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)

Prioritized bases, i.e., weakly ordered sets of sentences, have been used for specifying an agent’s ‘basic’ or ‘explicit’ beliefs, or alternatively for compactly encoding an agent’s belief state without the claim that the elements of a base are in any sense basic. This paper focuses on the second interpretation and shows how a shifting of priorities in prioritized bases can be used for a simple, constructive and intuitive way of representing a large variety of methods for the change of belief (...) states—methods that have usually been characterized semantically by a system-of-spheres modeling. Among the methods represented are ‘radical’, ‘conservative’ and ‘moderate’ revision, ‘revision by comparison’ in its raising and lowering variants, as well as various constructions for belief expansion and contraction. Importantly, none of these methods makes any use of numbers. (shrink)

En este artículo delineamos una propuesta para elaborar una lógica de las ficciones desde el enfoque lúdico del pragmatismo dialógico. En efecto, centrados en una de las críticas mayores al enfoque clásico de la lógica: la esquizofrenia estructural de su semántica, recorremos los compromisos ontológicos de las dos tradiciones mayores de la lógica para establecer sus posibilidades y límites en el análisis del discurso ficcional, y la superación desde una perspectiva lúdico pragmática.

L’insuffisance du logicisme pour la question des fondements des mathématiques nous invite à la poser sous un point de vue épistémologique, en termes de rationalité et non plus seulement de logique, et à l’étendre aux sciences formalisées portant sur la nature, comme la physique, voire à la pensée scientifique en général. On doit tenir compte, dans tous les cas, des changements qui correspondent à des constructions conceptuelles : ce sont eux qui assurent en même temps, après coup, le bien-fondé des (...) théories qui les ont préparés, en sorte que si des fondements rationnels peuvent être obtenus, ce n’est, en règle générale, que « vers l’avant ». Ces changements posent également la question de leurs conditions de possibilités. La conclusion est qu’il faut admettre, pour que ces changements, et avec eux un accroissement de la connaissance, soient possibles, des transformations corrélatives dans les formes mêmes de la rationalité, mathématique, physique et, d’une manière générale, de la rationalité scientifique. (shrink)

This book has two objectives: to be a contribution to the understanding of Frege's theory of truth – especially a defence of his notorious critique of the correspondence theory - and to be an introduction to the practice of interpreting philosophical texts.

The standard argument for the existence of distinctively mathematical objects like numbers has two main premises: some mathematical claims are true, and the truth of those claims requires the existence of distinctively mathematical objects. Most nominalists deny. Those who deny typically reject Quine’s criterion of ontological commitment. I target a different assumption in a standard type of semantic argument for. Benacerraf’s semantic argument, for example, relies on the claim that two sentences, one about numbers and the other about cities, have (...) the same grammatical form. He makes this claim on the grounds that the two sentences are superficially similar. I argue that these grounds are not sufficient. Other sentences with the same superficial form appear to have different grammatical forms. I offer two plausible interpretations of Benacerraf’s number sentence that make use of plural quantification. These interpretations appear not to incur ontological commitments to distinctively mathematical objects, even assuming Quine’s criterion. Such interpretations open a new, plural strategy for the mathematical nominalist. (shrink)

How does your information change when you learn that something might be the case, where the modal “might” is epistemic? On the orthodox view, a proposition is added to your information base; on the view defended here, no propositions are added to your information base but some are removed from it. I argue that Stephen Yablo’s recent attempt to define this removal operation as a kind of propositional subtraction fails, offer a definition of my own in terms of the part–whole (...) relations between the truthmakers of the propositions one accepts, and argue that a deontic analogue of this account solves a problem about permission posed long ago by David Lewis. (shrink)

The article addresses the works of three 19th-century philosophers: Herbart, Bolzano and Lotze. Despite their differences, i will analyze two essential features they share: their refusal to psychologism as the radicalization of psychology as the ultimate source of philosophy , and the positing of an ideal ‘reality’, independent from both sensibility and the mental and linguistic dimensions, upon which the refusal to psychologism is founded . As a conclusion, i will show how, according to these authors, this platonic, ideal background (...) constitutes the very foundation of every meaningful thought. (shrink)

Epistemic theories of objective chance hold that chances are idealised epistemic probabilities of some sort. After giving a brief history of this approach to objective chance, I argue for a particular version of this view, that the chance of an event E is its epistemic probability, given maximal knowledge of the possible causes of E. The main argument for this view is the demonstration that it entails all of the commonly-accepted properties of chance. For example, this analysis entails that chances (...) supervene on the physical facts, that the chance function is an expert probability, that the existence of stable frequencies results from invariant chances in repeated experiments, and that chances are probably close to long-run relative frequencies in repeated experiments. Despite these virtues, epistemic approaches to chance have been neglected in recent decades, on account of their conflict with accepted views about closely related topics such as causation, laws of nature, and epistemic probability. However, existing views on these topics are also very problematic, and I believe that the epistemic view of chance is a key piece of this puzzle that, once in place, allows all the other pieces to fit together into a new and coherent way. I also respond to some criticisms of the epistemic theory that I favour. (shrink)

In this paper six of the most important issues in the philosophy of logic are examined from a standpoint that rejects the First Commandment of empiricist analytic philosophy, namely, Ockham’s razor. Such a standpoint opens the door to the clarification of such fundamental issues and to possible new solutions to each of them.

This paper explores how words relate to concepts. It argues that in many cases, words get their meanings in part by associating with concepts, but only in conjunction with substantial input from language. Language packages concepts in grammatically determined ways. This structures the meanings of words, and determines which sorts of concepts map to words. The results are linguistically modulated meanings, and the extralinguistic concepts associated with words are often not what intuitively would be expected. The paper concludes by discussing (...) implications of this thesis for the relation of word to sentence meaning, and for issues of linguistic determinism. (shrink)

Deflationists about truth embrace the positive thesis that the notion of truth is useful as a logical device, for such purposes as blanket endorsement, and the negative thesis that the notion doesn’t have any legitimate applications beyond its logical uses, so it cannot play a significant theoretical role in scientific inquiry or causal explanation. Focusing on Christopher Hill as exemplary deflationist, the present paper takes issue with the negative thesis, arguing that, without making use of the notion of truth conditions, (...) we have little hope for a scientific understanding of human speech, thought, and action. For the reference relation, the situation is different. Inscrutability arguments give reason to think that a more-than-deflationary theory of reference is unattainable. With respect to reference, deflationism is the only game in town. (shrink)

_ Source: _Page Count 21 The “Context Principle”, as the author understands it in this paper, is the claim that sub-sentential linguistic expressions have meaning only in the context of complete meaningful sentences. The author reconstructs the version of the Context Principle that Wittgenstein holds in the _Tractatus_, shows that it has intuitive plausibility and can be defended against alleged counterexamples, and develops an argument for its truth that can be found in the _Tractatus_. In short, the author argues that (...) the “fit” between the meanings of sentences and the meanings of the expressions they contain can only be adequately explained if one acknowledges that for a sub-sentential expression to have meaning _is_ for it to make a contribution to the meaning of a sentence. (shrink)

This paper proposes an Interface Transparency Thesis concerning how linguistic meanings are related to the cognitive systems that are used to evaluate sentences for truth/falsity: a declarative sentence S is semantically associated with a canonical procedure for determining whether S is true; while this procedure need not be used as a verification strategy, competent speakers are biased towards strategies that directly reflect canonical specifications of truth conditions. Evidence in favor of this hypothesis comes from a psycholinguistic experiment examining adult judgments (...) concerning ‘Most of the dots are blue’. This sentence is true if and only if the number of blue dots exceeds the number of nonblue dots. But this leaves unsettled, e.g., how the second cardinality is specified for purposes of understanding and/or verification: via the nonblue things, given a restriction to the dots, as in ‘|{x: Dot(x) & ~Blue(x)}|’; via the blue things, given the same restriction, and subtraction from the number of dots, as in ‘|{x: Dot(x)}| – |{x: Dot(x) & Blue(x)}|’; or in some other way. Psycholinguistic evidence and psychophysical modeling support the second hypothesis. (shrink)

This is the first part of the essay devoted to the story of logicism, in particular to its Fregean version. Reviewing the classical period of Fregean studies, we first point out some critical moments of Frege‘s argumentation in the Grundla­gen, in order to be able later to differentiate between its salvageable and defec­tive features. We work on the presumption that there are no easy, catego­rical an­swers to questions like “Is logicism dead?“: Wittgenstein’s cri­tique of the foundational program as well as (...) the remarkable neo-Fregean dis­cov­eries of Boolos and Wright have to be confronted with the effects which the logi­cistic idea actually had on logico­matemati­cal practice. But that is another story, a sequel to this es­say, the purpose of which is systematic rather than criti­cal. (shrink)

This is the first part of the essay devoted to the story of logicism, in particular to its Fregean version. Reviewing the classical period of Fregean studies, we first point out some critical moments of Frege‘s argumentation in the Grundla­gen, in order to be able later to differentiate between its salvageable and defec­tive features. We work on the presumption that there are no easy, catego­rical an­swers to questions like “Is logicism dead?“: Wittgenstein’s cri­tique of the foundational program as well as (...) the remarkable neo-Fregean dis­cov­eries of Boolos and Wright have to be confronted with the effects which the logi­cistic idea actually had on logico­matemati­cal practice. But that is another story, a sequel to this es­say, the purpose of which is systematic rather than criti­cal. (shrink)

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

Since the appearance of Dummett’s monumental monograph in the early seventies of the last century there have been two interpretative traditions of reading Frege, one contradicting the other: the platonistic (or realistic) interpretation and the holistic (rationalistic) one. In this article, as a kind of intermediation, I firstly point out some pragmatic aspects of Frege’s work. Secondly, and more ambitiously, I outline the controversy between the most influential theories of truth (the correspondence, coherence, redundancy and pragmatic theory) in the light (...) of Frege’s philosophy. After all it was the concept of truth that Frege himself identified as the main objective of logic. (shrink)

Byeong-Uk Yi has argued that number words like ‘two’ primarily function as numerical predicates as in ‘Socrates and Hippias are two ’, and other grammatical uses of number words can be paraphrased in terms of the predicative use. This paper critically examines Yi’s paraphrase scheme and also some other alternative schemes, and argues that the adjectival use of number words as in ‘The Scots and the Irish are two peoples’ cannot be paraphrased in terms of the predicative use.

Husserl’s transcendental reduction admits of two motivations: the general methodological ban on begging the question, and the principle that a typology of objects ought to be based on a typology of my ways of cognizing them. As Husserl’s ‘transcendental phenomenology’ agrees with the ‘linguistic phenomenology’ of many analytic philosophers in being at bottom an effort to understand what precisely we mean to say by asserting that there ‘exists’ a ‘consciousness-independent’ or ‘transcendent’ world, the ‘residue’ of transcendental reduction is my subjective (...) consciousness. My cognitive approaches to the latter and to that of somebody else are not only entirely different but ‘complementary’ in the sense of Bohr’s. In the course of searching for an identity criterion for temporal entities lacking a spatial localisation in the world, we can show that the allegedly noetic phenomenon of, say, my now seeing a cat is a noema, to wit, the cat-as-now-beingseen-by-me-in-such-and-such-a-manner. There is no sound base for postulating, over and above the noematic nature of my consciousness, a second, noetic, aspect thereof. (shrink)

The paper critically examines an unpopular line of Frege’s view on numbers in the Foundations of Arithmetic. According to this view, which analyzes numbers in terms of properties and not in terms of extensions, numbers are properties of concepts vs. properties of objects. The latter view is held by Mill and is famously criticized in the Foundations. I argue that on the property account numbers cannot only be properties of concepts but they also have to be properties of objects. My (...) main argument rests on purely metaphysical grounds. It stems from the motivation that were numbers only properties of concepts we would not have been able to explain mathematical truths about the physical world or those truths would have been miraculous. On pains that we do have mathematical truths about the physical world that are not miraculous we cannot agree with Frege’s property line about the metaphysical nature of numbers. (shrink)

In his logical foundation of arithmetic, Frege faced the problem that the semantic interpretation of his system does not determine the reference of the abstract terms completely. The contextual definition of number, for instance, does not decide whether the number 5 is identical to Julius Caesar. In a late writing, Quine claimed that the indeterminacy of reference established by Frege’s Caesar problem is a special case of the indeterminacy established by his proxy-function argument. The present paper aims to show that (...) Frege’s Caesar problem does not really support the conclusions that Quine draws from the proxy-function argument. On the contrary, it reveals that Quine’s argument is a non sequitur: it does not establish that there are alternative interpretations of our terms that are equally correct, but only that these terms are ambiguous. The latter kind of referential indeterminacy implies that almost all sentences of our overall theory of the world are either false or neither true nor false, because they contain definite descriptions whose uniqueness presupposition is not fulfilled. The proxy-function argument must therefore be regarded as a reductio ad absurdum of Quine’s behaviorist premise that the reference of terms is determined only by our linguistic behavior. (shrink)

The article attempts to reconsider the relationship between Leibniz’s and Kant’s philosophy of geometry on the one hand and the nineteenth century debate on the foundation of geometry on the other. The author argues that the examples used by Leibniz and Kant to explain the peculiarity of the geometrical way of thinking are actually special cases of what the Jewish-German mathematician Felix Hausdorff called “transformation principle”, the very same principle that thinkers such as Helmholtz or Poincaré applied in a more (...) general form in their celebrated philosophical writings about geometry. The first two parts of the article try to show that Leibniz’s and Kant’s philosophies of geometry, despite their differences, appear to be preoccupied with the common problem of the impossibility to grasp conceptually the intuitive difference between two figures (such as a figure and its scaled, displaced or mirrored copy). In the third part, it is argued that from the perspective of Hausdorff’s philosophical-geometrical reflections, this very same problem seems to find a more radical application in Helmholtz’s or Poincaré’s thought experiments on the impossibility of distinguishing distorted copies of our universe from the original one. I draw the conclusion that in Hausdorff’s philosophical work, which has received scholarly attention only recently, one can find not only an original attempt to frame these classical arguments from a set-theoretical point of view, but also the possibility of considering the history of philosophy of geometry from an uncommon perspective, where especially the significance of Kant’s infamous appeal to “intuition” can be judged by more appropriate standards. (shrink)

Há muitos pontos de concordância entre Frege e os neokantianos. Isso vale especialmente para os representantes do neokantismo da teoria do valor ou do Sudoeste alemão na tradição de Hermann Lotze. Não discutiremos aqui todos os aspectos dessa proximidade; de acordo com o tema que propomos, ficaremos restritos à filosofia da matemática. A primeira parte do artigo tratará da relação entre aritmética e geometria, mostrando surpreendentes semelhanças entre Frege e o neokantiano Otto Liebmann. A segunda parte discutirá as diferentes recepções (...) do logicismo de Frege por Jonas Cohn, Paul Natorp, Heinrich Rickert, Ernst Cassirer e Bruno Bauch. (shrink)

The use of the three labels to denote the three foundational schools of the early twentieth century are now part of literature. Yet, neither their number nor the...

The paper has two parts: 1. A brief exposition of proof-theoretic semantics, not necessarily in connection to natural language. 2. A review, with a contrastive flavour, of some of the applications of PTS to NL with an indication of advantages of PTS as a theory of meaning for NL.

This unique anthology of new, contributed essays offers a range of perspectives on various aspects of ontic vagueness. It seeks to answer core questions pertaining to onticism, the view that vagueness exists in the world itself. The questions to be addressed include whether vague objects must have vague identity, and whether ontic vagueness has a distinctive logic, one that is not shared by semantic or epistemic vagueness. The essays in this volume explain the motivations behind onticism, such as the plausibility (...) of mereological vagueness and indeterminacy in quantum mechanics and they offer various arguments both for and against ontic vagueness; onticism is also compared with other, competing theories of vagueness such as semanticism, the view that vagueness exists only in our linguistic representation of the world. Gareth Evans’s influential paper of 1978, “Can There Be Vague Objects?” gave a simple but cogent argument against the coherence of ontic vagueness. Onticism was subsequently dismissed by many. However, in recent years, researchers have become aware of the logical gaps in Evans’s argument and this has triggered a new wave of interest in onticism. Onticism is now widely regarded as at least a coherent view. Reflecting this growing consensus, the present anthology for the first time puts together essays that are focused on onticism and its various facets and it fills in the lacuna in the literature on vagueness, a much-discussed subject in contemporary philosophy. (shrink)

Neo-Fregeanism is a combination of two ideas: logicism, according to which arithmetic can be derived from logic plus definitions, and Platonism, according to which there are mathematical objects. Neo-Fregeans propose a new interpretation of Frege’s principles of abstraction and of the role of reconceptualization and implicit definition for the introduction of numbers into our ontology. I analyze the ontological implications of neo-Fregeanism, not only for mathematics, but for abstract entities in general. After briefly introducing some of the main elements of (...) neo-Fregeanism, I present two possible readings of its ontological implications and I argue that none of them gives the desired results. (shrink)

We discuss the concept of a cardinal number and its history, focussing on Cantor's work and its reception. J'ay fait icy peu pres comme Euclide, qui ne pouvant pas bien >faire< entendre absolument ce que c'est que raison prise dans le sens des Geometres, definit bien ce que c'est que memes raisons. (Leibniz) 1.

Can identity itself be vague? Can there be vague objects? Does a positive answer to either question entail a positive answer to the other? In this paper we answer these questions as follows: No, No, and Yes. First, we discuss Evans’s famous 1978 argument and argue that the main lesson that it imparts is that identity itself cannot be vague. We defend the argument from objections and endorse this conclusion. We acknowledge, however, that the argument does not by itself establish (...) either that there cannot be vague objects or that there cannot be identity statements that are indeterminate for ontic reasons. And we further acknowledge that it does not by itself establish that there cannot be identity statements that are indeterminate in virtue of the existence of vague objects. We then go on to argue that, despite this, one who believes in vague objects cannot endorse Evans’s argument. To establish this we offer supplementary arguments that show that if vague objects exist then identity is vague, and that if identity is vague then vague objects exist. Finally we draw attention to an argument parallel to that of Evans’s, but safer, which can be employed against the putative ontic indeterminacy in identity of vague objects which can be differentiated by identity-free properties. (shrink)

Two topics figure prominently in recent discussions of mathematical structuralism: challenges to the purported metaphysical insight provided by sui generis structuralism and the significance of category theory for understanding and articulating mathematical structuralism. This article presents an overview of central themes related to these topics.

Megethology is the second-order theory of the part-whole relation developed by David Lewis, and it is obtained by combining plural quantification with classical extensional mereology. It can express some hypotheses about the size of the domain such as that there are inaccessibly many atoms. This will prove enough to get the orthodox set theory. Then, megethology is a possible foundation for mathematics. This paper is an introduction to megethology.

P.T. Geach has maintained (see, e.g., Geach (1967/1968)) that identity (as well as dissimilarity) is always relative to a general term. According to him, the notion of absolute identity has to be abandoned and replaced by a multiplicity of relative identity relations for which Leibniz's Law - which says that if two objects are identical they have the same properties - does not hold. For Geach relative identity is at least as good as Frege's cardinality thesis which he takes to (...) be strictly connected with relative identity - according to which an ascription of cardinality is always relative to a concept which specifies what, in any particular case, counts as a unit. The idea that there is a close connection between relative identity and Frege's cardinality thesis has been issued again quite recently by Alston and Bennett in (1984). In their opinion, Frege's cardinality thesis is not only similar to relative identity - as Geach maintains - but it implies it. Moreover, they agree with Geach in claiming that a commitment to Frege's cardinality thesis forces a parallel commitment to relative identity. Against Geach, Alston and Bennett we will claim that (Tl): "Frege's cardinality thesis is similar to relative identity" is false and that therefore (T2) "Frege's cardinality thesis implies relative identity" is false as well. (shrink)

In his last book about Locke’s philosophy, E. J. Lowe claims that Frege’s arguments against the Lockean conception of number are not compelling, while at the same time he painstakingly defines the Lockean conception Lowe himself espouses. The aim of this paper is to show that the textual evidence considered by Lowe may be interpreted in another direction. This alternative reading of Frege’s arguments throws light on Frege’s and Lowe’s different agendas. Moreover, in this paper, the problem of singular sentences (...) of number is presented, and Frege’s and Lowe’s views are confronted with it. (shrink)

Theory and Reality is about the connection between true theories and the world. A mathematical framefork for such connections is given, and it is shown how that framework can be used to infer facts about the structure of reality from facts about the structure of true theories, The book starts with an overview of various approaches to metaphysics. Beginning with Quine's programmatic "On what there is", the first chapter then discusses the perils involved in going from language to metaphysics. It (...) criticises contemporary intuition-driven metaphysics, comments on naturalistic approaches, and then presents the main proposition put forward in the thesis: we should base metaphysics on model theory. In chapters 2 to 5, mathematical treatments are given of concepts that we need: theories, metaphysics, necessitation and semantics. These are used in chapters 6 and 7 to prove that, seen from a certain informative view point, any true theory will give rise to an isomorphism between that theory and the world. This conclusion is similar to Wittgenstein's in the Tractatus, but differs in that it places the structural relationship on the level of whole theories, rather than single propositions. (shrink)