Let me start with a well-known story. Kant held that logic and conceptual analysis alone cannot account for our knowledge of arithmetic: “however we might turn and twist our concepts, we could never, by the mere analysis of them, and without the aid of intuition, discover what is the sum [7+5]” (KrV, B16). Frege took himself to have shown that Kant was wrong about this. According to Frege’s logicist thesis, every arithmetical concept can be defined in purely logical terms, and (...) every theorem of arithmetic can be proved using only the basic laws of logic. Hence, Kant was wrong to think that our grasp of arithmetical concepts and our knowledge of arithmetical truth depend on an extralogical source—the pure intuition of time (Frege 1884, §89, §109). Arithmetic, properly understood, is just a part of logic. (shrink)

Abstraction principles provide implicit definitions of mathematical objects. In this paper, an abstraction principle defining categories is proposed. It is unsatisfiable and inconsistent in the expected ways. Two restricted versions of the principle which are consistent are presented.

In this paper, we investigate (1) what can be salvaged from the original project of "logicism" and (2) what is the best that can be done if we lower our sights a bit. Logicism is the view that "mathematics is reducible to logic alone", and there are a variety of reasons why it was a non-starter. We consider the various ways of weakening this claim so as to produce a "neologicism". Three ways are discussed: (1) expand the conception of logic (...) used in the reduction, (2) allow the addition of analytic-sounding principles to logic so that the reduction is not to "logic alone" but to logic and truths knowable a priori, and (3) revise the conception of "reducible". We show how the current versions of neologicism fit into this classification scheme, and then focus on a kind of neologicism which we take to have the most potential for achieving the epistemological goals of the original logicist project. We argue that that the "weaker" the form of neologicism, the more likely it is to be a new form of logicism, and show how our preferred system, though mathematically weak, is metaphysically and epistemogically strong, and can "reduce" arbitrary mathematical theories to logic and analytic truths, if given a legitimate new sense of "reduction". (shrink)

Can there be objects that are ‘thin’ in the sense that very little is required for their existence? A number of philosophers have thought so. For instance, many Fregeans believe it suffices for the existence of directions that there be lines standing in the relation of parallelism; other philosophers believe it suffices for a mathematical theory to have a model that the theory be coherent. This article explains the appeal of thin objects, discusses the three most important strategies for articulating (...) and defending the idea of such objects, and outlines some problems that these strategies face. (shrink)

The neo-Fregean project of basing mathematics on abstraction principles faces “the bad company problem,” namely that a great variety of unacceptable abstraction principles are mixed in among the acceptable ones. In this paper I propose a new solution to the problem, based on the idea that individuation must take the form of a well-founded process. A surprising aspect of this solution is that every form of abstraction on concepts is permissible and that paradox is instead avoided by restricting what concepts (...) there are. (shrink)

Argument-places play an important role in our dealing with relations. However, that does not mean that argument-places should be taken as primitive entities. It is possible to give an account of relations in which argument-places play no role. But if argument-places are not basic, then what can we say about their identity? Can they, for example, be reconstructed in set theory with appropriate urelements? In this article, we show that for some relations, argument-places cannot be modeled in a neutral way (...) in V[A], the cumulative hierarchy with basic ingredients of the relation as urelements. We argue that a natural way to conceive of argument-places is to identify them with abstract, structureless points of a derivative structure exemplified by positional frames. In case the relation has symmetry, these points may be indiscernible. (shrink)

This paper aims to shed light on the broader significance of Frege’s logicism against the background of discussing and comparing Wittgenstein’s ‘showing/saying’-distinction with Brandom’s idiom of logic as the enterprise of making the implicit rules of our linguistic practices explicit. The main thesis of this paper is that the problem of Frege’s logicism lies deeper than in its inconsistency : it lies in the basic idea that in arithmetic one can, and should, express everything that is implicitly presupposed so that (...) nothing is left unsaid. This, in fact, is the target of Wittgenstein’s critique. Rather than the Tractatus, with its claim that logicism attempts to say something that can only be shown, it is the Philosophical Investigations, with its argument by regress against the thesis that every rule which one can follow must be of an explicit nature, that is of real significance here. (shrink)

Kim :1099–1112, 2013) defends a logicist theory of numbers. According to him, numbers are adverbial entities, similar to those denoted by “frequently” and “at 100 mph”. He even introduces new adverbs for numbers: “1-wise”, “2-wise”, and so on. For example, “Fs exist 2-wise” means that there are two Fs. Kim claims that, because we can derive Dedekind–Peano axioms from his definition of numbers as adverbial entities, it is a new form of logicism. In this paper, I will, however, argue that (...) his theory is vulnerable to an analogue of the so-called Bad Company objection to neo-Fregeanism. This means that we cannot be sure that numbers are actually given to us by Kim’s definition; for, we don’t know whether it is indeed a good definition. So, unless Kim, or somebody else, provides a demarcation criterion between good and bad adverbial definitions, Kim’s theory will remain incomplete. (shrink)

Paul Horwich (1990) once suggested restricting the T-Schema to the maximally consistent set of its instances. But Vann McGee (1992) proved that there are multiple incompatible such sets, none of which, given minimal assumptions, is recursively axiomatizable. The analogous view for set theory---that Naïve Comprehension should be restricted according to consistency maxims---has recently been defended by Laurence Goldstein (2006; 2013). It can be traced back to W.V.O. Quine(1951), who held that Naïve Comprehension embodies the only really intuitive conception of set (...) and should be restricted as little as possible. The view might even have been held by Ernst Zermelo (1908), who,according to Penelope Maddy (1988), subscribed to a ‘one step back from disaster’ rule of thumb: if a natural principle leads to contra-diction, the principle should be weakened just enough to block the contradiction. We prove a generalization of McGee’s Theorem, anduse it to show that the situation for set theory is the same as that for truth: there are multiple incompatible sets of instances of Naïve Comprehension, none of which, given minimal assumptions, is recursively axiomatizable. This shows that the view adumbrated by Goldstein, Quine and perhaps Zermelo is untenable. (shrink)

Neo-Fregean logicism seeks to base mathematics on abstraction principles. But the acceptable abstraction principles are surrounded by unacceptable ones. This is the "bad company problem." In this introduction I first provide a brief historical overview of the problem. Then I outline the main responses that are currently being debated. In the course of doing so I provide summaries of the contributions to this special issue.

Kripke’s notion of groundedness plays a central role in many responses to the semantic paradoxes. Can the notion of groundedness be brought to bear on the paradoxes that arise in connection with abstraction principles? We explore a version of grounded abstraction whereby term models are built up in a ‘grounded’ manner. The results are mixed. Our method solves a problem concerning circularity and yields a ‘grounded’ model for the predicative theory based on Frege’s Basic Law V. However, the method is (...) poorly behaved unless the background second-order logic is predicative. (shrink)

Neo-Fregeanism is a family of positions in the philosophy of mathematics that combines a certain type of platonism about mathematical abstracta with a certain type of logicism about the foundations and epistemology of mathematics. This paper addresses the following question: what sort of theory of reference can/should NF be committed to? The theory of reference I propose for NF comes in two parts. First, an alethic account of referential success: the fact that a term ‘a’ succeeds in referring to something (...) depends on facts about truth. Second, a deflationary account of referential specification: given that ‘a’ refers to something, the fact that ‘a’ refers to b specifically follows from the disquotational schema for reference together with the fact that b = a. In the first section of the paper I argue that NF should be committed to the first part of this theory. This a point on which there is already agreement. The bulk of the paper is therefore devoted to arguing that NF should be committed to the second part, given that it is committed to the first part. I close the paper by indicating some significant implications and a possible problem for this theory of reference. (shrink)

Neo-Fregeanism in the philosophy of mathematics consists of two main parts: the logicist thesis, that mathematics (or at least branches thereof, like arithmetic) all but reduce to logic, and the platonist thesis, that there are abstract, mathematical objects. I will here focus on the ontological thesis, platonism. Neo-Fregeanism has been widely discussed in recent years. Mostly the discussion has focused on issues specific to mathematics. I will here single out for special attention the view on ontology which underlies the neo-Fregeans’ (...) claims about mathematical objects, and discuss this view in a broader setting. (shrink)

A central element in neo-Fregean philosophy of mathematics is the focus on abstraction principles, and the use of abstraction principles to ground various areas of mathematics. But as is well known, not all abstraction principles are in good standing. Various proposals for singling out the acceptable abstraction principles have been presented. Here I investigate what philosophical underpinnings can be provided for these proposals; specifically, underpinnings that fit the neo-Fregean's general outlook. Among the philosophical ideas I consider are: general views on (...) a priori justification; the idea of abstraction as reconceptualization, the idea that truth is prior to reference in the sense associated with Frege's context principle; and various broadly relativistic views. The conclusions are by and large negative. (shrink)

Objects appear to fall into different sorts, each with their own criteria for identity. This raises the question of whether sorts overlap. Abstractionists about numbers—those who think natural numbers are objects characterized by abstraction principles—face an acute version of this problem. Many abstraction principles appear to characterize the natural numbers. If each abstraction principle determines its own sort, then there is no single subject-matter of arithmetic—there are too many numbers. That is, unless objects can belong to more than one sort. (...) But if there are multi-sorted objects, there should be cross-sortal identity principles for identifying objects across sorts. The going cross-sortal identity principle, ECIA2 of Cook and Ebert (2005), solves the problem of too many numbers. But, I argue, it does so at a high cost. I therefore propose a novel cross-sortal identity principle, based on embeddings of the induced models of abstracts developed by Walsh (2012). The new criterion matches ECIA2’s success, but offers interestingly different answers to the more controversial identifications made by ECIA2. (shrink)

The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. In this period, there were also lively exchanges between the various schools culminating in (...) the famous Hilbert-Brouwer controversy in the 1920s. -/- The purpose of this anthology is to review the programmes in the foundations of mathematics from the classical period and to assess their possible relevance for contemporary philosophy of mathematics. What can we say, in retrospect, about the various foundational programmes of the classical period and the disputes that took place between them? To what extent do the classical programmes of logicism, intuitionism and formalism represent options that are still alive today? These questions are addressed in this volume by leading mathematical logicians and philosophers of mathematics. (shrink)

This entry introduces the reader to the main ideas in Frege's philosophy of logic, mathematics, and language.

A number of important philosophical problems are problems in the overlap of logic and ontology. Both logic and ontology are diverse fields within philosophy, and partly because of this there is not one single philosophical problem about the relation between logic and ontology. In this survey article we will first discuss what different philosophical projects are carried out under the headings of "logic" and "ontology" and then we will look at several areas where logic and ontology overlap.

I argue that we can and should extend Tarski's model-theoretic criterion of logicality to cover indefinite expressions like Hilbert's ɛ operator, Russell's indefinite description operator η, and abstraction operators like 'the number of'. I draw on this extension to discuss the logical status of both abstraction operators and abstraction principles.

Some central philosophical issues concern the use of mathematics in putatively non-mathematical endeavors. One such endeavor, of course, is philosophy, and the philosophy of mathematics is a key instance of that. The present article provides an idiosyncratic survey of the use of mathematical results to provide support or counter-support to various philosophical programs concerning the foundations of mathematics.

I develop a technique for specifying truth-conditions.

This paper is divided in five sections. Section 11.1 sketches the history of the distinction between speech act with negative content and negated speech act, and gives a general dynamic interpretation for negated speech act. “Downdate semantics” for AGM contraction is introduced in Section 11.2. Relying on semantically interpreted contraction, Section 11.3 develops the dynamic semantics for constative and directive speech acts, and their external negations. The expressive completeness for the formal variants of natural language utterances, none of which is (...) a retraction, has been proved in Section 11.4. The last section gives a laconic answer to the question posed in the title of the paper. (shrink)

Philosophers love a priori knowledge: we delight in truths that can be known from the comfort of our armchairs, without the need to venture out in the world for confirmation. This is due not to laziness, but to two different considerations. First, it seems that many philosophical issues aren’t settled by our experience of the world — the nature of morality; the way concepts pick out objects; the structure of our experience of the world in which we find ourselves — (...) these issues seem to be decided not on the basis of our experience, but in some manner by things prior to (or independently of) that experience. Second, even when we are deeply interested in how our experience lends credence to our claims about the world, the matter remains of the remainder: we learn more about how experience contributes to knowledge when we see what knowledge is available independent of that experience. (shrink)

This thesis comprises three main chapters—each comprising one relatively standalone paper. The unifying theme is fragmentalism about truth, which is the view that the predicate “true” either expresses distinct concepts or expresses distinct properties. -/- In Chapter 1, I provide a formal development of alethic pluralism. Pluralism is the view that there are distinct truth properties associated with distinct domains of subject matter, where a truth property satisfies certain truth-characterizing principles. On behalf of pluralists, I propose an account of logic (...) and semantics that shows how they can answer central conceptual and logical challenges for their view. -/- In Chapter 2, I motivate and develop a modal account of propositions on the basis of an iterative conception of propositions, where the modality is logico-mathematical. The modal account of propositions takes the conception to motivate an inherently potential hierarchy of propositions. I show that the account helps provide satisfying solutions to the intensional paradoxes of Russell-Myhill, Kaplan, and Prior. -/- In Chapter 3, I propose that “true” is polysemous. I suggest that “true” is initially polysemous between correspondence truth and disquotational truth, and further polysemous between the meanings corresponding to the subconcepts of the concept truth generated by the indefinite extensibility of that concept. I show that the proposal provides satisfying solutions to the semantic paradoxes. (shrink)