Abstractionism, which is a development of Frege's original Logicism, is a recent and much debated position in the philosophy of mathematics. This volume contains 16 original papers by leading scholars on the philosophical and mathematical aspects of Abstractionism. After an extensive editors' introduction to the topic of abstractionism, the volume is split into 4 sections. The contributions within these sections explore the semantics and meta-ontology of Abstractionism, abstractionist epistemology, the mathematics of Abstractionis, and finally, Frege's application (...) constraint within an abstractionist setting. (shrink)

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)

Both traditional and naturalistic epistemologists have long assumed that the examination of human psychology has no relevance to the prescriptive goal of traditional epistemology, that of providing first-person guidance in determining the truth. Contrary to both, I apply insights about the psychology of human perception and concept-formation to a very traditional epistemological project: the foundationalist approach to the epistemic regress problem. I argue that direct realism about perception can help solve the regress problem and support a foundationalist account of justification, (...) but only if it is supplemented by an abstractionist theory of concept-formation, the view that it is possible to abstract concepts directly from the empirically given. Critics of direct realism like Laurence BonJour are correct that an account of direct perception by itself does not provide an adequate account of justification. However a direct realist account of perception can inform the needed theory of concept-formation, and leading critics of abstractionism like McDowell and Sellars, direct realists about perception themselves, fail to appreciate the ways in which their own views about perception help fill gaps in earlier accounts of abstractionism. Recognizing this undercuts both their objections to abstractionism and (therefore) their objections to foundationalism. (shrink)

The thesis of this paper is that Thomas Aquinas offers an alternative model of abstraction (the Active Principle Model) that overcomes the standard objections to abstractionism and expands our view of what an abstractionist theory might look like. I contend that this alternative model of abstraction has been invisible in plain sight, in Aquinas’s references to the mind’s abstractive mechanism as an “intellectual light.” Such language is not metaphorical but rather technical, signaling that intellectual abstraction is to be modeled (...) on the activity of physical light as he understood it from theories proposed by Avicenna and Averroes. The Active Principle Model requires us to rethink Aquinas’s account of how we come to know essences—a process that turns out to be much more tentative and incremental than previously thought. (shrink)

This chapter covers number of Abstractionist views of modality. It considers three ways that Abstractionists might account for how possible worlds represent possibilities, rather than in terms of the categorial nature of worlds. First, there is Magical Abstractionism, according to which that question has no informative answer. Second, there is Linguistic Abstractionism, according to which possible worlds represent in the way that languages do. And finally, there is Pictorial Abstractionism, according to which possible worlds represent in the (...) way that pictures do. Pictorial Abstractionists might respond by trying to identify a feature that Concretist worlds have that their worlds don't, or vice versa. One way for something to be possible is for actual things to have the power to make it so. Since the notion of power is central to Aristotelian Modality, we get sub‐varieties of the view depending on one's metaphysical account of powers. (shrink)

Abstractionism is a philosophical account of the ontology of mathematics according to which abstract objects are grounded in a process of abstraction. Abstraction involves arranging a domain of underlying objects into classes and then identifying … Continue reading Abstractionism →.

Are there objects that are “thin” in the sense that their existence does not make a substantial demand on the world? Frege famously thought so. He claimed that the equinumerosity of the knives and the forks suffices for there to be objects such as the number of knives and the number of forks, and for these objects to be identical. The idea of thin objects holds great philosophical promise but has proved hard to explicate. This book attempts to develop the (...) needed explanations by drawing on some Fregean ideas. First, to be an object is to be a possible referent of a singular term. Second, singular reference can be achieved by providing a criterion of identity for the would-be referent. The second idea enables a form of easy reference and thus, via the first idea, also a form of easy being. Paradox is avoided by imposing a predicativity restriction on the criteria of identity. But the abstraction based on a criterion of identity may result in an expanded domain. By iterating such expansions, a powerful account of dynamic abstraction is developed. (shrink)

ionism as a foundational programme in the philosophy of mathematics traditionally originates with Gottlob Frege. According to it, significant portions of mathematics (arithmetic, possibly r...

An artistic discussion on the critical potential of African American expressive culture In a major reassessment of African American culture, Phillip Brian Harper intervenes in the ongoing debate about the “proper” depiction of black people. He advocates for African American aesthetic abstractionism—a representational mode whereby an artwork, rather than striving for realist verisimilitude, vigorously asserts its essentially artificial character. Maintaining that realist representation reaffirms the very social facts that it might have been understood to challenge, Harper contends that (...) class='Hi'>abstractionism shows up the actual constructedness of those facts, thereby subjecting them to critical scrutiny and making them amenable to transformation. Arguing against the need for “positive” representations, Abstractionist Aesthetics displaces realism as the primary mode of African American representational aesthetics, re-centers literature as a principal site of African American cultural politics, and elevates experimental prose within the domain of African American literature. Drawing on examples across a variety of artistic production, including the visual work of Fred Wilson and Kara Walker, the music of Billie Holiday and Cecil Taylor, and the prose and verse writings of Ntozake Shange, Alice Walker, and John Keene, this book poses urgent questions about how racial blackness is made to assume certain social meanings. In the process, African American aesthetics are upended, rendering abstractionism as the most powerful modality for Black representation. (shrink)

A discussion of the philosophical prospects for basing a neo-Fregean theory of classes on a principle that attempts to articulate the limitation-of-size conception.

In Thin Objects: An Abstractionist Account, Linnebo offers what he describes as a “simple and definitive” solution to the bad company problem facing abstractionist accounts of mathematics. “Bad” abstraction principles can be rendered “good” by taking abstraction to have a predicative character. But the resulting predicative axioms are too weak to recover substantial portions of mathematics. Linnebo pursues two quite different strategies to overcome this weakness in the case of set theory and arithmetic. I argue that neither infinitely iterated abstraction (...) nor abstraction on possible specifications fully resolves the bad company problem. (shrink)

In this paper, I present two crucial problems for Wolff’s metaphysics of quantities: 1) The structural identification problem and 2) the Pythagorean problem. The former is the problem of uniquely defining a general algebraic structure for all quantities; the latter is the problem of distinguishing physical quantitative structure from mathematical quantities. While Wolff seems to have a consistent and elegant solution to the first problem, the second problem may put in jeopardy his metaphysical view on quantities as spaces. After drawing (...) a parallelism between Wolff’s treatment of quantitative structures and Frege’s conception of quantitative domain, I propose a solution to the Pythagorean problem based on the idea that mathematical structures are the result of applying an abstraction principle on physical quantitative structures. In particular, I propose the view that abstraction may be seen as the operation of structure determination which transforms concrete physical quantities (i.e. undetermined structures) into abstract mathematical quantities (fully determined structures of thin and shallow objects). Keywords: Metaphysics of Quantities, Abstraction Principles, Locationism. (shrink)

I discuss certain representative accounts of metaphysical emergence falling into three broad categories, assessing their prospects for satisfying certain criteria; the ensuing dialectic has a bit of the Goldilocks fable about it. At one end of the spectrum are what I call ‘scientistic’ accounts, which characterize metaphysical emergence by appeal to one or another specific feature commonly registered in scientific descriptions of seeming cases of emergence; such accounts, I argue, typically fail to provide a clear basis for ensuring incompatibility with (...) ontological reduction, and thus fail to guarantee satisfaction of the criterion of appropriate contrast (§2). At the other end of the spectrum are what I call ‘abstractionist’ accounts, which characterize emergence in terms floating free of scientific notions; such accounts, I’ll argue, typically fail to guarantee satisfaction of the criterion of illuminating contrast (§3). I’ll then look at several accounts of metaphysical emergence that are what I call ‘substantive’, in appealing to familiar relations or posits which are properly metaphysical, while being intelligibly connected to scientific relations and posits. As we’ll see, the resources of such accounts are up to the tasks of blocking ontological reduction and of enabling investigations into metaphysical emergence to proceed in an illuminating way (§4). I close with some methodological morals (§5). (shrink)

We discuss abstraction principles in the context of modal and temporal logic. It is argued that abstractionism conflicts with both serious presentism and serious actualism.

George Berkeley maintains both anti-abstractionism (that abstract ideas are impossible) and idealism (that physical objects and their qualities are mind-dependent). Some scholars (including Atherton, Bolton, and Pappas) have argued, in different ways, that Berkeley uses anti-abstractionism as a premise in a simple argument for idealism. In this paper, I argue that the relation between anti-abstractionism and idealism in Berkeley's metaphysics is more complex than these scholars acknowledge. Berkeley distinguishes between two kinds of abstraction, singling abstraction and generalizing (...) abstraction. He then rests his case for idealism, not on the denial of the possibility of generalizing abstraction, but rather on the denial of the possibility of singling abstraction. Moreover, Berkeley's argument does not rest on a blanket rejection of all forms of singling abstraction. Rather, the fundamental anti-abstractionist assumption, for his purposes, is the claim that primary qualities cannot be mentally singled out from secondary qualities. Crucially, the claim that the existence of physical objects cannot be mentally singled out from their being perceived is not a premise in, but rather a consequence of, Berkeley's argument for idealism. Berkeley's argument therefore avoids circularity inasmuch as it appeals to the impossibility of singly abstracting one idea in order to establish the impossibility of singly abstracting another. (shrink)

Richard Kimberly Heck and Paolo Mancosu have claimed that the possibility of non-Cantorian assignments of cardinalities to infinite concepts shows that Hume's Principle (HP) is not implicit in the concept of cardinal number. Neologicism would therefore be threatened by the ‘good company' HP is kept by such alternative assignments. In his review of Mancosu's book, Bob Hale argues, however, that ‘getting different numerosities for different countable infinite collections depends on taking the groups in a certain order – but it is (...) of the essence of cardinal numbers that the cardinal size of a collection does not depend upon how its members are ordered'. This paper's goal is to implement Hale's response to the Good Company problem by producing a Cantorian argument for HP. In Section 2, we present the Heck-Mancosu argument against neologicism. In Section 3, we discuss Hale's defence of Hume's Principle. In Section 4, we discuss Cantor's abstractionist definitions of number. In Section 5, we argue that good abstraction must comply with what we call ‘Gödel’s Minimal Account of Abstraction’ (GMAA). We finally show (Sections 5 and 6) that non-Cantorian theories of cardinality fail to satisfy GMAA. (shrink)

This chapter outlines Kant’s account of empirical concept formation and discusses two objections that have been advanced against it. Kant holds that we form empirical concepts, such as colour concepts, by comparing sensory representations of individuals, identifying shared features, and abstracting from the differences between them. According to the first objection, we cannot acquire colour concepts in this way because there is no feature that all and only the instances of a given colour share and the boundary between colours is (...) conventional. According to the second objection, assuming that all instances of a given colour share certain features, we can identify them only if we already possess a concept of that colour. None of the two objections is convincing as it stands. Kant can offer replies to both objections that are consistent with his views and with empirical evidence concerning the perception and representation of colours. (shrink)

According to St. Thomas, the natures of material things are the proper objects of human understanding.Thomas claims only that the natures of things are the proper objects of the intellect, not that they are its only objects: he does not deny that we have intellective cognition also of the contingent states and situations of particular material things. And he holds that, at least in this life, humans cognize these natures, not through innate species or by perceiving the divine exemplars, but (...) only by abstraction from phantasms (ST Ia, 84.7, 85.1).This claim applies to the exercising of concepts already acquired, as well as their initial acquisition (ST Ia 84.7). Here and throughout, I use “cognition” to translate “cognitio.” As Scott MacDonald (“Theory of Knowledge” in Cambridge Companion to Aquinas, ed., N. Kretzmann and E. Stump [Cambridge: Cambridge University Press, 1993], p. 162) points out, translating “cognitio” with “knowledge” is misleading, given that cognitiones can, on Thomas’s account, be false (see, for example, ST Ia 17.3). I discuss Aquinas’s conception of cognition in section I.1. More precisely, the human intellect’s active component, the agent intellect, produces cognition of the natures of material things by abstracting intelligible forms from phantasms and informing them on its passive component, the possible intellect, to actualize the latter’s potency to understand.This division of the intellect into an active and a passive component originates in Aristotle’s cryptic remark that in the soul “there is a mind for becoming all things” and “a mind for producing all things” (DA III 5, 430a10). This passage has been subject to myriad interpretations. Aristotle’s Arabic commentators read him as saying that (one or both) of these intellects are single and separate from individual human souls. In opposition to these interpretations, Aquinas holds that the agent and possible intellects are both immanent powers of each individual soul. Since these interpretations enjoyed considerable popularity in Aquinas’s day, we often find him developing his account of the human intellect in explicit opposition to them (cf. OUIAA). The aim of the present piece is to clarify Thomas’s account of this intellective abstraction, and thereby the precise force of the conceptual empiricism it asserts.Important recent work by several scholars has advanced our understanding of many central aspects of Aquinas’s philosophy of mind and epistemology, including his account of intellective abstraction. See Eleonore Stump, “Aquinas on the Foundations of Knowledge,” Canadian Journal of Philosophy (Supplementary Volume) 17 (1992); “Aquinas’s Account of the Mechanisms of Intellective Cognition,” Revenue Internationale de Philosophie 52 (1998): 287–307; “Aquinas on Sensory Cognition,” in Medieval Analyses in Language and Cognition, eds. Sten Ebbesen and Russell L. Friedman, volume 77 in the series Det Kongelige Danske Videnskabernes Selskab, Historisk-Filosofiske Meddelelser (Copenhagen: Munksgaard, 1999), pp. 377–395. Norman Kretzmann, “Infallibility, Error, and Ignorance,” Canadian Journal of Philosophy 17 (1992); Kretzmann, “Philosophy of Mind,” in Cambridge Companion to Aquinas; Robert Pasnau, Theories of Cognition in the Later Middle Ages (Cambridge: Cambridge University Press, 1997); and Scott MacDonald (“Theory of Knowledge,” in Cambridge Companion to Aquinas).Many of these discussions share a salutary focus on Aquinas’s claims regarding the reliability of our faculties, so that their treatments of intellective abstraction tend to center on his prima faciae wildly implausible claim to its infallibility: “The proper object of intellect is the quiddity of a thing, and this is why intellect is not fallible regarding the quiddity of a thing, speaking of it just as such” (ST Ia 85.6). Kretzmann provides an interesting and detailed treatment of this claim. Drawing on a range of texts, he argues that the intellect’s apprehension of the quiddity of a thing is such that one acquires an initial, crude concept of that thing, and that this apprehension must be distinguished from the intellect’s act of compounding, in judgments, the aspects of things so apprehended: it is only through a fallible, and extended process of making such judgments that we arrive at more adequate concepts of things (see the final section of “Infallibility, Error, and Ignorance”). MacDonald also makes insightful, and complementary, observations along similar lines (pp. 183–84). However, though helpful, these points do not suffice to explain, let alone render defensible, Thomas’s claim of infallibility: why can’t some of our acts of intellective abstraction yield false cognitions (however rudimentary), in the sense Thomas specifies?; in other words; why can’t the agent intellect sometimes produce in the possible intellect forms which do not correspond to the forms that actually inhere, or even could inhere, in material things? I think that Stump is right to take this optimism to be one striking instance of Thomas’s general epistemic optimism rooted in his conviction that God designed our cognitive faculties (Stump, “Foundations of Knowledge,” pp. 145–48).The interpretation of his intellective abstraction provided here complements Stump’s insight. For it, in effect, elaborates how Aquinas’s theistic metaphysics grounds his epistemic optimism about the first operation of our intellect. In particular, it explains how the agent intellect, as the human participating likeness in the divine light of understanding, has the active potency to order phantasms in such a way that it can produce in the possible intellect intelligible species which are determinate likenesses of the particular material things of which we have experience (cf. Section III iv, below). Examining his distinction between sensible and intellectual cognition, and his account of the way the former, in phantasms, supplies the data for intellective abstraction, will lead us to reassess the nature of Thomas’s antinativism—arguably the most important historical and philosophical legacy of his cognitive psychology. (shrink)

According to St. Thomas, the natures of material things are the proper objects of human understanding. 1 And he holds that, at least in this life, humans cognize these natures, not through innate species or by perceiving the divine exemplars, but only by abstraction from phantasms (ST Ia, 84.7, 85.1). 2 More precisely, the human intellects potency to understand. 3 The aim of the present piece is to clarify Thomass antinativism—arguably the most important historical and philosophical legacy of his cognitive (...) psychology. (shrink)

From the point of view of proof-theoretic semantics, we examine the logical background invoked by Neil Tennant's abstractionist realist account of mathematical existence. To prepare the way, we must first look closely at the rule of existential elimination familiar from classical and intuitionist logics and at rules governing identity. We then examine how well free logics meet the harmony and uniqueness constraints familiar from the proof-theoretic semantics project. Tennant assigns a special role to atomic formulas containing singular terms. This, we (...) find, secures harmony and uniqueness but militates against the putative realism. (shrink)

Thin Objects: Anionist Account, by LinneboØystein. Oxford: Oxford University Press, 2018. Pp. xviii + 238.

A thriving literature has developed over logical and mathematical pluralism – i.e. the views that several rival logical and mathematical theories can be equally correct. These have unfortunately grown separate; instead, they both could gain a great deal by a closer interaction. Our aim is thus to present some novel forms of abstractionist mathematical pluralism which can be modeled on parallel ways of substantiating logical pluralism (also in connection with logical anti-exceptionalism). To do this, we start by discussing the Good (...) Company Problem for neo-logicists recently raised by Paolo Mancosu (2016), concerning the existence of rival abstractive definitions of cardinal number which are nonetheless equally able to reconstruct Peano Arithmetic. We survey Mancosu’s envisaged possible replies to this predicament, and suggest as a further path the adoption of some form of mathematical pluralism concerning abstraction principles. We then explore three possible ways of substantiating such pluralism—Conceptual Pluralism, Domain Pluralism, Pluralism about Criteria—showing how each of them can be related to analogous proposals in the philosophy of logic. We conclude by considering advantages, concerns, and theoretical ramifications for these varieties of mathematical pluralism. (shrink)

The paper discusses Berkeley's classical critique of Locke's theory of generating concepts by abstraction, rebuts it, and shows that endorses Lockean abstractionism concerning the formation of empirical concepts.

The idea that some objects are metaphysically “cheap” has wide appeal. An influential version of the idea builds on abstractionist views in the philosophy of mathematics, on which numbers and other mathematical objects are abstracted from other phenomena. For example, Hume’s Principle states that two collections have the same number just in case they are equinumerous, in the sense that they can be correlated one-to-one: (HP) #xx=#yy iff xx≈yy. The principal aim of this article is to use the notion of (...) grounding to develop this sort of abstractionism. The appeal to grounding enables a unified response to the two main challenges that confront abstractionism. First, we must explicate the metaphor of meta- physical “cheapness.” Second, we must rebut the “bad company” objection, which rejects abstraction principles like (HP) as tarnished by their similarity to inconsistent principles like Frege’s Basic Law V. By enforcing a simple requirement that all abstraction be properly grounded, we propose a unified solution to these two hard, and prima facie unrelated, problems. On our view, grounded abstraction simultaneously ensures “cheap” abstracta and permissible abstraction. (shrink)

In a narrow sense, the formalistic conception of art cannot range and explain a wide variety of contemporary art phenomena. In a wide sense, formalism – rejecting the reductionist definition of the essence of art, involving the conceptual dimension of the artwork, and expanding the dimension of sensibility – becomes a productive philosophical art theory allowing one to explain and investigate some part of the contemporary art (modern and postmodern) phenomena. In the expanded conception, the basic principles of aesthetic formalism (...) include a wide scale of the principles of the creation and comprehension of the artwork – from an abstract form (perceived by senses) to a conceptual form. The article gives an illustration of these principles discussing abstractionism and surrealism as two different types of the contemporary formalistic visual art. (shrink)

This chapter considers various views about the precise nature of possible worlds, but each view is compatible with this initial characterization. It considers modality, particularly focusing on metaphysical possibility, necessity, and impossibility, that broadest kind of modality. The chapter offers an example of why one might care about this issue, an example of why the study of modality matters to philosophy more generally. It is plausible that modality is importantly connected to understanding. The chapter focuses on two contrasting views about (...) the nature of possible worlds, Concretism and Abstractionism. The fundamental difference between Concretism and Abstractionism is whether possible worlds are concrete or abstract. Concretism is the view that possible worlds are like parallel universes. Concretism requires Modal Indexicalism and denies Modal Anti‐Indexicalism. The chapter describes the ways that the Concretism carries out its reduction of possibility and actuality. (shrink)

This chapter focuses mainly on how possible worlds relate to the truth and falsity of modal claims (or propositions), and therefore to whether claims are necessarily true, necessarily false, possibly true, possibly false, and so on. This issue is that of modality de dicto, modality concerning propositions. But there is another type of modality, namely modality de re. This has to do with the modal status of relations between things and their properties, with whether things possess properties necessarily, contingently or (...) not even possibly. This chapter introduces the distinction between de dicto and de re necessity, and then explores different ways of making sense of de re necessity‐necessities involving particular individuals. It talks about Strong Transworld Identity, and explores how Abstractionists respond to the Leibniz's Law argument. Some metaphysicians go farther and identify what is possible with what is conceivable, or perhaps with what we are justified in believing to be possible. (shrink)

In his "Two concepts of possible worlds", Peter Van Inwagen explores two kinds of views about the nature of possible worlds : abstractionism and concretism. The latter is the view defended by David Lewis who claims that possible worlds are concrete spatio-temporal universes, very much like our own, causally and spatio-temporally disconnected from each other. The former is the view of the majority who claims that possible worlds are some kind of abstract objects – such as propositions, properties, states (...) of affairs, or sets of numbers. In this paper, I will develop this view in an 'extreme abstractionist' way, appealing to a 'modal bundle theory', and I will try to show that it is preferable to the standard abstractionist ones. Finally, I will compare this kind of abstractionism to concretism, only to find that the difference between the two is minimal. (shrink)

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)

Mainstream psychologists have not only ignored the unique and radical character of practice; they have generally misunderstood it. A major reason for this ignorance and misunderstanding is mainstream psychology's assumption of a particular ontology--abstractionism. With abstractionism, psychologists have generally assumed that abstractions, such as theories, techniques, and principles, capture and embody the fundamentally real. Most pertinently, abstractions are believed to precede and lay the foundation for good and thoughtful practice. Indeed, practices do not exist, in an important ontological (...) sense, except in relation to the concrete and particular situations and cultures that give rise to them, implying what we might call a relational ontology. The purpose of this article is to explore the possible meanings and implications of a relational ontology for psychology. I first attempt to clarify the important features of relational conceptions in general, distinguishing most importantly between their weak and strong forms. Then, I situate strong or ontological relationality historically by describing its initial embrace and eventual rejection by early scientists. To bring these historical lessons into the present, ontological relationality is next compared to the abstractionist understandings of the real and fundamental in mainstream psychology. These two forms of ontology are then illustrated by exploring their contrasting implications for individual and community identity. 2012 APA, all rights reserved). (shrink)

In the treatise On the Spiritual in Art, Wassily Kandinsky divided the creation of art into three categories, the ultimate one of which is called Composition. In this article, I argue that Kandinsky’s classification is similar and comparable to the principle of semi-abstract Chinese freehand brushwork summarized by Zheng Xie in the Inscriptions on Painting - Bamboo. In an attempt to clarify the core of Kandinsky’s strategy of abstraction, i.e. the transformation from painting to writing, I then connect it with (...) the evolution of Chinese characters and the aesthetic theory of Chinese calligraphy, which is also in accord with Kandinsky’s idea of internal necessity and the spirit in art. Through this aesthetic dialogue across time and space, this article aims to bring a new dimension to the understanding of Western abstractionism from the perspective of the oriental philosophy of art. (shrink)

"The main aim of the paper is to compare two types of abstractionistic accounts of fictional objects, and to analyze their consequences for interpretation of existential quantification. According to a proponent of general abstractionistic theory, fictional objects have abstract nature in a way similar to contracts, marriages, and the likes. This view is an alternative to strongly realistic accounts of fictional objects, defended by Terence Parsons or David Lewis. Within abstractionistic theories, as in all philosophical areas, one can find divergences (...) of opinions. The main differences between two of them – Peter van Inwagen's and Edward Zalta's – are connected with the interpretation of existential quantification. According to van Inwagen, “being” is the same as “existence” and its sense is captured by the existential quantifier. Edward Zalta's theory is much closer to the Meinongian Theory of Objects. He argues for the need of distinguishing between “being” and “existence” and for invoking nonexistent objects. Because of that he suggests an alternative interpretation of quantification. Admittedly, there is one abstractionistic source for both theories, but their ontological consequences are different - van Inwagen is a staunch opponent of nonexistent objects, and Zalta describes his own theory as "Meinongian".". (shrink)

Abstractionism in the philosophy of mathematics aims at deriving large fragments of mathematics by combining abstraction principles (i.e. the abstract objects $\S e_1, \S e_2$, are identical if, and only if, an equivalence relation $Eq_\S$ holds between the entities $e_1, e_2$) with logic. Still, as highlighted in work on the semantics for relevant logics, there are different ways theories might be combined. In exactly what ways must logic and abstraction be combined in order to get interesting mathematics? In this (...) paper, we investigate the matter by deriving the axioms of second-order Peano Arithmetic from Frege's Basic Law V (the extension of $F$ is identical with the extension of $G$ if, and only if, $F$ and $G$ are extensionally equivalent) in the presence of a relevant higher-order logic. The results are interesting. Not only must we take on logic as true, and not only must we apply our logic to abstraction principles, but also we have to apply our theory of abstraction back to the logic in order to arrive at arithmetic. Thus, what Abstractionism gives us is not simply what we get from abstraction via logic, but also what we get from logic via abstraction. (shrink)

This is a technical supplement to "Abstraction and Grounding", forthcoming in /Philosophy and Public Affairs/.

APA PsycNET abstract: This is the first volume of a two-volume work on Probability and Induction. Because the writer holds that probability logic is identical with inductive logic, this work is devoted to philosophical problems concerning the nature of probability and inductive reasoning. The author rejects a statistical frequency basis for probability in favor of a logical relation between two statements or propositions. Probability "is the degree of confirmation of a hypothesis (or conclusion) on the basis of some given evidence (...) (or premises)." Furthermore, all principles and theorems of inductive logic are analytic, and the entire system is to be constructed by means of symbolic logic and semantic methods. This means that the author confines himself to the formalistic procedures of word and symbol systems. The resulting sentence or language structures are presumed to separate off logic from all subjectivist or psychological elements. Despite the abstractionism, the claim is made that if an inductive probability system of logic can be constructed it will have its practical application in mathematical statistics, and in various sciences. 16-page bibliography. (PsycINFO Database Record (c) 2016 APA, all rights reserved). (shrink)