This book offers an historically-informed critical assessment of Dummett's account of abstract objects, examining in detail some of the Fregean presuppositions whilst also engaging with recent work on the problem of abstract entities.

ABSTRACT Is it possible to effect singular reference to mathematical objects in the abstractionist framework? I will argue that even if mathematical expressions pass the relevant syntactic and inferential tests to qualify as singular terms, that does not mean that their semantic function is to refer to a particular object. I will defend two arguments leading to this claim: the permutation argument for the referential indeterminacy of mathematical terms, and the argument from the semantic idleness of the terms introduced by (...) abstraction principles. (shrink)

This paper aims to provide hyperintensional foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of properties endorsed by (...) Hale and Wright and examined in Hale (2013), and demonstrate how a two-dimensional approach to the epistemology of mathematics is consistent with Hale and Wright's notion of there being non-evidential epistemic entitlement rationally to trust that abstraction principles are true. A choice point that I flag is that between availing of intensional or hyperintensional semantics. The hyperintensional semantic approach that I advance is a topic-sensitive epistemic two-dimensional truthmaker semantics. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest that observational type theory can be applied to first-order abstraction principles in order to make abstraction principles recursively enumerable. Epistemic and metaphysical states and possibilities may thus be shown to play a constitutive role in vindicating the reality of mathematical objects and truth, and in providing a conceivability-based route to the truth of abstraction principles as well as other axioms and propositions in mathematics. (shrink)

Some platonists truly agonize over the ontological commitments which their platonism demands of them. Peter van Inwagen, for example, confesses candidly,I am happy to admit that I am uneasy about believing in the existence of ‘causally irrelevant’ objects. The fact that abstract objects, if they exist, can be neither causes or [sic] effects is one of the many features of abstract objects that make nominalism so attractive. I should very much like to be a nominalist, but I don't see how (...) to be one …. (shrink)

Provides a comprehensive overview and introduction to the Routledge Handbook of Propositions.

I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and variants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. -/- It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit quantification, but (...) on the logical conception of set which motivates naive set theory. The accepted solution is to replace this with the iterative conception of set. -/- I show that this picture is doubly mistaken. After a close examination of the paradoxes in chapters 2--3, I argue in chapters 4 and 5 that it is possible to rescue naive set theory by restricting quantification over sets and that the resulting restrictivist set theory is expressible. In chapters 6 and 7, I argue that it is the iterative conception of set and the thesis of absolutism that should be rejected. (shrink)

Nominalizations of modal predicates have received little, if any, attention in the semantic or philosophical literature. This paper will argue that nominalizations of modal predicates require recognizing a novel ontological category of modal objects and it will outline a new semantics of modals based on modal objects.

Nominalizations are expressions that are particularly challenging philosophically in that they help form singular terms that seem to refer to abstract or derived objects often considered controversial. The three standard views about the semantics of nominalizations are [1] that they map mere meanings onto objects, [2] that they refer to implicit arguments, and [3] that they introduce new objects, in virtue of their compositional semantics. In the second case, nominalizations do not add anything new but pick up objects that would (...) be present anyway in the semantic structure of a corresponding sentence without a nominalization. In the first and third case, nominalizations in a sense ‘create’ new objects’, enriching the ontology on the basis of the meaning of expressions. I will argue that there is a fourth kind of nominalization which requires a quite different treatment. These are nominalizations that introduce ‘new’ objects, but only partially characterize them. Such nominalizations generally refer to events or tropes. I will explore an account according on which such nominalizations refer to truth makers. (shrink)

We talk as if there are natural kinds and in particular we quantify over them. We can count the number of elements discovered by Sir Humphrey Davy, or the number of kinds of particle in the standard model. Consequently, it looks at first sight at least, that natural kinds are entities of a sort. In the light of this we may ask certain questions: is the apparent existence of natural kinds real or an illusion? And if real, what sort of (...) entity are natural kinds? Are they sui generis? Or can they be identified with or reduced to some other kind of entity? In this essay I shall look at possible reasons for asserting that either kinds are no sort of entity, or, if they are entities, their existence is equivalent to some fact not involving kinds. Richard Boyd seems to take the view that the apparent existence of natural kinds is an illusion. (shrink)

The distinction between a type and its tokens is a useful metaphysical distinction. In §1 it is explained what it is, and what it is not. Its importance and wide applicability in linguistics, philosophy, science and everyday life are briefly surveyed in §2. Whether types are universals is discussed in §3. §4 discusses some other suggestions for what types are, both generally and specifically. Is a type the sets of its tokens? What exactly is a word, a symphony, a species? (...) §5 asks what a token is. §6 considers the relation between types and their tokens. Do the type and all its tokens share the same properties? Must all the tokens be alike in some or all respects? §7 explains some problems for the view that types exist, and some problems for the view that they don't. §8 elucidates a distinction often confused with the type-token distinction, that between a type (or token) and an occurrence of it. It also discusses some problems that occurrences might be thought to give rise to, and one way to resolve them. (shrink)

Though the divide between reason-based and causal-explanatory approaches in psychiatry and psychopathology is old and deeply rooted, current trends involving multi-factorial explanatory models and evidence-based approaches to interpersonal psychotherapy, show that it has already been implicitly bridged. These trends require a philosophical reconsideration of how reasons can be causes. This paper contributes to that trajectory by arguing that Donald Davidson’s classic paradigm of 1963 is still a valid option.

This paper discusses three approaches to the semantics of event nominalizations and adverbial modification: the Davidsonian account, the Kimian account, and the truthmaker account. It argues that a combination of all three accounts is needed for the semantics of the full range of event, trope, and state nominalizations in English.

This is my entry on natural language ontology in the Stanford Encyclopedia of Philosophy.

This paper is concerned with an intuitive contrast that arises when we consider sentences containing empty definite descriptions. A sentence like ‘The king of France is bald’ appears neither true nor false, while a sentence like ‘My friend was visited by the king of France’ appears false. Recently, Stephen Yablo has suggested an account of this intuitive contrast. Yablo’s account is particularly interesting, since it has important consequences for the ontological commitments of number sentences like ‘The number of planets is (...) even’. However, the paper argues that Yablo’s account is not convincing and that it can thus not establish these consequences. Further, it develops a Strawsonian account of the intuitive contrast. The developed account allows us to draw important conclusions regarding the correct analysis of definite descriptions and of existence sentences containing definite descriptions. (shrink)

In this excellent book Sebastien Gandon focuses mainly on Russell's two major texts, Principa Mathematica and Principle of Mathematics, meticulously unpicking the details of these texts and bringing a new interpretation of both the mathematical and the philosophical content. Winner of The Bertrand Russell Society Book Award 2013.

What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...) argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims, as well as our logical and mathematical knowledge. (shrink)

It is a standard assumption in contemporary metaphysics that concrete objects come with a location in space and time. This applies not only to material objects and events, but also modes (such as the roundness of the apple, the softness of the pillow, Socrates' wisdom) and entities that have been called 'disturbances' (e.g. holes, folds, faults, and scratches). Taking the approach of descriptive metaphysics, I will show that modes and disturbances fail to have a bearer-independent spatial location. This allows for (...) a metaphysical explanation of the Chomskyan contrast between 'There is a fly believed to be in the bottle' and the unacceptable 'There is a flaw believed to be in the argument'. A subsidiary point this paper makes is that in their lack of a direct spatial location, modes need to be sharply distinguished from tropes as a category of foundationalist metaphysics that has been at the center of a pursuit of a one-category ontology since Williams (1953). (shrink)

This dissertation concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The dissertation demonstrates how phenomenal consciousness and gradational possible-worlds models in Bayesian perceptual psychology relate to epistemic modal space. The dissertation demonstrates, then, how epistemic modality relates to the computational theory of mind; metaphysical modality; deontic modality; logical modality; the types of mathematical modality; to the (...) epistemic status of undecidable propositions and abstraction principles in the philosophy of mathematics; to the apriori-aposteriori distinction; to the modal profile of rational propositional intuition; and to the types of intention, when the latter is interpreted as a modal mental state. Examining the nature of epistemic logic itself, I develop a novel approach to conditions of self-knowledge in the setting of the modal μ-calculus, as well as novel epistemicist solutions to Curry's, the liar, and the knowability paradoxes. Solutions to previously intransigent issues concerning the first-person concept; the distinction between fundamental and derivative truths; and the unity of intention and its role in decision theory, are developed along the way. (shrink)

Here is Frege in Foundations of Arithmetic, § 64:The judgment 'Line a is parallel to line b', in symbols: ab, can be taken as an identity. If we do this, we obtain the concept of direction, and say: 'The direction of line a is equal to the direction of line b.' Thus we replace the symbol by the more generic symbol =, through removing what is specific in the content of the former and dividing it between a and b. We (...) carve up the content in a way different from the original way, and this yields us a new concept.Something important is going on in this passage. But at the same time it borders on incoherent. For Frege is saying at least the following:'dir(a ) = dir(b )' has the same content as 'ab'reflecting on that .. (shrink)

Anthony Everett () argues that those who embrace the reality of fictional entities run into trouble when it comes to specifying criteria of character identity. More specifically, he argues that realists must reject natural principles governing the identity and distinctness of fictional characters due to the existence of fictions which leave it indeterminate whether certain characters are identical and the existence of fictions which say inconsistent things about the identities of their characters. Everett's critique has deservedly drawn much attention and (...) a number of defensive moves have been made by, or on the behalf of, fictional realists. My goal in this paper is to move this debate on a further step. I have three goals: to clarify the importance of Everett's discussion of identity criteria within the context of fictional realism, to reassess Everett's objections to realism in light of the resultant literature, and to develop a novel strategy for responding to Everett's concerns. On the approach to be developed, the problems emerge due to an indeterminacy inherent in the concept of a fictional character itself. (shrink)

This essay clarifies quantifier variance and uses it to provide a theory of indefinite extensibility that I call the variance theory of indefinite extensibility. The indefinite extensibility response to the set-theoretic paradoxes sees each argument for paradox as a demonstration that we have come to a different and more expansive understanding of ‘all sets’. But indefinite extensibility is philosophically puzzling: extant accounts are either metasemantically suspect in requiring mysterious mechanisms of domain expansion, or metaphysically suspect in requiring nonstandard assumptions about (...) mathematical objects. Happily, the view of quantifier meanings that underwrites quantifier variance can be used to provide an account of indefinite extensibility that is both metasemantically and metaphysically satisfying. Section 1 introduces the puzzle of indefinite extensibility; section 2 develops and clarifies the metasemantics of quantifier variance; section 3 solves section 1's puzzle of indefinite extensibility by applying section 2's account of quantifier meanings; and section 4 compares the theory developed in section 3 to several other theories in the literature. (shrink)

This paper investigates the determinacy of mathematics. We begin by clarifying how we are understanding the notion of determinacy before turning to the questions of whether and how famous independence results bear on issues of determinacy in mathematics. From there, we pose a metasemantic challenge for those who believe that mathematical language is determinate, motivate two important constraints on attempts to meet our challenge, and then use these constraints to develop an argument against determinacy and discuss a particularly popular approach (...) to resolving indeterminacy, before offering some brief closing reflections. We believe our discussion poses a serious challenge for most philosophical theories of mathematics, since it puts considerable pressure on all views that accept a non-trivial amount of determinacy for even basic arithmetic. (shrink)

A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions of representation (...) from contemporary mathematical logic. It is argued that the theory-based versions of such logicism are either too liberal (the plethora problem) or are committed to intuitively incorrect closure conditions (the consistency problem). Structure-based versions must on the other hand respond to a charge of begging the question (the circularity problem) or explain how one may have a knowledge of structure in advance of a knowledge of axioms (the signature problem). This discussion is significant because it gives us a better idea of what a notion of representation must look like if it is to aid in realizing some of the traditional epistemic aims of logicism in the philosophy of mathematics. (shrink)

The specification and implementation of computational artefacts occurs throughout the discipline of computer science. Consequently, unpacking its nature should constitute one of the core areas of the philosophy of computer science. This paper presents a conceptual analysis of the central role of specification in the discipline.

Concepts of infinity have been subjects of dispute since antiquity. The main problems of this paper are: is the mind able to acquire a concept of infinity? and: how are concepts of infinity acquired? The aim of this paper is neither to say what the meanings of the word “infinity” are nor what infinity is and whether it exists. However, those questions will be mentioned, but only in necessary extent.

Paul Benacerraf’s argument that mathematical realism is apparently incompatible with mathematical knowledge has been widely thought to also show that a priori knowledge in general is problematic. Although many philosophers have rejected Benacerraf’s argument because it assumes a causal theory of knowledge, some maintain that Benacerraf nevertheless put his finger on a genuine problem, even though he didn’t state the problem in its most challenging form. After diagnosing what went wrong with Benacerraf’s argument, I argue that a new, more challenging, (...) version of Benacerraf’s problem can be constructed. The new version—what I call the Defeater Version—of Benacerraf’s problem makes use of a no-defeater condition on knowledge and justification. I conclude by arguing that the best way to avoid the problem is to construct a theory of how a priori judgments reliably track the facts. I also suggest four different kinds of theories worth pursuing. (shrink)

Segundo um influente argumento contra a existência de conhecimento a priori, não há conhecimento a priori porque (i) nenhuma crença é imune à revisão, e (ii) se houvesse conhecimento a priori, algumas crenças seriam irrevisíveis. Uma versão deste argumento foi celebremente defendida por W. V. Quine e ainda é popular entre filósofos naturalistas. O objectivo deste artigo é examinar e rejeitar este argumento contra o a priori. O artigo começa por discutir a tese (i) e o seu papel no modelo (...) da Teia de Crenças de Quine. Defende-se que (i) enfrenta importantes desafios que colocam em risco o seu uso no argumento supra. A premissa (ii) do argumento é então discutida. Philip Kitcher é conhecido pela sua defesa de uma versão da premissa (ii). Os seus argumentos são analisados e rejeitados. A conclusão é que não temos uma boa razão para aceitar (ii), e consequentemente este argumento contra o a priori. O artigo termina com a proposta de uma caracterização do a priori perfeitamente compatível com (i). (shrink)

I argue against the standard view that ontological debates can be fully described as disagreements about what we should believe to exist. The central thesis of the paper is that believing in Fs in the ontologically relevant sense requires more than merely believing that Fs exist. Believing in Fs is not even a propositional attitude; it is rather an attitude one bears to the term expressed by 'Fs'. The representational correctness of such a belief requires not only that there be (...) Fs, but also that the term expressed by 'Fs' should not misrepresent them. In certain cases we might believe that there are Fs without believing our conception of Fs applies to them. This may well be the situation we are in with regard to abstract entities of various sorts. (shrink)

The Caesar problem arises for abstractionist views, which seek to secure reference for terms such as ‘the number of Xs’ or #X by stipulating the content of ‘unmixed’ identity contexts like ‘#X = #Y’. Frege objects that this stipulation says nothing about ‘mixed’ contexts such as ‘# X = Julius Caesar’. This article defends a neglected response to the Caesar problem: the content of mixed contexts is just as open to stipulation as that of unmixed contexts.

Neologicists have sought to ground mathematical knowledge in abstraction. One especially obstinate problem for this account is the bad company problem. The leading neologicist strategy for resolving this problem is to attempt to sift the good abstraction principles from the bad. This response faces a dilemma: the system of ‘good’ abstraction principles either falls foul of the Scylla of inconsistency or the Charybdis of being unable to recover a modest portion of Zermelo–Fraenkel set theory with its intended generality. This article (...) argues that the bad company problem is due to the ‘static’ character of abstraction on the neologicist's account and develops a ‘dynamic’ account of abstraction that avoids both horns of the dilemma. 1 Introduction2ion Reconstructed3 From Bad to Worse4 Abstraction Reconceived5 Bad Company Made GoodA AppendixA.1SemanticsA.2LogicA.3ConsistencyA.4Strength. (shrink)

One of the more distinctive features of Bob Hale and Crispin Wright’s neologicism about arithmetic is their invocation of Frege’s Constraint – roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. In particular, they maintain that, if adopted, Frege’s Constraint adjudicates in favor of their preferred foundation – Hume’s Principle – and against alternatives, such as the Dedekind-Peano axioms. In what follows we establish two main claims. First, we show (...) that, if sound, Hale and Wright’s arguments for Frege’s Constraint at most establish a version on which the relevant application of the naturals is transitive counting – roughly, the counting procedure by which numerals are used to answer “how many”-questions. Second, we show that this version of Frege’s Constraint fails to adjudicate in favor of Hume’s Principle. If this is the version of Frege’s Constraint that a foundation for arithmetic must respect, then Hume’s Principle no more – and no less – meets the requirement than the Dedekind-Peano axioms do. (shrink)

NeoFregeanism is an intriguing but elusive philosophy of mathematical existence. At crucial points, it goes cryptic and metaphorical. I want to put forward an interpretation of neoFregeanism—perhaps not one that actual neoFregeans will embrace—that makes sense of much of what they say. NeoFregeans should embrace quantifier variance.

Around the turn of the century, Poincare and Hilbert each published an account of geometry that took the discipline to be an implicit definition of its concepts. The terms ‘point’, ‘line’, and ‘plane’ can be applied to any system of objects that satisfies the axioms. Each mathematician found spirited opposition from a different logicist—Russell against Poincare' and Frege against Hilbert— who maintained the dying view that geometry essentially concerns space or spatial intuition. The debates illustrate the emerging idea of mathematics (...) as the science of structure. The article closes with some remarks on structuralist and nonstructuralist themes in Frege's own development of arithmetic. (shrink)

The purpose of this paper is to assess the prospects for a neo‐logicist development of set theory based on a restriction of Frege's Basic Law V, which we call (RV): ∀P∀Q[Ext(P) = Ext(Q) ≡ [(BAD(P) & BAD(Q)) ∨ ∀x(Px ≡ Qx)]] BAD is taken as a primitive property of properties. We explore the features it must have for (RV) to sanction the various strong axioms of Zermelo–Fraenkel set theory. The primary interpretation is where ‘BAD’ is Dummett's ‘indefinitely extensible’.1 Background: what (...) and why?2 Framework3 GOOD candidates, indefinite extensibility4 The framework of (RV) alone, or almost alone5 The axioms6 Brief closing. (shrink)

Neo-Fregeans take their argument for arithmetical realism to depend on the availability of certain, so-called broadly syntactic tests for whether a given expression functions as a singular term. The broadly syntactic tests proposed in the neo-Fregean tradition are the so-called inferential test and the Aristotelian test. If these tests are to subserve the neo-Fregean argument, they must be at least adequate, in the sense of correctly classifying paradigm cases of singular terms and non-singular terms. In this paper, I pursue two (...) main goals. On the one hand, I show that the tests’ current state-of-the-art formulations are inadequate and, hence, cannot subserve the neo-Fregean argument. On the other hand, I propose revisions that are adequate and, hence, can subserve this argument. (shrink)

Adherents of Ockham’s fundamental razor contend that considerations of ontological parsimony pertain primarily to fundamental objects. Derivative objects, on the other hand, are thought to be quite unobjectionable. One way to understand the fundamental vs. derivative distinction is in terms of the Aristotelian distinction between ontologically independent and dependent objects. In this paper I will defend the thesis that every natural number greater than 0 is an ontologically dependent object thereby exempting the natural numbers from Ockham’s fundamental razor.

Neo-logicism is, not least in the light of Frege’s logicist programme, an important topic in the current philosophy of mathematics. In this essay, I critically discuss a number of issues that I consider to be relevant for both Frege’s logicism and neo-logicism. I begin with a brief introduction into Wright’s neo-Fregean project and mention the main objections that he faces. In Sect. 2, I discuss the Julius Caesar problem and its possible Fregean and neo-Fregean solution. In Sect. 3, I raise (...) what I take to be a central objection to the position of neo-logicism. In Sect. 4, I attempt to clarify how we should understand Frege’s stipulation that the two sides of an abstraction principle qua contextual definition of a term-forming operator shall be “gleichbedeutend”. In Sect. 5, I consider the options that Frege might have had to establish the analyticity of Hume’s Principle: The number that belongs to the concept F is equal to the number that belongs to the concept G if and only if F and G are equinumerous. Section 6 is devoted to Frege’s two criteria of thought identity. In Sects. 7 and 8, I defend the position of the neo-logicist against an alleged “knock-down argument”. In Sect. 9, I comment on Frege’s description of abstraction in Grundlagen, §64 and the use of the terms “recarving” and “reconceptualization” in the relevant literature on Fregean abstraction and neo-logicism. I argue that Fregean abstraction has nothing to do with the recarving of a sentence content or its decomposition in different ways. I conclude with remarks on global logicism versus local logicisms. (shrink)

In Die Grundlagen der Arithmetik, Frege attempted to introduce cardinalnumbers as logical objects by means of a second-order abstraction principlewhich is now widely known as ``Hume's Principle'' (HP): The number of Fsis identical with the number of Gs if and only if F and G are equinumerous.The attempt miscarried, because in its role as a contextual definition HP fails tofix uniquely the reference of the cardinality operator ``the number of Fs''. Thisproblem of referential indeterminacy is usually called ``the Julius Caesar (...) problem''.In this paper, Frege's treatment of the problem in Grundlagen is critically assessed. In particular, I try to shed new light on it by paying special attention to the framework of his logicism in which it appears embedded. I argue, among other things, that the Caesar problem, which is supposed to stem from Frege's tentative inductive definition of the natural numbers, is only spurious, not genuine; that the genuine Caesar problem deriving from HP is a purely semantic one and that the prospects of removing it by explicitly defining cardinal numbers as objects which are not classes are presumably poor for Frege. I conclude by rejecting two closely connected theses concerning Caesar put forward by Richard Heck: (i) that Frege could not abandon Axiom V because he could not solve the Julius Caesar problem without it; (ii) that (by his own lights) his logicist programme in Grundgesetze der Arithmetik failed because he could not overcome that problem. (shrink)

Penelope Maddy advances a purportedly naturalistic account of mathematical methodology which might be taken to answer the question 'What justifies axioms of set theory?' I argue that her account fails both to adequately answer this question and to be naturalistic. Further, the way in which it fails to answer the question deprives it of an analog to one of the chief attractions of naturalism. Naturalism is attractive to naturalists and nonnaturalists alike because it explains the reliability of scientific practice. Maddy's (...) account, on the other hand, appears to be unable to similarly explain the reliability of mathematical practice without violating one of its central tenets. (shrink)

Philip Kitcher has argued against the apriority of mathematical knowledge in a number of places. His arguments rely on a conception of mathematical knowledge as embedded in a historical tradition and the claim that this sort of embedding compromises apriority. In this paper, I argue that tradition dependence of mathematical knowledge does not compromise its apriority. I further identify the factors which appear to lead Kitcher to argue as he does.

As introduction to the special issue on the semantics of cardinals, we offer some background on the relevant literature, and an overview of the contributions to this volume. Most of these papers were presented in earlier form at an interdisciplinary workshop on the topic at The Ohio State University, and the contributions to this issue reflect that interdisciplinary character: the authors represent both fields in the title of this journal.

One prominent criticism of the abstractionist program is the so-called Bad Company objection. The complaint is that abstraction principles cannot in general be a legitimate way to introduce mathematical theories, since some of them are inconsistent. The most notorious example, of course, is Frege’s Basic Law V. A common response to the objection suggests that an abstraction principle can be used to legitimately introduce a mathematical theory precisely when it is stable: when it can be made true on all sufficiently (...) large domains. In this paper, we raise a worry for this response to the Bad Company objection. We argue, perhaps surprisingly, that it requires very strong assumptions about the range of the second-order quantifiers; assumptions that the abstractionist should reject. (shrink)

In section 10 of Grundgesetze, Frege confronts an indeterm inacy left by his stipulations regarding his ‘smooth breathing’, from which names of valueranges are formed. Though there has been much discussion of his arguments, it remains unclear what this indeterminacy is; why it bothers Frege; and how he proposes to respond to it. The present paper attempts to answer these questions by reading section 10 as preparatory for the (fallacious) proof, given in section 31, that every expression of Frege's formal (...) language denotes. (shrink)

For Quine, the ontological commitments of a discourse are what fall under its (objectual) quantifiers. The recent literature, however, is beginning to move away from this picture. There are direct challenges to Quine's criterion, and there are also attempts to provide alternatives. Azzouni suggests that the ontological commitments of a discourse should be determined by an existence predicate instead. The availability of this alternative forces an adjudication between Qune's criterion and the predicate approach to ontological commitment. I argue that to (...) adjudicate between these criteria for ontological commitment, we need first to adjudicate between criteria for what exists. (shrink)

Neo-Fregeans such as Bob Hale and Crispin Wright seek a foundation of mathematics based on abstraction principles. These are sentences involving a relation called the abstraction relation. It is usually assumed that abstraction relations must be equivalence relations, so reflexive, symmetric and transitive. In this article I argue that abstraction relations need not be reflexive. I furthermore give an application of non-reflexive abstraction relations to restricted abstraction principles.

This paper introduces a persistence puzzle involving two sports clubs with a somewhat intertwined history. As one might assume, the implications of the puzzle go far beyond a mere plea for a precise metaphysical analysis of certain perplexing quandaries regarding sports clubs and represents a challenge for our everyday understanding of social groups. To overcome the supposed impediment to the puzzle, as a starting point, I will accept the assumption that these entities are both social groups and abstract artifacts and (...) will further argue that they are _temporal abstract objects_. Since temporal objects are in time, it will be determined that the cause of the conundrum is the seeming inability to correctly account for the I-relation between the temporal parts, and I will elaborate on why such an explanation is required. Ultimately, I’ll introduce the notion of _I-shifting_, which will explain how temporal abstract objects persist through time and consequently solve the puzzle. (shrink)