Neo-Fregeans argue that substantial mathematics can be derived from a priori abstraction principles, Hume's Principle connecting numerical identities with one:one correspondences being a prominent example. The embarrassment of riches objection is that there is a plurality of consistent but pairwise inconsistent abstraction principles, thus not all consistent abstractions can be true. This paper considers and criticizes various further criteria on acceptable abstractions proposed by Wright settling on another one—stability—as the best bet for neo-Fregeans. However, an analogue of the embarrassment of (...) riches objection resurfaces in the metatheory and I conclude by arguing that the neo-Fregean program, at least insofar as it includes a platonistic ontology, is fatally wounded by it. (shrink)

In this important study D. M. Armstrong offers a comprehensive system of analytical metaphysics that synthesises but also develops his thinking over the last twenty years. Armstrong's analysis, which acknowledges the 'logical atomism' of Russell and Wittgenstein, makes facts the fundamental constituents of the world, examining properties, relations, numbers, classes, possibility and necessity, dispositions, causes and laws. All these, it is argued, find their place and can be understood inside a scheme of states of affairs. This is a comprehensive and (...) rigorously this-worldly account of the most general features of reality, argued from a distinctive philosophical perspective, and it will appeal to a wide readership in analytical philosophy. (shrink)

Editorial preface to the fourth edition and modified translation -- The text of the Philosophische Untersuchungen -- Philosophische untersuchungen = Philosophical investigations -- Philosophie der psychologie, ein fragment = Philosophy of psychology, a fragment.

This paper contains a close analysis of Frege's proofs of the axioms of arithmetic §§70-83 of Die Grundlagen, with special attention to the proof of the existence of successors in §§82-83. Reluctantly and hesitantly, we come to the conclusion that Frege was at least somewhat confused in those two sections and that he cannot be said to have outlined, or even to have intended, any correct proof there. The proof he sketches is in many ways similar to that given in (...) Grundgesetze der Arithmetik, but fidelity to what Frege wrote in Die Grundlagen and in Grundgesetze requires us to reject the charitable suggestion that it was this (beautiful) proof that he had in mind in §§82-83. (shrink)

Perhaps the most important work of philosophy written in the twentieth century, the Tractatus Logico-Philosophicus first appeared in 1921 and was the only philosophical work that Ludwig Wittgenstein (1889-1951) published during his lifetime. Written in short, carefully numbered paragraphs of extreme compression and brilliance, it immediately convinced many of its readers and captivated the imagination of all. Its chief influence, at first, was on the Logical Positivists of the 1920s and 30s, but many other philosophers were stimulated by its philosophy (...) of language, finding attractive, if ultimately unsatisfactory, its view that propositions were pictures of reality. Perhaps most of all, Wittgenstein himself, after his return to philosophy in the late 1920s, was fascinated by its vision of an inexpressible, crystalline world of logical relationships. The posthumous publication of other writings has, therefore, only served to reawaken interest in the Tractatus and to illuminate its more neglected aspects. David Pears and Brian McGuinness revised their translation in the light of Wittgenstein's own suggestions and comments in his correspondence with C.K. Ogden about the first translation. In addition, this edition contains the introduction by Bertrand Russell which appeared in the original English translation. (shrink)

Paul Boghossian defends a meaning‐based approach to the apriority of the propositions of logic. His model is based on the idea that the logical constants are implicitly defined by some of the axioms and inference rules in which they are involved, thereby offering an alternative to those theories that deny that grasp of meaning can contribute to the explanation of a thinker's entitlement to a particular type of transition or belief.

This chapter aims to provide materials with which to substantiate the claim that, under the appropriate circumstances, the notion of analyticity can help explain how one might have a priori knowledge even in the strong sense. It argues that Implicit Definition, properly understood, is completely independent of any form of irrealism about logic. The chapter defends the thesis of Implicit Definition against Quine's criticisms, and examines the sort of account of the apriority of logic that this doctrine is able to (...) provide. The chapter shows that, against the background of a rejection of indeterminacy, its insolubility cannot be conceded. It also argues that neither a non‐factualism about Frege‐analyticity, nor an error thesis about it, can plausibly fall short of an outright rejection of meaning itself. The chapter shows how the doctrine that appears to offer the most promising account of how we grasp the meanings of the logical constants. (shrink)

Discourse carries thin commitment to objects of a certain sort iff it says or implies that there are such objects. It carries a thick commitment to such objects iff an account of what determines truth-values for its sentences say or implies that there are such objects. This paper presents two model-theoretic semantics for mathematical discourse, one reflecting thick commitment to mathematical objects, the other reflecting only a thin commitment to them. According to the latter view, for example, the semantic role (...) of number-words is not designation but rather the encoding of cardinality-quantifiers. I also present some reasons for preferring this view. (shrink)

Robert Brandom; XII*—The Significance of Complex Numbers for Frege's Philosophy of Mathematics1, Proceedings of the Aristotelian Society, Volume 96, Issue 1, 1.

Frege's analysis of Zahlangaben is expounded and evaluated.

Frege, famously, held that there is a close connection between our concept of cardinal number and the notion of one-one correspondence, a connection enshrined in Hume's Principle. Husserl, and later Parsons, objected that there is no such close connection, that our most primitive conception of cardinality arises from our grasp of the practice of counting. Some empirical work on children's development of a concept of number has sometimes been thought to point in the same direction. I argue, however, that Frege (...) was close to right, that our concept of cardinal number is closely connected with a notion like that of one-one correspondence, a more primitive notion we might call just as many. (shrink)

This paper argues that that Caesar problem had a technical aspect, namely, that it threatened to make it impossible to prove, in the way Frege wanted, that there are infinitely many numbers. It then offers a solution to the problem, one that shows Frege did not really need the claim that "numbers are objects", not if that claim is intended in a form that forces the Caesar problem upon us.

Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.

Divided into two parts, the first concentrates on the logical properties of propositions, their relation to facts and sentences, and the parallel objects of commands and questions. The second part examines theories of intentionality and discusses the relationship between different theories of naming and different accounts of belief.

pt. 1. The proposition and its sense.--pt. 2. On logic and mathematics.

This volume examines the notion of an analytic proof as a natural deduction, suggesting that the proof's value may be understood as its normal form--a concept with significant implications to proof-theoretic semantics.

Here, Bob Hale and Crispin Wright assemble the key writings that lead to their distinctive neo-Fregean approach to the philosophy of mathematics. In addition to fourteen previously published papers, the volume features a new paper on the Julius Caesar problem; a substantial new introduction mapping out the program and the contributions made to it by the various papers; a section explaining which issues most require further attention; and bibliographies of references and further useful sources. It will be recognized as the (...) most powerful presentation yet of a neo-Fregean program. (shrink)

In this exciting new collection, a distinguished international group of philosophers contribute new essays on central issues in philosophy of language and logic, in honor of Michael Dummett, one of the most influential philosophers of the late twentieth century. The essays are focused on areas particularly associated with Professor Dummett. Five are contributions to the philosophy of language, addressing in particular the nature of truth and meaning and the relation between language and thought. Two contributors discuss time, in particular the (...) reality of the past. The last four essays focus on Frege and the philosophy of mathematics. The volume represents some of the best work in contemporary analytical philosophy. (shrink)

Michael Dummett is a leading contemporary philosopher whose work on the logic and metaphysics of language has had a lasting influence on how these subjects are conceived and discussed. This volume contains some of the most provocative and widely discussed essays published in the last fifteen years, together with a number of unpublished or inaccessible writings. Essays included are: "What is a Theory of Meaning?," "What do I Know When I Know a Language?," "What Does the Appeal to Use Do (...) for the Theory of Meaning?," "Language and Truth," "Truth and Meaning," "Language and Communication," "The Source of the Concept of Truth," "Mood, Force, and Convention," "Frege and Husserl on Reference," "Realism," "Existence," "Does Quantification Involve Identity?," "Could there be Unicorns?," "Causal Loops," "Common Sense and Physics," "Testimony and Memory," "What is Mathematics About?," "Wittgenstein on Necessity: Some Reflections," and "Realism and Anti-Realism." Serving well as a companion to Dummett's other collections, the essays in this volume are not forbiddingly technical or specialized, and have relevance to many areas of analytic philosophy. (shrink)

Recasting important questions about truth and objectivity in new and helpful terms, his book will become a focus in the contemporary debates over realism, and ...

The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify (...) many concepts in contemporary mathematics, and thus that both first- and higher-order logics are needed to fully reflect current work. Throughout, the emphasis is on discussing the associated philosophical and historical issues and the implications they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic comparable to that provided in a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in the field today. (shrink)

Anti-realism is a doctrine about logic, language, and meaning that is based on the work of Wittgenstein and Frege. In this book, Professor Tennant clarifies and develops Dummett's arguments for anti-realism and ultimately advocates a radical reform of our logical practices.

Gottlob Frege and Ludwig Wittgenstein (the later Wittgenstein) are often seen as polar opposites with respect to their fundamental philosophical outlooks: Frege as a paradigmatic "realist", Wittgenstein as a paradigmatic "anti-realist". This opposition is supposed to find its clearest expression with respect to mathematics: Frege is seen as the "arch-platonist", Wittgenstein as some sort of "radical anti-platonist". Furthermore, seeing them as such fits nicely with a widely shared view about their relation: the later Wittgenstein is supposed to have developed his (...) ideas in direct opposition to Frege. The purpose of this paper is to challenge these standard assumptions. I will argue that Frege's and Wittgenstein's basic outlooks have something crucial in common; and I will argue that this is the result of the positive influence Frege had on Wittgenstein. (shrink)

In this paper, we describe "metaphysical reductions", in which the well-defined terms and predicates of arbitrary mathematical theories are uniquely interpreted within an axiomatic, metaphysical theory of abstract objects. Once certain (constitutive) facts about a mathematical theory T have been added to the metaphysical theory of objects, theorems of the metaphysical theory yield both an analysis of the reference of the terms and predicates of T and an analysis of the truth of the sentences of T. The well-defined terms and (...) predicates of T are analyzed as denoting abstract objects and abstract relations, respectively, in the background metaphysics, and the sentences of T have a reading on which they are true. After the technical details are sketched, the paper concludes with some observations about the approach. One important observation concerns the fact that the proper axioms of the background theory abstract objects can be reformulated in a way that makes them sound more like logical axioms. Some philosophers have argued that we should accept (something like) them as being logical. (shrink)

Explains why Bob Hale's proposed notion of weak sense cannot explain the analyticity of Hume's principle as he claims. Argues that no other notion of the sort Hale wants could do the job either.

Atomism denies that complexes exist. Common-sense metaphysics may posit masses, composite individuals and sets, but atomism says there are only simples. In a singularist logic, it is difficult to make a plausible case for atomism. But we should accept plural logic, and then atomism can paraphrase away apparent reference to complexes. The paraphrases require unfamiliar plural universals, but these are of independent interest; for example, we can identify numbers and sets with plural universals. The atomist paraphrases would fail if plurals (...) presuppose complexes: but an Appendix shows that reference to complexes is not required in the semantics of plurals. (shrink)

Realism concerning a given subject-matter is characterised as a semantic doctrine with metaphysical consequences, namely as the adoption, for the relevant class of statements, of a truth-conditional theory of meaning resting upon the classical two-valued semantics. it is argued that any departure from classical semantics may, though will not necessarily, be seen as in conflict with some variety of realism. a sharp distinction is drawn between the rejection of realism and the acceptance of a reductionist thesis; though intimately related, neither (...) entails the other. realism is to be classified as "naive", "semi-naive" or "sophisticated": the first of these involves an all but unintelligible epistemological component. (shrink)

We examine the arguments on both sides of the recent debate (Hale and Wright v. Field) on the existence, and modal status, of the natural numbers. We formulate precisely, with proper attention to denotational commitments, the analytic conditionals that link talk of numbers with talk of numerosity and with counting. These provide conceptual controls on the concept of number. We argue, against Field, that there is a serious disanalogy between the existence of God and the existence of numbers. We give (...) stronger reasons than those advanced by Wright for resisting Field's analogy. We argue that the rules governing the basic numerical notions commit us to the natural numbers as necessary existents. We also show that the latest twist in the debate involving 'surdons' leaves both sides in a stalemate. (shrink)

Crispin Wright and Bob Hale have defended the strategy of defining the natural numbers contextually against the objection which led Frege himself to reject it, namely the so-called ‘Julius Caesar problem’. To do this they have formulated principles (called sortal inclusion principles) designed to ensure that numbers are distinct from any objects, such as persons, a proper grasp of which could not be afforded by the contextual definition. We discuss whether either Hale or Wright has provided independent motivation for a (...) defensible version of the sortal inclusion principle and whether they have succeeded in showing that numbers are just what the contextual definition says they are. (shrink)

The purpose of this article is to delimit what can and cannot be claimed on behalf of second-order logic. The starting point is some of the discussions surrounding my Foundations without Foundationalism: A Case for Secondorder Logic.