • Accepting a Logic, Accepting a Theory
    1 To appear in Romina Padró and Yale Weiss (eds.), Saul Kripke on Modal Logic. New York: Springer. Accepting a Logic, Accepting a Theory Timothy Williamson Abstract: This chapter responds to Saul Kripke’s critique of the idea of adopting an alternative logic. It defends an anti-exceptionalist view of logic, on which coming to accept a new logic is a special case of coming to accept a new scientific theory. The approach is illustrated in detail by debates on quantified modal logic. A distinction between folk logic and scientific logic is modelled on the distinction between folk physics and scientific physics. The importance of not confusing logic with metalogic in applying this distinction is emphasized. Defeasible inferential dispositions are shown to play a major role in theory acceptance in logic and mathematics as well as in natural and social science. Like beliefs, such dispositions are malleable in response to evidence, though not simply at will. Consideration is given to the Quinean objection that accepting an alternative logic involves changing the subject rather than denying the doctrine. The objection is shown to depend on neglect of the social dimension of meaning determination, akin to the descriptivism about proper names and natural kind terms criticized by Kripke and Putnam. Normal standards of interpretation indicate that disputes between classical and non-classical logicians are genuine disagreements. Keywords: Modal logic, intuitionistic logic, alternative logics, Kripke, Quine, Dummett, Putnam Author affiliation: Oxford University, U.K. Email: [email protected] 2 1. Introduction I first encountered Saul Kripke in my first term as an undergraduate at Oxford University, studying mathematics and philosophy, when he gave the 1973 John Locke Lectures (later published as Kripke 2013).
    [Show full text]
  • Modality in Medieval Philosophy Stephen Read
    Modality in Medieval Philosophy Stephen Read The Metaphysics of Modality The synchronic conception of possibility and necessity as describing simultaneous alternatives is closely associated with Leibniz and the notion of possible worlds.1 The notion seems never to have been noticed in the ancient world, but is found in Arabic discussions, notably in al-Ghazali in the late eleventh century, and in the Latin west from the twelfth century onwards, elaborated in particular by Scotus. In Aristotle, the prevailing conception was what (Hintikka 1973: 103) called the “statistical model of modality”: what is possible is what sometimes happens.2 So what never occurs is impossible and what always occurs is necessary. Aristotle expresses this temporal idea of the possible in a famous passage in De Caelo I 12 (283b15 ff.), and in Metaphysics Θ4 (1047b4-5): It is evident that it cannot be true to say that this is possible but nevertheless will not be. (Aristotle 2006: 5) However, Porphyry. in his Introduction (Isagoge) to Aristotle’s logic (which he took to be Plato’s), included the notion of inseparable accidents, giving the example of the blackness of crows and Ethiopians.3 Neither, he says, is necessarily black, since we can conceive of a white crow, even if all crows are actually black. Whether this idea is true to Aristotle is uncertain.4 Nonetheless, Aristotle seems to have held that once a possibility is 1 realized any alternatives disappear. So past and present are fixed and necessary, and what never happens is impossible. If the future alone is open—that is, unfixed and possible— then once there is no more future, nothing is possible.
    [Show full text]
  • Essence and Necessity, and the Aristotelian Modal Syllogistic: a Historical and Analytical Study Daniel James Vecchio Marquette University
    Marquette University e-Publications@Marquette Dissertations (2009 -) Dissertations, Theses, and Professional Projects Essence and Necessity, and the Aristotelian Modal Syllogistic: A Historical and Analytical Study Daniel James Vecchio Marquette University Recommended Citation Vecchio, Daniel James, "Essence and Necessity, and the Aristotelian Modal Syllogistic: A Historical and Analytical Study" (2016). Dissertations (2009 -). 686. http://epublications.marquette.edu/dissertations_mu/686 ESSENCE AND NECESSITY, AND THE ARISTOTELIAN MODAL SYLLOGISTIC: A HISTORICAL AND ANALYTICAL STUDY by Daniel James Vecchio, B.A., M.A. A Dissertation submitted to the Faculty of the Graduate School, Marquette University, in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Milwaukee, Wisconsin December 2016 ABSTRACT ESSENCE AND NECESSITY, AND THE ARISTOTELIAN MODAL SYLLOGISTIC: A HISTORICAL AND ANALYTICAL STUDY Daniel James Vecchio, B.A., M.A. Marquette University, 2016 The following is a critical and historical account of Aristotelian Essentialism informed by recent work on Aristotle’s modal syllogistic. The semantics of the modal syllogistic are interpreted in a way that is motivated by Aristotle, and also make his validity claims in the Prior Analytics consistent to a higher degree than previously developed interpretative models. In Chapter One, ancient and contemporary objections to the Aristotelian modal syllogistic are discussed. A resolution to apparent inconsistencies in Aristotle’s modal syllogistic is proposed and developed out of recent work by Patterson, Rini, and Malink. In particular, I argue that the semantics of negation is distinct in modal context from those of assertoric negative claims. Given my interpretive model of Aristotle’s semantics, in Chapter Two, I provide proofs for each of the mixed apodictic syllogisms, and propose a method of using Venn Diagrams to visualize the validity claims Aristotle makes in the Prior Analytics.
    [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]
  • The Logical Analysis of Key Arguments in Leibniz and Kant
    University of Mississippi eGrove Electronic Theses and Dissertations Graduate School 2016 The Logical Analysis Of Key Arguments In Leibniz And Kant Richard Simpson Martin University of Mississippi Follow this and additional works at: https://egrove.olemiss.edu/etd Part of the Philosophy Commons Recommended Citation Martin, Richard Simpson, "The Logical Analysis Of Key Arguments In Leibniz And Kant" (2016). Electronic Theses and Dissertations. 755. https://egrove.olemiss.edu/etd/755 This Thesis is brought to you for free and open access by the Graduate School at eGrove. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of eGrove. For more information, please contact [email protected]. THE LOGICAL ANALYSIS OF KEY ARGUMENTS IN LEIBNIZ AND KANT A Thesis presented in partial fulfillment of requirements for the degree of Master of Arts in the Department of Philosophy and Religion The University of Mississippi By RICHARD S. MARTIN August 2016 Copyright Richard S. Martin 2016 ALL RIGHTS RESERVED ABSTRACT This paper addresses two related issues of logic in the philosophy of Gottfried Leibniz. The first problem revolves around Leibniz’s struggle, throughout the period of his mature philosophy, to reconcile his metaphysics and epistemology with his antecedent theological commitments. Leibniz believes that for everything that happens there is a reason, and that the reason God does things is because they are the best that can be done. But if God must, by nature, do what is best, and if what is best is predetermined, then it seems that there may be no room for divine freedom, much less the human freedom Leibniz wished to prove.
    [Show full text]
  • A Brief Introduction to Modal Logic
    A BRIEF INTRODUCTION TO MODAL LOGIC JOEL MCCANCE Abstract. Modal logic extends classical logic with the ability to express not only `P is true', but also statements like `P is known' or `P is necessarily true'. We will define several varieties of modal logic, providing both their semantics and their axiomatic proof systems, and prove their standard soundness and completeness theorems. Introduction Consider the statement `It is autumn,' thinking in particular of the ways in which we might intend its truth or falsity. Is it necessarily autumn? Is it known that it is autumn? Is it believed that it is autumn? Is it autumn now, or will it be autumn in the future? If I fly to Bombay, will it still be autumn? All these modifications of our initial assertion are called by logicians `modalities', indicating the mode in which the statement is said to be true. These are not easily han- dled by the truth tables and propositional variables of the propositional calculus (PC) taught in introductory logic and proof courses, so logicians have developed an augmented form called `modal logic'. This provides mathematicians, computer scientists, and philosophers with the symbols and semantics needed to allow for rigorous proofs involving modalities, something that until recently was thought to be at best pointless, and at worst impossible. This paper will provide an introductory discussion of the subject that should be clear to those with a basic grasp of PC. We will begin by discussing the history and motivations of modal logic, including a few examples of such logics. We will then introduce the definitions and concepts needed for a rigorous discussion of the subject.
    [Show full text]
  • Kripke on Modality
    KRIPKE ON MODALITY Saul Kripke’s published contributions to the theory of modality include both mathematical material (1959a, 1963a, 1963b, and more) and philosophical material (1971, henceforth N&I; 1980, henceforth N&N). Here the latter must be given primary emphasis, but the former will not be entirely neglected. APOSTERIORI NECESSITY Kripke’s overarching philosophical point is that the necessary is not to be confused with the apriori, but preliminary explanations are needed before his contribution can be specified more precisely. To begin with, the linguistics literature (Palmer 1986) distinguishes three flavors of modality: deontic, epistemic/evidential, and dynamic. The pertinent notions of necessity are the obligatory or what must be if obligations are fulfilled, the known or what must be given what is known, and the inevitable in the sense of what cannot fail and could not have failed to be, no matter what. This last is the only notion of necessity of interest in our present context. As Kripke emphasizes, this notion of the necessary and its correlative notion of the contingent are metaphysical concepts. They are as such at least conceptually distinct from the epistemological notions of the apriori and aposteriori and the semantical notions of the analytic and synthetic. Before Kripke this was never completely forgotten, but its importance was arguably grossly underestimated. Prior to Kripke’s work it was the near-unanimous opinion that the analytic is included in the apriori, which in turn is included in the necessary (Quine 1960: 59). It was the majority opinion that these inclusions reverse, making the three classifications coextensive; but there were dissenters, conscious of being in the minority (Kneale and Kneale 1962: 637-639).
    [Show full text]