Bertrand Russell on Modality and Logical Relevance

Jan Dejnožka

• Publisher: Routledge
• Year: 2018
• Print ISBN: 9781138625563
• eBook ISBN: 9780429861710
• Edition: 1
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

First published in 1999, this volume re-examines Bertrand Russell's views on modal logic and logical relevance, arguing that Russell does in fact accommodate modality and modal logic. The author, Jan Dejnožka, draws together Russell's comments and perspectives from throughout his canon in order to demonstrate a coherent view on logical modality and logical relevance. To achieve this, Dejnožka explores questions including whether Russell has a possible worlds logic, Rescher's case against Russell, Russell's three levels of modality and the motives and origins of Russell's theory of modality.

1 Introduction

This book is the only exhaustive study of Russell on logical modality and logical relevance ever written. It is a book on how one can think about and interpret logic. Thus it is a book in the philosophy of logic.

The study of modality is the study of necessity, possibility, impossibility, and contingency. In our case, it is primarily the study of logical necessity, but also of causal necessity, epistemic necesssity, and moral necessity.

The study of logical relevance is the study of how premises logically relate to conclusions in arguments. It is the study of logical inference with emphasis on how premises connect, or ought to connect, to conclusions.

Many philosophers seem unaware that Russell had any views on the subject of modality at all, or think he disliked it. There are three reasons for this. First, Russell’s remarks on modality are scattered throughout his works, including many works which are not primarily about logic. A full portrait of Russell on modality can be drawn only by those who have read all or nearly all of those works. And most philosophers, even many of those who write on Russell, have read very few of those works. This book will show that the time is past when logicians could write about Russell’s logic in ignorance of Russell’s nonlogical works. Second, while perhaps logic without metaphysics or ontology is possible, adequate scholarship on Russell’s modal logic without adequate scholarship on his metaphysics and ontology is not possible. Third, Russell’s fundamental paper on modality, “Necessity and Possibility” (Russell c. 1903–5), which Russell read to the Oxford Philosophical Society on October 22, 1905 (Urquhart 1994: 507), was not published during his lifetime. It appeared in a volume of the Collected Papers of Bertrand Russell only in 1994 (Russell 1994a). Had it been published in 1905 as the companion piece to “On Denoting” I believe it was, the course of modal logic—not to mention the course of Russell studies—might well have been different.

Modal logicians have criticized Russell’s logic for being too limited to accommodate their ideas. I show how Russell accommodates ideas about modality. Many logicians have criticized Russell for impeding the development of modal logic. I show how Russell developed his own modal logic. But to see these things, we must read Russell sympathetically and thoroughly.

Russell’s idea is simple: to use notions of ordinary quantificational logic to define and analyze away modal notions. Modal notions are eliminated across the board. The individual (“existential”) and universal quantifiers are used to simulate and replace modal notions. These quantifiers are interpreted as functioning as if they had modal meanings-in-use. They do not in fact have modal meanings-in-use. Literally speaking, Russell has banished modality from logic. Yet functionally speaking, Russell has achieved a modal logic based on a rich and sophisticated theory of modality. And all this without having to assume any modal entities or even modal notions. The modern moral is that a modal logic is as a modal logic does. This is modal logic without modal metaphysics.

Another way to put it is that it is a fallacy of composition to suppose that just because a logic has no modal elements or parts, it cannot be a modal logic as a whole.

But the best way to put it is that Russell is engaged in dialectical accommodation. Russell refuses to allow ontological status to modal entities, and refuses to admit modal notions as logically primitive. But if that were the whole story, then there would be no point in writing this book. The philosophical world would be justified in saying that Russell rejected modality and he had his reasons—end of story. But this is only the beginning of the story. Russell finds modality important enough not only to give a philosophical theory of modality, but also to show how to formalize it as a logic. His approach is economical, even elegant: he eliminates and formalizes possibility in the very same way he eliminates and formalizes existence. Thus the book is really an object lesson in Russell as master of dialectical thinking. It is about the sense in which he functionally accepts and assimilates modality into his philosophical system even as he rejects what he considers certain more primitive accounts of modality. It is about a synthesis Hegel himself might have admired.

Thus Russell inverts Nathan U. Salmon’s metaphor of pulling a rabbit out of a hat. Salmon is criticizing those who pull the rabbit of modal entities out of the hat of mere considerations of language (Salmon 1981, picturing a rabbit in a hat on the dust cover). But Russell pulls the rabbit of modal logic out of the hat of a world devoid of modal entities.

Russell’s idea is so simple, one wonders why it has been so hard for scholars to notice it for over ninety years. Indeed, a few scholars have noticed it. But they have been unsympathetic, and so have helped keep it hidden under a dismissive cloud. This is understandable. In chapter 2, I describe five logical howlers Russell’s theory of modality seems to commit, including several Alfred Jules Ayer, Nicholas Rescher, Raymond Bradley, and Timothy Sprigge actually accuse Russell of. It will take some work to show how Russell’s theory accommodates all these howlers.

The three definitions on which Russell’s modal logic is based are:

F(x) is necessary with respect to x =Df F(x) is always true (true universal generalization)

F(x) is possible with respect to x =Df F(x) is not always false (true “existential” generalization)

F(x) is impossible with respect to x =Df F(x) is always false (true universal generalization over the negation of F)

I call this group of definitions “MDL”.

MDL is the stepping-stone from which Russell develops his modal logics. Russell’s first modal logic analyzes logically necessary truths as fully general true propositions, where a fully general proposition is a proposition which contains only logical constants and bound individual and predicate variables. I call this first modal logic “FG–MDL”. Russell’s idea is that a fully general true proposition is necessary (i.e. always true) not just with respect to some one of its variables, but is necessary (i.e. always true) with respect to all of its variables. Such a proposition may be called fully necessary. As early as 1905, Russell defines “analytic propositions” as fully general true propositions (Russell c. 1903–05: 519), equating analytic truth with fully generalized truth. I accept Gregory Landini’s formalization of this: Analytically

” (Russell 1994a; c. 1903–5; Landini 1993). Thus for example, “(x)(F)(Fx ∨ ¬Fx)” belongs to FG–MDL, while “(x)(Red(x) ∨ ¬(Red(x))” does not, since “Red()” is a predicate constant. We today might wish to include the second of these statements as a logical truth, since it is an instance of the first statement. But for Russell, full generality is always a requirement of logical truth because “pure logic” is universal; it contains only logical constants (PLA 139, 240–41). Logicians should not be interested in empirical questions of what the real world contains (PLA 199). Even those timeless universals which can only be empirically known, such as the color red, are not part of pure logic. In Principia terms, fully general truths are second-order propositions (PM 163). We must never model Russellian logical necessity in terms of Kripkean arbitrary non-empty sub-sets of the set of all possible worlds with accessibility relations among the worlds in the sub-set, since that would import description into the semantics for second-order logical truths, ruining Russell’s treatment of modality (see Cocchiarella 1975).

Russell eventually finds that full generality is not a sufficient condition, but only a necessary condition, of logically necessary truth. The 1914–19 Russell therefore adds a second requirement to that of full generality, namely, that a logically true proposition be true in virtue of logical form, or tautologous. This gives us Russell’s second and more mature modal logic, which I call “FG–MDL*” (pronounced FG–MDL star). In 1913 FG–MDL* is characterized only in terms of logical form, where a form is some sort of special entity. By 1919 FG–MDL* is in effect characterized alternatively in terms of logical form and in terms of tautology, as if these features were not significantly different (IMP 199–205). This is so even though Russell finds that he can easily define “form” in terms of what remains the same in a proposition through replacements of its constants:

The “form” of a proposition is that, in it, that remains unchanged when every constituent of the proposition is replaced by another. (IMP 199; see PLA 238)

but does not know how to define “tautologous” (IMP 205), and can only say it means what used to be intended by the old term “analytic” (IMP 203–4). It seems clear that Russell would find a proposition tautologously true if and only if it is true in virtue of its form. Evidently he wishes to use truth in virtue of form as a technical notion to explain the intuitive notion of tautology. In any case, FG–MDL* is Russell’s mature and final modal logic. In both FG–MDL and FG–MDL*, Russell intends that there be no distinction between logical truth and logically necessary truth. That is, logical necessity is nothing over and above logical truth. Note how close FG–MDL* is to being a mere generality theory in the block-indented quotation just above. The only new thing is the addition of form. I shall discuss the reason for Russell’s move from FG–MDL to FG–MDL* in more detail shortly.

MDL occurs in at least nine works over a period of at least thirty-six years (1905–40):

c. 1903–05 “Necessity and Possibility” (Russell c. 1903–05: 518)

1906 Review of MacColl’s Symbolic Logic and Its Applications (Russell 1906: 257)

1908 ‘“If’ and ‘Imply’, A Reply to Mr. MacColl” (Russell 1908: 301);

1908 “Mathematical Logic as Based on the Theory of Types” (LK 66n)

1913 “On the Notion of Cause,” in Mysticism and Logic (ONC 176)

1918 “The Philosophy of Logical Atomism” (PLA 231, 232, 233, 240, 242, 254–55)

1919 Introduction to Mathematical Philosophy (IMP 165)

1927 The Analysis of Matter (AMA 170)

1940 Inquiry (IMT 37)

Talk of possible worlds occurs in at least nine works over a period of forty-two years (1907–48):

1907 “The Study of Mathematics,” in Mysticism and Logic (MAL 65)

1912 The Problems of Philosophy (PP 78)

1914 Our Knowledge of the External World (KEW 145)

1918 “The Philosophy of Logical Atomism” (PLA 240)

1919 Introduction to Mathematical Philosophy (IMP 141, 192–93, 203–4)

1921 The Analysis of Mind (AMI 268)

1927 The Analysis of Matter (AMA 200)

1938 The Principles of Mathematics, 2d. ed. (POM viii)

1948 Human Knowledge (HK 157)

These two periods of time overlap for some thirty-four years. In fact, there are four works which contain both MDL and possible worlds talk:

1918 “The Philosophy of Logical Atomism”

1918 Mysticism and Logic

1919 Introduction to Mathematical Philosophy

1927 The Analysis of Matter

These four works constitute a very direct overlap for some ten years. The plain suggestion is that MDL is consistent with and is deliberately intended as at least a basic part of Russell’s logical analysis or eliminative paraphrase of his more casual possible worlds talk, contra Raymond Bradley, who argues that MDL and the possible worlds talk are mutually inconsistent. Bradley seems unaware of the magnitude of the overlap, and correspondingly unaware of the magnitude of the inconsistency he is attributing to Russell.

From the dates given above, it seems reasonable to infer not only that Russell held MDL continuously from at least 1905 to 1940, but that Russell held MDL continuously from at least 1905 until he died. It is trivial to add that since MDL evidently persists unchanged throughout this period of at least thirty-six years, MDL evidently persists unchanged through all the vicissitudes of Russell’s logical theorizing during those years. This is not surprising, since MDL has nothing to do with Russell’s paradox and is not even a formal innovation. MDL is merely a definition of possibility and necessity as respectively being the same basic notions of ‘not always false’ and ‘always true’ which Russell defines his first-level individual and universal quantifiers as being. Russell says, “It will be out of this notion of sometimes, which is the same as the notion of possible, that we get the notion of existence” (PLA 232).

It would be beyond the scope of this book to describe all of Russell’s logical phases in any detail—Russell claims “to have tried at least a hundred theories” to resolve his paradox by October 20, 1903 (Russell 1994: xxiii). But a brief summary of twenty main phases might be helpful:

(1) MDL is not to be found in Russell’s 1903 Principles of Mathematics. This book was mostly written in 1900, though Appendix B, giving Russell’s first theory of types, was written in late 1902. There is no hint of MDL anywhere in the book, not even in Appendix B. Instead we find a rival theory, Moore’s theory of degrees of implicative necessity (POM 454 and 454n). Evidently Russell abandons Moore’s theory when Russell develops MDL. There are some brief remarks about propositional form, but nothing like a theory that logical truths are true in virtue of their form.

(2) Russell may or may not have developed MDL at the time of his May 1903 no-classes logic, which uses propositional functions to eliminate classes (Russell 1994: xx–xxii; 1973a: 129–30).

(3) Russell may or may not have developed MDL at the time of Russell’s 1903–04 return to a Platonic realism of classes (Russell 1994: xxv; 1973a: 129–30), which was reflected in his 1904 paper, “The Axiom of Infinity” (Russell 1973).

(4) Russell’s c. 1903–05 “On Necessity and Possibility” (Russell 1994a) is the first and the main appearance of MDL and FG–MDL. This paper seems to be a companion piece to Russell’s 1905 “On Denoting,” since it defines possibility as ‘not always false’, and “On Denoting” defines existence as ‘not always false’.

(5) Russell almost certainly held MDL by the time of his 1905 “On Some Difficulties in the Theory of Transfinite Numbers and Order Types.” This paper discusses three possible solutions of Russell’s paradox: zig-zag theory, limitation of classes theory, and no-classes theory. Russell declines to choose a specific theory (Russell 1973a: 129–30). But the Feb 5, 1906 note appended to the paper shows Russell settled for the no-classes theory (Russell 1973c: 164n; see Cocchiarella 1980: 78).

(6) Russell held MDL during the period of his May 1906 paper, “On the Substitutional Theory of Classes and Relations,” which advocates a no-classes theory, this time using not propositional functions but a propositions-constants-and-substitution method of eliminating classes, following Maxime Bôcher. This substitution method was to prove too hard to combine with theory of types (Lackey 1973a: 129–30). Note that substitutional quantification for the 1906 Russell is not quantification over expressions in the contemporary sense of substitutional quantification (Cocchiarella 1980: 87). Substitution is replacement of one constant by another (Russell 1973d: 167), while determination is assigning a constant as value of a variable (Russell 1973d: 166). These operations lead to different results in many cases (Russell 1973d: 167–68). This is a no-classes theory. No classes, relations, or numbers are assumed. (Russell 1973d: 166). But this is not merely not assuming. There are really no such things as classes (Russell 1973d: 166, 179). The reason they cannot exist is Russell’s paradox (Russell 1973d: 171).

(7) Russell held MDL during the period of his September 1906 paper, “Les Paradoxes de la Logique” (“On ‘Insolubilia’ and their Solution by Symbolic Logic”) (Russell 1973b). Here Russell claims to prove there are infinitely many complex but single entities, namely, propositions (Landin...

Table of contents

• Cover
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Foreword
• Preface
• 1. Introduction
• 2. Propositional Functions and Possible Worlds
• 3. Russell’s Three Levels of Modality
• 4. The Ontological Foundation
• 5. Rescher’s Case Against Russell
• 6. The Strength of Russell’s Modal Logic
• 7. Does Russell Have a Possible Worlds Logic?
• 8. The Motives and Origins of Russell’s Theory of Modality
• 9. Russell’s Relevance Logic
• 10. Russell, Keynes, and the Legal Origins of Logical Relevance
• Notes
• Bibliography
• Index of Names
• Index of Subjects
• Glossary of Logical Terms
• About the Author