This article improves two existing theorems of interest to neologicist philosophers of mathematics. The first is a classification theorem due to Fine for equivalence relations between concepts definable in a well-behaved second-order logic. The improved theorem states that if an equivalence relation E is defined without nonlogical vocabulary, then the bicardinal slice of any equivalence class—those equinumerous elements of the equivalence class with equinumerous complements—can have one of only three profiles. The improvements to Fine’s theorem allow for an analysis of (...) the well-behaved models had by an abstraction principle, and this in turn leads to an improvement of Walsh and Ebels-Duggan’s relative categoricity theorem. (shrink)

In this paper, I present a class of ‘structural’ abstraction principles, and describe how they are suggested by some features of Cantor's and Dedekind's approach to abstraction. Structural abstraction is a promising source of mathematically tractable new axioms for the neo-logicist. I illustrate this by showing, first, how a theorem of Shelah gives a sufficient condition for consistency in the structural setting, solving what neo-logicists call the ‘bad company’ problem for structural abstraction. Second, I show (...) how, in the structural setting, we can measure the logical strength of abstraction principles using categories of interpretations between theories. (shrink)

Neo-Fregean logicism attempts to base mathematics on abstraction principles. Since not all abstraction principles are acceptable, the neo-Fregeans need an account of which ones are. One of the most promising accounts is in terms of the notion of stability; roughly, that an abstraction principle is acceptable just in case it is satisfiable in all domains of sufficiently large cardinality. We present two counterexamples to stability as a sufficient condition for acceptability and argue that these counterexamples (...) can be avoided only by major departures from the existing neo-Fregean programme. (shrink)

Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory. Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles. In earlier work, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this (...) paper, we show that most other abstraction principles are not naturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. To better understand the precise demands of relative categoricity in the context of abstraction principles, we compare and contrast these constraints to stability-like acceptability criteria on abstraction principles, the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli and Fine, and supervaluational ideas coming out of Hodes' work. (shrink)

The logical status of abstraction principles, and especially Hume’s Principle, has been long debated, but the best currently availeble tool for explicating a notion’s logical character—permutation invariance—has not received a lot of attention in this debate. This paper aims to fill this gap. After characterizing abstraction principles as particular mappings from the subsets of a domain into that domain and exploring some of their properties, the paper introduces several distinct notions of permutation invariance for such (...) class='Hi'>principles, assessing the philosophical significance of each. (shrink)

Frege's views regarding analysis and synomymy have long been the subject of critical discussion. Some commentators, led by Dummett, have argued that Frege was committed to the view that each thought admits of a unique ultimate analysis. However, this interpretation is in apparent conflict with Frege's criterion of synonymy, according to which two sentence express the same thought if one cannot understand them without regarding them as having the same truth–value. In a recent article in this journal, Drai attempts to (...) reconcile Frege's criterion of synonymy with unique ultimate analysis by holding that, for Frege, if two sentences satisfy the criterion without being intensionally isomorphic, at most one of them is a privileged representation of the thought expressed. I argue that this proposal fails, because it conflicts not only with Frege's views of abstraction principles but also with slingshot arguments (including one presented by Drai herself) that accurately reflect Frege's commitment to the view that sentences alike in truth–value have the same Bedeutung. While Drai helpfully connects Frege's views of abstraction principles with such slingshot arguments, this connection cannot become fully clear until we recognise that Frege rejects unique ultimate analysis. (shrink)

I extend theorems due to Roy Cook on third- and higher-order versions of abstraction principles and discuss the philosophical importance of results of this type. Cook demonstrated that the satisfiability of certain higher-order analogues of Hume's Principle is independent of ZFC. I show that similar analogues of Boolos's new v and Cook's own ordinal abstraction principle soap are not satisfiable at all. I argue, however, that these results do not tell significantly against the second-order versions of these (...) principles. (shrink)

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)

Introduction to a special issue on abstraction principles.

This brief note aims to reply to a comment made by Menno Rol abou...

In the history of philosophy, philosophers have defined philosophy based on their own perspectives. This situation shows that it is a difficult and meaningless endeavor to put forward an understanding of philosophy that has clear boundaries and is accepted by everyone. However, despite these differences in definitions, there are some common points. Perhaps the most important of these common points is that philosophy is a stance, a positioning activity in the face of truth. In this positioning, William James tried to (...) find the moderate path between the currents of rationalism and empiricism and took a stance based on concrete differences in life. Like the empiricists, he tried to keep people's relations with facts alive, but at the same time, he did not exclude religion by adopting a rationalist attitude. However, James argued that abstract concepts cannot have meaning unless they find their counterpart in life. According to James, a life based only on abstract principles would not be qualified. In this study, I will try reconciling William James’ attitude towards abstract principles with practical life. (shrink)

Since Carol Gilligan's analysis of the “Heinz dilemma,” many philosophers working on care have articulated critiques of abstraction and principles in ethics. Their objections to abstraction and principles have not always been systematically set out. In this paper, I try to clarify the debate. I begin by distinguishing several aspects of the care critique. I then consider the strengths of each from a Kantian perspective. I conclude that, although some of these objections point out potential misuses (...) of abstraction and principle, and in doing so, suggest strategies and cautions for their correct and careful use in ethics, they do not present a successful challenge to abstraction or principles as such. (shrink)

A standard understanding of abstraction principles elicits two opposite readings: Intolerant Reductionism, where abstractions are seen as reducing talk of abstract objects to talk about non-problematic domains, and Robustionism, where newly introduced terms genuinely refer to abstract objects. Against this dichotomy between such “austere” and “robust” readings, Dummett suggested ways to steer intermediate paths. We explore different options for intermediate stances, by reviewing metaontological strategies and semantic ones. Based on Dummett’s and Picardi’s understanding of the Context Principle, the (...) paper acknowledges that Frege’s Grundlagen are open to several interpretations of contextual definitions, and argues that the prospects for broadly Fregean intermediate views are all but foreclosed. (shrink)

Recent literature in experimental philosophy has postulated the existence of the abstract/concrete paradox : the tendency to activate inconsistent intuitions depending on whether a problem to be analyzed is framed in abstract terms or is described as a concrete case. One recent study supports the thesis that this effect influences judicial decision-making, including decision-making by professional judges, in areas such as interpretation of constitutional principles and application of clear-cut rules. Here, following the existing literature in legal theory, we argue (...) that the susceptibility to such an effect might depend on whether decision-makers operate in a legal system characterized by the formalist or particularist approach to legal interpretation, with formalist systems being less susceptible to the effect. To test this hypothesis, we compare the results of experimental studies on ACP run on samples from two countries differing in legal culture: Poland and Brazil. The lack of significant differences between those results suggests that ACP is a robust effect in the legal context. (shrink)

The neo-Fregean program in the philosophy of mathematics seeks a foundation for a substantial part of mathematics in abstraction principles—for example, Hume’s Principle: The number of Fs D the number of Gs iff the Fs and Gs correspond one-one—which can be regarded as implicitly definitional of fundamental mathematical concepts—for example, cardinal number. This paper considers what kind of abstraction principle might serve as the basis for a neo- Fregean set theory. Following a brief review of the main (...) difficulties confronting the most widely discussed proposal to date—replacing Frege’s inconsistent Basic Law V by Boolos’s New V which restricts concepts whose extensions obey the principle of extensionality to those which are small in the sense of being smaller than the universe—the paper canvasses an alternative way of implementing the limitation of size idea and explores the kind of restrictions which would be required for it to avoid collapse. (shrink)

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)

Neo-Fregeans have been troubled by the Nuisance Principle, an abstraction principle that is consistent but not jointly satisfiable with the favored abstraction principle HP. We show that logically this situation persists if one looks at joint consistency rather than satisfiability: under a modest assumption about infinite concepts, NP is also inconsistent with HP.

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.

Which abstraction principles are acceptable? A variety of criteria have been proposed, in particular irenicity, stability, conservativeness, and unboundedness. This note charts their logical relations. This answers some open questions and corrects some old answers.

Paolo Mancosu provides an original investigation of historical and systematic aspects of the notions of abstraction and infinity and their interaction. A familiar way of introducing concepts in mathematics rests on so-called definitions by abstraction. An example of this is Hume's Principle, which introduces the concept of number by stating that two concepts have the same number if and only if the objects falling under each one of them can be put in one-one correspondence. This principle is at (...) the core of neo-logicism. In the first two chapters of the book, Mancosu provides a historical analysis of the mathematical uses and foundational discussion of definitions by abstraction up to Frege, Peano, and Russell. Chapter one shows that abstraction principles were quite widespread in the mathematical practice that preceded Frege's discussion of them and the second chapter provides the first contextual analysis of Frege's discussion of abstraction principles in section 64 of the Grundlagen. In the second part of the book, Mancosu discusses a novel approach to measuring the size of infinite sets known as the theory of numerosities and shows how this new development leads to deep mathematical, historical, and philosophical problems. The final chapter of the book explore how this theory of numerosities can be exploited to provide surprisingly novel perspectives on neo-logicism. (shrink)

Human participants and recurrent (“connectionist”) neural networks were both trained on a categorization system abstractly similar to natural language systems involving irregular (“strong”) classes and a default class. Both the humans and the networks exhibited staged learning and a generalization pattern reminiscent of the Elsewhere Condition (Kiparsky, 1973). Previous connectionist accounts of related phenomena have often been vague about the nature of the networks’ encoding systems. We analyzed our network using dynamical systems theory, revealing topological and geometric properties that can (...) be directly compared with the mechanisms of non-connectionist, rule-based accounts. The results reveal that the networks “contain” structures related to mechanisms posited by rule-based models, partly vindicating the insights of these models. On the other hand, they support the one mechanism (OM), as opposed to the more than one mechanism (MOM), view of symbolic abstraction by showing how the appearance of MOM behavior can arise emergently from one underlying set of principles. The key new contribution of this study is to show that dynamical systems theory can allow us to explicitly characterize the relationship between the two perspectives in implemented models. (shrink)

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)

We develop a functional abstraction principle for the type-free algorithmic logic introduced in our earlier work. Our approach is based on the standard combinators but is supplemented by the novel use of evaluation trees. Then we show that the abstraction principle leads to a Curry fixed point, a statement C that asserts C ⇒ A where A is any given statement. When A is false, such a C yields a paradoxical situation. As discussed in our earlier work, this (...) situation leaves one no choice but to restrict the use of a certain class of implicational rules including modus ponens. (shrink)

It is widely thought that the acceptability of an abstraction principle is a feature of the cardinalities at which it is satisfiable. This view is called into question by a recent observation by Richard Heck. We show that a fix proposed by Heck fails but we analyze the interesting idea on which it is based, namely that an acceptable abstraction has to “generate” the objects that it requires. We also correct and complete the classification of proposed criteria for (...) acceptable abstraction. (shrink)

Frege's theorem says that second-order Peano arithmetic is interpretable in Hume's Principle and full impredicative comprehension. Hume's Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law V from Frege's Grundgesetze. In this paper we study the strength of abstraction principles in the presence of predicative restrictions on the comprehension schema, and in particular we study a predicative Fregean theory which contains all the abstraction principles whose underlying equivalence relations can (...) be proven to be equivalence relations in a weak background second-order logic. We show that this predicative Fregean theory interprets second-order Peano arithmetic. (shrink)

One of the important questions in the interpretation of Rawls’s philosophy is the connection between the two problems he wrote about throughout his life – justice and legitimacy. In this paper, I take the difference principle as a special feature of Rawls’s theory of justice, while I take the idea of the public reason as a special aspect of his theory of legitimacy, and I try to show that both aspects are connected, that is, that we should not see them (...) as two separate projects. I am trying to show that we should formulate the justification of the difference principle as a justification of a particular socio-economic arrangement rather than as an abstract principle, one that is better than other legitimate arrangements. For Rawls, property-owning democracy and liberal socialism are socio-economic arrangements that realize the difference principle. The main difference between the two systems is that in the first one, private ownership of the means of production is allowed, but it is widely dispersed. In this paper, I try to show how such a system can be publicly justified. The justification that we need to use is precisely the one that Rawls formulates through the idea of the public mind, by referring to the political values contained in our public political culture. (shrink)

Abstraction principles and grounding can be combined in a natural way Modality: metaphysics, logic, and epistemology, Oxford University Press, Oxford, pp 109–136, 2010; Schwartzkopff in Grazer philosophische studien 82:353–373, 2011). However, some ground-theoretic abstraction principles entail that there are circles of partial ground :775–801, 2017). I call this problem auto-abstraction. In this paper I sketch a solution. Sections 1 and 2 are introductory. In Sect. 3 I start comparing different solutions to the problem. In Sect. (...) 4 I contend that the thesis that the right-hand side of an abstraction principle is prior to its left-hand side motivates an independence constraint, and that this constraint leads to predicative restrictions on the acceptable instances of ground-theoretic abstraction principles. In Sect. 5 I argue that auto-abstraction is acceptable unless the left-hand side is essentially grounded by the right-hand side. In Sect. 6 I highlight several parallelisms between auto-abstraction and the puzzles of ground. I finally compare my solution with the strategies listed in Sect. 3. (shrink)

This paper analyses the abstractionist account of quantity championed by Leibniz, especially in the 1680s. Leibniz introduced the notion of quantity in an indirect way, via an abstraction principle. In the first part of the paper, I identify the context in which this approach arose in light of Leibniz's criticism of his earlier dream of an ‘alphabet of human thought’. Recognising the impossibility of such a project led him to realise that, when dealing with terms referring to abstract objects, (...) we should always consider them within the true sentences in which they occur. In the second part, I describe this approach in detail. This allows us to look at some key concepts of Leibniz's theory of quantity. In particular, I raise the problem of the relationship between the two sides of the abstraction principle: how should we think of the relation between the claim that a and b are equal, and the claim that the quantity of a is identical to the quantity of b? I argue that we can find a positive answer to this problem in Leibniz. (shrink)

The injective version of Cantor’s theorem appears in full second-order logic as the inconsistency of the abstraction principle, Frege’s Basic Law V (BLV), an inconsistency easily shown using Russell’s paradox. This incompatibility is akin to others—most notably that of a (Dedekind) infinite universe with the Nuisance Principle (NP) discussed by neo-Fregean philosophers of mathematics. This paper uses the Burali–Forti paradox to demonstrate this incompatibility, and another closely related, without appeal to principles related to the axiom of choice—a result (...) hitherto unestablished. It discusses both the general interest of this result, its interest to neo-Fregean philosophy of mathematics, and the potential significance of the Burali–Fortian method of proof. (shrink)

This paper presents a formalization of first-order arithmetic characterizing the natural numbers as abstracta of the equinumerosity relation. The formalization turns on the interaction of a nonstandard cardinality quantifier with an abstraction operator assigning objects to predicates. The project draws its philosophical motivation from a nonreductionist conception of logicism, a deflationary view of abstraction, and an approach to formal arithmetic that emphasizes the cardinal properties of the natural numbers over the structural ones.

Higher courts sometimes assess the constitutionality of law by working through a concrete case, other times by reasoning about the underlying question in a more abstract way. Prior research has found that the degree of concreteness or abstraction with which an issue is formulated can influence people's prescriptive views: For instance, people often endorse punishment for concrete misdeeds that they would oppose if the circumstances were described abstractly. We sought to understand whether the so-called ‘abstract/concrete paradox’ also jeopardizes the (...) consistency of judicial reasoning. In a series of experiments, both lay and professional judges sometimes reached opposite conclusions when reasoning about concrete cases versus the underlying issues formulated in abstract terms. This effect emerged whether participants reasoned with broad principles, such as human dignity, or narrow rules, and was largest among individuals high in trait empathy. Finally, to understand whether people reflectively endorse the discrepancy between abstract and concrete resolutions, we examined their reactions when evaluating both, either simultaneously or sequentially. These approaches revealed no single pattern across lay and expert populations, or exploratory and confirmatory studies. Taken together, our studies suggest that empathic concern plays a greater role in guiding the judicial resolution of concrete cases than in illuminating judges' professed standards—which may result in concrete decisions in violation of their own abstract principles. (shrink)

Wright claims that “the epistemology of good abstraction principles should be assimilated to that of basic principles of logical inference”. In this paper I follow Wright’s recommendation, but I consider a different epistemology of logic, namely anti-exceptionalism. Anti-exceptionalism’s main contention is that logic is not a priori, and that the choice between rival logics should be based on abductive criteria such as simplicity, adequacy to the data, strength, fruitfulness, and consistency. This paper’s goal is to lay down (...) the foundations for an application of the anti-exceptionalist methodology to abstraction principles. In Sect. 1 I outline my strategy. I show that anti-exceptionalism has a bearing on the so-called Bad Company problem, and explore some versions of anti-exceptionalism about abstraction. I then consider new criteria for choosing between rival abstraction principles, and, in particular, between consistent revisions of Frege’s Basic Law V. I finally consider how the abstractionist can compare between competing criteria, and argue that anti-exceptionalists should be pluralists about abstraction. (shrink)

A famous passage in Section 64 of Frege’s Grundlagen may be seen as a justification for the truth of abstraction principles. The justification is grounded in the procedureofcontent recarvingwhich Frege describes in the passage. In this paper I argue that Frege’sprocedure of content recarving while possibly correct in the case of first-order equivalencerelations is insufficient to grant the truth of second-order abstractions. Moreover, I propose apossible way of justifying second-order abstractions by referring to the operation of contentrecarving and (...) I show that the proposal relies to a certain extent on the Basic Law V. Therefore,if we are to justify the truth of second-order abstractions by invoking the content recarvingprocedure we are committed to a special status of some instances of the Basic Law V and thusto a special status of extensions of concepts as abstract objects. (shrink)

In this paper I explore a position on which it is possible to eliminate the need for postulating abstract objects through abstraction principles by treating terms for abstracta as ‘incomplete symbols’, using Russell's no-classes theory as a template from which to generalize. I defend views of this stripe against objections, most notably Richard Heck's charge that syntactic forms of nominalism cannot correctly deal with non-first-orderizable quantifcation over apparent abstracta. I further discuss how number theory may be developed in (...) a system treating apparent terms for numbers using these definitions. (shrink)

The existence of mereological sums can be derived from an abstraction principle in a way analogous to numbers. I draw lessons for the thesis that “composition is innocent” from neo-Fregeanism in the philosophy of mathematics.

Philosophers often have tried to either reduce "disagreeable" objects or concepts to (more) acceptable objects or concepts. Reduction is regarded attractive by those who subscribe to an ideal of ontological parsimony. But the topic is not just restricted to traditional metaphysics or ontology. In the philosophy of mathematics, abstraction principles, such as Hume's principle, have been suggested to support a reconstruction of mathematics by logical means only. In the philosophy of language and the philosophy of science, the logical (...) analysis of language has long been regarded to be the dominating paradigm, and liberalized projects of logical reconstruction remain to be driving forces of modern philosophy. (shrink)

Linnebo and Pettigrew have recently developed a version of non-eliminative mathematical structuralism based on Fregean abstraction principles. They recognize that this version of structuralism is vulnerable to the well-known problem of non-rigid structures. This paper offers a solution to the problem for this version of structuralism. The solution involves expanding the languages used to describe mathematical structures. We then argue that this solution is philosophically acceptable to those who endorse mathematical structuralism based on Fregean abstraction principles.

The paper takes thePrincipal Principle to be a norm demanding that subjective degrees of belief of a Bayesian agent be equal to the objective probabilities once the agent has conditionalized his subjective degrees of beliefs on the values of the objective probabilities, where the objective probabilities can be not only chances but any other quantities determined objectively. Weak and strong consistency of the Abstract Principal Principle are defined in terms of classical probability measure spaces. It is proved that the Abstract (...) Principal Principle is weakly consistent and that it is strongly consistent in the category of probability measure spaces where the Boolean algebra representing the objective random events is finite. It is argued that it is desirable to strengthen the Abstract Principal Principle by adding a stability requirement to it. Weak and strong consistency of the resulting Stable Abstract Principal Principle are defined, and the strong consistency of the Abstract Principal Principle is interpreted as necessary for a non-omniscient Bayesian agent to be able to have rational degrees of belief in all epistemic situations. It is shown that the Stable Abstract Principal Principle is weakly consistent, but the strong consistency of the Stable Abstract Principal principle remains an open question. We conclude that we do not yet have proof that Bayesian agents can have rational degrees of belief in every epistemic situation. (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)

Knowledge aims to fit the world, and action to change it. In this collection of essays, Onora O'Neill explores the relationship between these concepts and shows that principles are not enough for ethical thought or action: we also need to understand how practical judgement identifies ways of enacting them and of changing the way things are. Both ethical and technical judgement are supported, she contends, by bringing to bear multiple considerations, ranging from ethical principles to real-world constraints, and (...) while we will never find practical algorithms - let alone ethical algorithms - that resolve moral and political issues, good practical judgement can bring abstract principles to bear in situations that call for action. Her essays thus challenge claims that all inquiry must use either the empirical methods of scientific inquiry or the interpretive methods of the humanities. They will appeal to a range of readers in moral and political philosophy. (shrink)