This paper examines the potential for abstracting propositions – an as yet untested way of defending the realist thesis that propositions as abstract entities exist. I motivate why we should want to abstract propositions and make clear, by basing an account on the neo-Fregean programme in arithmetic, what ontological and epistemological advantages a realist can gain from this. I then raise a series of problems for the abstraction that ultimately have serious repercussions for realism about propositions in general. I first (...) identify problems about the number of entities able to be abstracted using these techniques. I then focus on how issues of language relativity result in problems akin to the Caesar problem in arithmetic by exposing circularity and modal concern over the status of the criterion of identity for propositions. (shrink)

The paper is concerned with the bad company problem as an instance of a more general difficulty in the philosophy of mathematics. The paper focuses on the prospects of stability as a necessary condition on acceptability. However, the conclusion of the paper is largely negative. As a solution to the bad company problem, stability would undermine the prospects of a neo-Fregean foundation for set theory, and, as a solution to the more general difficulty, it would impose an unreasonable constraint on (...) mathematical practice. (shrink)

That expressions should have their contents can seem paradigmatically contingent. But it can also seem a priori that expressions in one's own language should have their contents to the extent that instances of disquotation, such as "Socrates" refers to Socrates' and "cat" refers to cats', are trivially true. I attempt to reconcile these conflicting intuitions about meaningfulness by examining semantic and metasemantic details of linguistic reflexivity. I argue that instances of disquotation are contingent analytic in Kaplan's sense, and bring this (...) lesson to bear on semantic strategies for responding to skepticism, such as Putnam's Brains-in-a-Vat argument. (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.

In this paper, I shall discuss several topics related to Frege’s paradigms of second-order abstraction principles and his logicism. The discussion includes a critical examination of some controversial views put forward mainly by Robin Jeshion, Tyler Burge, Crispin Wright, Richard Heck and John MacFarlane. In the introductory section, I try to shed light on the connection between logical abstraction and logical objects. The second section contains a critical appraisal of Frege’s notion of evidence and its interpretation by Jeshion, the introduction (...) of the course-of-values operator and Frege’s attitude towards Axiom V, in the expression of which this operator occurs as the key primitive term. Axiom V says that the course-of-values of the function f is identical with the course-of-values of the function g if and only if f and g are coextensional. In the third section, I intend to show that in Die Grundlagen der Arithmetik (1884) Frege hardly could have construed Hume’s Principle (HP) as a primitive truth of logic and used it as an axiom governing the cardinality operator as a primitive sign. HP expresses that the number of Fs is identical with the number of Gs if and only if F and G are equinumerous. In the fourth section, I argue that Wright falls short of making a convincing case for the alleged analyticity of HP. In the final section, I canvass Heck’s arguments for his contention that Frege knew he could deduce the simplest laws of arithmetic from HP without invoking Axiom V. I argue that they do not carry conviction. I conclude this section by rejecting an interpretation concerning HP suggested by MacFarlane. (shrink)

In this paper, I shall discuss several topics related to Frege's paradigms of second-order abstraction principles and his logicism. The discussion includes a critical examination of some controversial views put forward mainly by Robin Jeshion, Tyler Burge, Crispin Wright, Richard Heck and John MacFarlane. In the introductory section, I try to shed light on the connection between logical abstraction and logical objects. The second section contains a critical appraisal of Frege's notion of evidence and its interpretation by Jeshion, the introduction (...) of the course-of-values operator and Frege's attitude towards Axiom V, in the expression of which this operator occurs as the key primitive term. Axiom V says that the course-of-values of the function f is identical with the course-of-values of the function g if and only if f and g are coextensional. In the third section, I intend to show that in Die Grundlagen der Arithmetik Frege hardly could have construed Hume's Principle as a primitive truth of logic and used it as an axiom governing the cardinality operator as a primitive sign. HP expresses that the number of Fs is identical with the number of Gs if and only if F and G are equinumerous. In the fourth section, I argue that Wright falls short of making a convincing case for the alleged analyticity of HP. In the final section, I canvass Heck's arguments for his contention that Frege knew he could deduce the simplest laws of arithmetic from HP without invoking Axiom V. I argue that they do not carry conviction. I conclude this section by rejecting an interpretation concerning HP suggested by MacFarlane. (shrink)

I argue that Neo-Fregean accounts of arithmetical language and arithmetical knowledge tacitly rely on a thesis I call [Success by Default]—the thesis that, in the absence of reasons to the contrary, we are justified in thinking that certain stipulations are successful. Since Neo-Fregeans have yet to supply an adequate defense of [Success by Default], I conclude that there is an important gap in Neo-Fregean accounts of arithmetical language and knowledge. I end the paper by offering a naturalistic remedy.

I propose a way of thinking aboout content, and a related way of thinking about ontological commitment. (This is part of a series of four closely related papers. The other three are ‘On Specifying Truth-Conditions’, ‘An Actualist’s Guide to Quantifying In’ and ‘An Account of Possibility’.).

After describing the philosophical background of Kerry's work, an account is given of the way Kerry proposed to supplement Bolzano's conception of logic with a psychological account of the mental acts underlying mathematical judgements.In his writings Kerry criticized Frege's work and Kerry's views were then attacked by Frege.The following two issues were central to this controversy: (a) the relation between the content of a concept and the object of a concept; (b) the logical roles of the definite article. Not only (...) did Frege in 1892 offer an unconvincing solution to Kerry's puzzle concerning 'the concept horse' but he also overlooked the many criticisms levelled by Kerry against the notion of an (indefinite) extension on which his own definition of number was based. (shrink)

Discussion of moral explanation has reached animpasse, with proponents of contemporaryethical naturalism upholding the explanatoryintegrity of moral facts and properties, andopponents – including both anti-realists andnon-naturalistic realists – insisting thatsuch robustly explanatory pretensions as moraltheory has be explained away. I propose thatthe key to solving the problem lies in thequestion whether instances of moral propertiesare causally efficacious. It is argued that,given the truth of contemporary ethicalnaturalism, moral properties are causallyefficacious if the properties of the specialsciences are. Certain objections are rebuttedinvolving (...) the nature of causation, on the onehand, and putative special features of themoral realm, on the other. (shrink)

It is widely accepted that physicalism faces its most serious challenge when it comes to making room for the phenomenal character of psychological experience, its so-called what-it-is-like aspect. The challenge has surfaced repeatedly over the past two decades in a variety of forms. In a particularly striking one, Frank Jackson considers a situation in which Mary, a brilliant scientist who knows all the physical facts there are to know about psychological experience, has spent the whole of her life in a (...) black and white room. He asks, What will happen when Mary is released from her black and white room or is given a colour television monitor? Will she learn anything or not? It seems just obvious that she will learn something about the world and our visual experience of it. But then it is inescapable that her previous knowledge was incomplete. But she had all the physical information. (Jackson 1986: 130). (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)

As is well known, Frege gave an explicit definition of number (belonging to some concept) in ?68 of his Die Grundlagen der Arithmetik.

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)

A number of formal constraints on acceptable abstraction principles have been proposed, including conservativeness and irenicity. Hume’s Principle, of course, satisfies these constraints. Here, variants of Hume’s Principle that allow us to count concepts instead of objects are examined. It is argued that, prima facie, these principles ought to be no more problematic than HP itself. But, as is shown here, these principles only enjoy the formal properties that have been suggested as indicative of acceptability if certain constraints on the (...) size of the continuum hold. As a result, whether or not these higher-order versions of Hume’s Principle are acceptable seems to be independent of standard (ZFC) set theory. This places the abstractionist in an uncomfortable dilemma: Either there is some inherent difference between counting objects and counting concepts, or new criteria for acceptability will need to be found. It is argued that neither horn looks promising. (shrink)

Fine and Antonelli introduce two generalizations of permutation invariance — internal invariance and simple/double invariance respectively. After sketching reasons why a solution to the Bad Company problem might require that abstraction principles be invariant in one or both senses, I identify the most fine-grained abstraction principle that is invariant in each sense. Hume’s Principle is the most fine-grained abstraction principle invariant in both senses. I conclude by suggesting that this partially explains the success of Hume’s Principle, and the comparative lack (...) of success in reconstructing areas of mathematics other than arithmetic based on non-invariant abstraction principles. (shrink)

Frege attempted to provide arithmetic with a foundation in logic. But his attempt to do so was confounded by Russell's discovery of paradox at the heart of Frege's system. The papers collected in this special issue contribute to the on-going investigation into the foundations of mathematics and logic. After sketching the historical background, this introduction provides an overview of the papers collected here, tracing some of the themes that connect them.

Die gegenwärtig unter dem Titel ›Realismus‹ geführten Debatten in der Philosophie befinden sich nach allgemeiner Ansicht in einem Zustand größter Verwirrung, so daß es nützlich erscheint, ein wenig Ordnung in die theoretischen Optionen zu bringen bevor man für die eine oder andere Auffassung Partei ergreift. In der vorliegenden Arbeit wird dafür argumentiert, daß sich ein systematisch zusammenhängendes Zentrum dieser Debatten mit Hilfe des Begriffes der Referenz ordnen läßt. Nach der Analyse einiger klassischer Positionen soll ein Rahmen erstellt werden, innerhalb dessen (...) die Positionen eingeordnet und die zentralen Probleme fruchtbar diskutiert werden können. Zu diesem Zwecke ist es erforderlich, für die Einordnung der Positionen theoretische Kriterien zu benennen, die sich an den Problemen orientieren von denen hier argumentiert wird, das sie zentral seien. Sehr knapp ausgedrückt wird hier die Auffassung vertreten, Realismus sei als metaphysische These aufzufassen, welche eine von menschlichen epistemischen und semantischen Möglichkeiten unabhängige Existenz behauptet; in den gegenwärtigen Debatten typischerweise die Existenz einer Art von Dingen, nicht die eines individuellen Gegenstandes. Diese metaphysische These wird jedoch in der gegenwärtigen Debatte mit semantischen Argumenten untermauert, bzw. angegriffen und jene semantischen Argumente wiederum verwenden epistemische Erwägungen – die Frage betreffend, was man wissen kann und was nicht. Der Beginn der gegenwärtigen Realismusdebatten mit der soeben skizzierten zentralen Stellung semantischer Argumente ist, der hier vertretenen Auffassung zufolge, die Kritik am traditionellen Fregeschen Referenzbegriff durch Kripke und den frühen Putnam um 1970 (Kap. 2). Aus dieser neu zu bewertenden Kritik und der von den Autoren daraus entwickelten externalistischen realistischen Semantik für Artausdrücke läßt sich das erste Kriterium für eine Position in einer Realismusdebatte ableiten und klären (Kap. 3): Hält man die fragliche Art für eine natürliche Art und meint also, sie habe ihren ›Zusammenhalt‹ von Natur aus? Wenn ja, dann ist die fragliche Position eine realistische (das Kriterium ist dank des engen Zusammenhangs von Realismus und Externalismus sowohl hinreichend als auch notwendig). Hält man die Art nicht für eine natürliche Art, ist man Antirealist. Damit geht jeweils not- wendig eine bestimmte Semantik für den auf die Art referierenden Ausdruck einher. Es zeigt sich zugleich, daß es zwei Varianten des Realismus zu unterscheiden gilt, die hier als klassischer bzw. als moderater Realismus bezeichnet werden. Im folgenden (Kap. 4.1) wird argumentiert, daß der Begriff der Wahrheit nicht eigentlich der zentrale Punkt in den in Frage stehenden Realismusdebatten sein sollte, wie vielfach behauptet wurde, sondern seine Brisanz vielmehr aus zugrundeliegenden semantischen Fragen gewinnt die also in der weiteren Aufdeckung der Kriterien die zentrale Rolle spielen müssen. In der Analyse der Kritik an der in Kap. 3 entwickelten Position des ›klassischen Realismus‹ läßt sich ein zweites unterscheidendes Kriterium für Positionen in Realismusdebatten entwickeln: begriffliche Relativität. Nach der Ablehnung von Putnams Auffassungen zu diesem Thema, werden zwei Varianten vorgeschlagen, starke und schwache begriffliche Relativität (Kap. 4.2). Die Anwendung dieses Merkmals zwingt, so wird argumentiert, in einigen Fällen zu einer Kombination einer realistischen Auffassung von Arten als natürliche mit begrifflicher Relativität. Diese mittlere Position zwischen klassischem Realismus und Antirealismus wird als »moderater Realismus« bezeichnet. Der im ersten Kriterium verwendete Begriff der natürlichen Art, und damit die Optionen in den Debatten, wird schließlich mittels einer Dis- kussion des Phänomens der Vagheit noch weiter verdeutlicht (Kap. 4.3). Ab- schließend werden die drei Optionen zusammenfassend dargestellt und ein Versuch unternommen, ihre Fruchtbarkeit für diverse Realismusdebatten anzudeuten. (shrink)