Replies to Part II Intuitionism and the Sorites

This chapter provides a reply to the chapters in Part II of this book. It looks at the philosophical challenges presented by vagueness. Philosophers of language from Frege on had been for the most part content to theorize in ways that ignored vagueness, or to focus on idealized languages in which there was none. And no one writing before 1970 seemed fully to have taken the measure of the awkwardness of the Sorites paradox, or the depth of its roots in our intuitive thinking about what kind of ability mastery of a language is. The chapter concludes that the intuitionistic approach, with its integral repudiation of any idea of vagueness as constituted in semantic facts that somehow underlie and explain the distinctive patterns of use of vague expressions, and its consequent commitment to liberalism about verdicts in the borderline region, does exactly that.

Replies to Part I Frege and Logicism

This chapter presents a reply to the chapters that constitute Part I of this book. It looks at each contribution and provides analysis and argument based on the ideas presented in those chapters. It looks in detail at Frege’s theorem and also (neo-)logicism and higher-order logic.

CCCP

This chapter is concerned with Crispin Wright’s critique, in his 2002 “The Conceivability of Naturalism,” of the well-known argument developed in Saul Kripke’s Naming and Necessity against the identity of pain with C-fibre firing. Kripke argued that if the identity held it would do so necessarily, so that the identity theorist would have the task of explaining away the apparent conceivability of pain without C-fibre firing and C-fibre firing without pain. Wright identified a principle underlying Kripke’s argument (the “Counter-Conceivability Principle,” to the effect that a clear and distinct conception of a situation is the best possible evidence of its possibility), and suggested that Kripke’s deployment of it against the identity theory resulted in failure. The present chapter raises some doubts about the details of Wright’s diagnosis of the flaw in Kripke’s argument, and makes a contribution of its own to our understanding of the aetiology of modal illusion.

Inferentialism, Logicism, Harmony, and a Counterpoint

Inferentialism is explained as an attempt to provide an account of meaning that is more sensitive (than the tradition of truth-conditional theorizing deriving from Tarski and Davidson) to what is learned when one masters meanings. The logically reformist inferentialism of Dummett and Prawitz is contrasted with the more recent quietist inferentialism of Brandom. Various other issues are highlighted for inferentialism in general, by reference to which different kinds of inferentialism can be characterized. Inferentialism for the logical operators is explained, with special reference to the Principle of Harmony. The statement of that principle in the author’s book Natural Logic is fine-tuned here in the way obviously required in order to bar an interesting would-be counterexample furnished by Crispin Wright, and to stave off any more of the same.

Quandary and Intuitionism

This chapter offers a sustained commentary on Crispin Wright’s paper “On Being in a Quandary: Relativism, Vagueness, Logic Revisionism.” It begins by giving a brief introduction to the issues surrounding vagueness and the sorites paradox before going on to reconstruct the main argument of Wright’s paper “Vagueness: A Fifth Column Approach”. It then proceeds to endorse Wright’s principal reasons for discounting epistemicist and supervaluationist treatments of vagueness. The chapter then develops a critique of Wright’s technical notion of quandary and of his attempt to solve the Sorites Paradox along intuitionistic lines. Despite venturing some criticisms, the chapter concludes that Wright is correct in thinking that vagueness needs to be explained in terms of the sort of quandary state one is in when one takes a proposition to be borderline.

Foreword

This chapter is divided into four parts, corresponding to the partitioning of the essays in the volume. Part I, on neo-Fregeanism in the philosophy of mathematics develops replies to Demopolous, Heck, Rosen and Yablo, Boolos and Edwards; Part II, on vagueness, intuitionistic logic and the Sorites Paradox develops replies to Rumfitt and Schiffer; Part III, on revisionism in the philosophy of logic develops replies to Shieh and Tennant; and Part IV, on the epistemology of metaphysical possibility develops a reply to Hale. In each section, Crispin Wright offers an overview of the relevant area and outlines and refines his views on the relevant topics. Inter alia, he offers detailed replies to each of the ten contributed essays in the volume.

Wright and Revisionism

Do considerations in the theory of meaning pose a challenge to classical logic, and in particular to the law of excluded middle? Michael Dummett suggested an affirmative answer to this question, and advocated a form of logical revisionism. In his 1981 study “Anti-Realism and Revisionism,” Crispin Wright developed a critique of Dummett’s case for logical revisionism, but in more recent work (e.g., his 1992 book Truth and Objectivity), Wright has advanced an argument in favour of logical revisionism. This chapter investigates the nature and limitations of anti-realist revisionism, and offers a critique of Wright’s arguments in favour of logical revisionism. It also develops an alternative proposal about how revisionism might proceed.

Solving the Caesar Problem—with Metaphysics

According to neo-Fregean Platonism, abstraction principles—such as the principle that the direction of line a is identical to the direction of line b iff a and b are parallel—may in some cases be regarded as introducing new singular terms (e.g., “the direction of line a”) and as fixing the truth-conditions of genuine identity statements featuring them. If neo-Fregeanism is to vindicate Frege’s idea that a plausible philosophy of arithmetic can and should treat the natural numbers as a species of object, it must address the so-called “Caesar Problem”: the problem of explaining in general terms which objects given in other terms they are to be distinguished from. This chapter pilots a novel solution to the Caesar Problem via the notion of a real definition: a definition whose purpose is not to explain a meaning, but to characterize the essential nature of the thing introduced.

Generality and Objectivity in Frege’s Foundations of Arithmetic

This chapter argues for two principal contentions, both of which mark points of divergence from the neo-Fregean position first developed in Crispin Wright’s monograph Frege’s Conception of Numbers as Objects, and developed further in an extended series of works by Wright and Bob Hale. First, that Frege can be regarded as addressing the apriority of arithmetic in a manner that is independent of the ideas that numbers are logical objects or that arithmetic is analytic or a part of logic. Second, that Frege can secure the objectivity of arithmetic in a way that is independent of the idea that numbers are logical objects.

Logicism and Second-Order Logic

This chapter outlines an argument to the effect that there is no reduction of arithmetical truth to logical truth, where “logic” is understood to be elementary (first-order) logic, or any system of logic whose theses form an effectively generable set. It suggests, however, that it leaves open the possibility of a significant reduction of arithmetic to something that might be called a system of logic. By investigating metatheoretic differences between first- and second-order logic, it explores the extent to which second-order logic might play a role in facilitating such a reduction.