In the section “Validity and Existence in Logik, Book III,” I explain Lotze’s famous distinction between existence and validity in Book III of Logik. In the following section, “Lotze’s Platonism,” I put this famous distinction in the context of Lotze’s attempt to distinguish his own position from hypostatic Platonism and consider one way of drawing the distinction: the hypostatic Platonist accepts that there are propositions, whereas Lotze rejects this. In the section “Two Perspectives on Frege’s Platonism,” I argue that this (...) is an unsatisfactory way of reading Lotze’s Platonism and that the Ricketts-Reck reading of Frege is in fact the correct way of thinking about Lotze’s Platonism. (shrink)

Both Frege's Grundgesetze, and Lagrange's treatises on analytical functions pursue a foundational purpose. Still, the former's program is not only crucially different from the latter's. It also depends on a different idea of what foundation of mathematics should be like . Despite this contrast, the notion of function plays similar roles in their respective programs. The purpose of my paper is emphasising this similarity. In doing it, I hope to contribute to a better understanding of Frege's logicism, especially in relation (...) to its crucial differences with a set-theoretic foundational perspective. This should also spread some light on a question arisen by J. Hintikka and G. Sandu in a widely discussed paper, namely whether Frege should or should not be credited with the notion of arbitrary function underlying our standard interpretation of second-order logic. In section 1, I account for Lagrange's notion of function. In section 2, I advance some remarks on connected historical matters. This will provide an appropriate framework for discussing the role played by the notion of function in Frege's Grundgesetze. Section 3 is devoted to this. Some concluding remarks will close the paper. (shrink)

This book proposes a novel theory of truth and falsity. It argues that truth is a form of reference and falsity is a form of reference failure. -/- Most of the philosophical literature on truth concentrates on certain ontological and epistemic problems. This book focuses instead on language. By utilizing the Fregean idea that sentences are singular referring expressions, the author develops novel connections between the philosophical study of truth and falsity and the huge literature in in the philosophy of (...) language on the notion of reference. The first part of the book constructs the author’s theory and argues for it in length. Part II addresses the ways in which the theory relates to, and is different from, some of the basic theories of truth. Part III takes up how to account for the truth of sentences with logical operators and quantifiers. Finally, Part IV discusses the applications and implications of the theory for longstanding problems in philosophy of language, metaphysics, and epistemology. -/- A Referential Theory of Truth and Falsity will appeal to researchers and advanced students working in philosophy of language, epistemology, metaphysics, and linguistics. -/- . (shrink)

Frege is celebrated as an arch-Platonist and arch-realist. He is renowned for claiming that truths of arithmetic are eternally true and independent of us, our judgments and our thoughts; that there is a third realm containing nonphysical objects that are not ideas. Until recently, there were few attempts to explicate these renowned claims, for most philosophers thought the clarity of Frege's prose rendered explication unnecessary. But the last ten years have seen the publication of several revisionist interpretations of Frege's writings (...) — interpretations on which these claims receive a very different reading. In Frege on Knowing the Third Realm, Tyler Burge attempts to undermine this trend. Burge argues that Frege is the very Platonist most have thought him — that revisionist interpretations of Frege's Platonism, mine among them, run afoul of the words on Frege's pages. This paper is a response to Burge's criticisms. I argue that my interpretation is more faithful than Burge's to Frege's texts. (shrink)

I investigate why Frege rejected the theory of types, as Russell presented it to him in their correspondence. Frege claims that it commits one to violations of the law of excluded middle, but this complaint seems to rest on a dogmatic refusal to take Russell’s proposal seriously on its own terms. What is at stake is not so much the truth of a law of logic, but the structure of the hierarchy of the logical categories, something Frege seems to neglect. (...) To come to a better understanding of Frege’s response, I proceed to investigate his conception of the nature of the logical categories, and how it differs from Russell’s. I argue that, for Frege, our grasp of the logical categories cannot be severed from our grasp of the Begriffsschrift notation itself. Russell, on the other hand, attaches no such importance to notation. From Frege’s point of view, Russell has not succeeded in presenting an alternative conception of the logical hierarchy, since such a conception must go in tandem with the development of a notation. Moreover, Frege has good reasons to think that Russell’s proposal does not admit of a suitable notation. (shrink)

In the opening to his late essay, Der Gedanke, Frege asserts without qualification that the word "true" points the way for logic. But in a short piece from his Nachlass entitled "My Basic Logical Insights", Frege writes that the word true makes an unsuccessful attempt to point to the essence of logic, asserting instead that "what really pertains to logic lies not in the word "true" but in the assertoric force with which the sentence is uttered". Properly understanding what Frege (...) takes to be at issue here is crucial for understanding his conception of logic and, in particular, what he takes to be its normative status vis-à-vis judgement, assertion, and inference. In this paper, I focus my attention on clarifying the latter claim and Frege's motivations for making it, exposing what I take to be a fundamental tension in Frege's conception of logic. Finally, I discuss whether Frege's deployment of the horizontal in his mature Begriffsschrift helps to resolve this tension. (shrink)

This paper develops some respects in which the philosophy of mathematics can fruitfully be informed by mathematical practice, through examining Frege's Grundlagen in its historical setting. The first sections of the paper are devoted to elaborating some aspects of nineteenth century mathematics which informed Frege's early work. (These events are of considerable philosophical significance even apart from the connection with Frege.) In the middle sections, some minor themes of Grundlagen are developed: the relationship Frege envisions between arithmetic and geometry and (...) the way in which the study of reasoning is to illuminate this. In the final section, it is argued that the sorts of issues Frege attempted to address concerning the character of mathematical reasoning are still in need of a satisfying answer. (shrink)

One of the most striking differences between Frege's Begriffsschrift (logical system) and standard contemporary systems of logic is the inclusion in the former of the judgement stroke: a symbol which marks those propositions which are being asserted , that is, which are being used to express judgements . There has been considerable controversy regarding both the exact purpose of the judgement stroke, and whether a system of logic should include such a symbol. This paper explains the intended role of the (...) judgement stroke in a way that renders it readily comprehensible why Frege insisted that this symbol was an essential part of his logical system. The key point here is that Frege viewed logic as the study of inference relations amongst acts of judgement , rather than – as in the typical contemporary view – of consequence relations amongst certain objects (propositions or well-formed formulae). The paper also explains why Frege's use of the judgement stroke is not in conflict with his avowed anti-psychologism, and why Wittgenstein's criticism of the judgement stroke as 'logically quite meaningless' is unfounded. The key point here is that while the judgement stroke has no content , its use in logic and mathematics is subject to a very stringent norm of assertion. (shrink)

The central premise of Michael Dummett's global argument for anti-realism is the thesis that a speaker's grasp of the meaning of a declarative, indexical-free sentence must be manifested in her uses of that sentence. This enigmatic thesis has been the subject of a great deal of discussion, and something of a consensus has emerged about its content and justification. The received view is that the manifestation thesis expresses a behaviorist and reductive theory of meaning, essentially in agreement with Quine's view (...) of language, and motivated by worries about the epistemology of communication. In the present paper I begin by arguing that this standard interpretation of the manifestation thesis is neither particularly faithful to Dummett's writings nor philosophically compelling. I then continue by reconstructing, from Dummett's texts, an account of the manifestation thesis, and of its justification, that differ sharply from the received view. On my reading, the thesis is motivated not epistemologically, but conceptually. I argue that connections among our conceptions of meaning, assertion, and justification lead to a conclusion about the metaphysics of meaning: we cannot form a clearly coherent conception of how two speakers can attach different meanings to a sentence without at the same time differing in what they count as justifying assertions made with that sentence. I conclude with some suggestions about how Dummett's argument for global anti-realism should be understood, given my account of the manifestation thesis. (shrink)

This article treats three aspects of Frege's discussions of definitions. First, I survey Frege's main criticisms of definitions in mathematics. Second, I consider Frege's apparent change of mind on the legitimacy of contextual definitions and its significance for recent neo-Fregean logicism. In the remainder of the article I discuss a critical question about the definitions on which Frege's proofs of the laws of arithmetic depend: do the logical structures of the definientia reflect the understanding of arithmetical terms prevailing prior to (...) Frege's analyses? Unless they do, it is unclear how Frege's proofs demonstrate the analyticity of the arithmetic in use before logicism. Yet, especially in late writings, Frege characterizes definitions as arbitrary stipulations of the senses or references of expressions unrelated to pre-definitional understanding. I conclude by examining some options for conceiving of the status of Frege's logicism in light of this apparent tension, and outline a suggestion for a philosophically fruitful way of resolving this tension. (shrink)

In this paper, I address some puzzles about Frege’s conception of how we “grasp” thoughts. I focus on an enigmatic passage that appears near the end of Frege’s great essay “The Thought.” In this passage Frege refers to a “non-sensible something” without which “everyone would remain shut up in his inner world.” I consider and criticize Wolfgang Malzkorn’s interpretation of the passage. According to Malzkorn, Frege’s view is that ideas [Vorstellungen] are the means by which we grasp thoughts. My counter-proposal (...) is that language enables us to grasp thoughts (ideas are merely their baggage or “trappings,” as Frege puts it). One significant consequence of my interpretation is that it helps challenge the standard reading of Frege according to which he is a metaphysical platonist about thoughts. My interpretation thus provides support for the deflationary, anti-ontological reading spelled out by readers like Thomas Ricketts and Wolfgang Carl. As Ricketts puts it, Frege’s distinction between the objective and the subjective, rather than being an ontological doctrine, “lodges in the contrast between asserting something and giving vent to a feeling.”. (shrink)

Gottlob Frege is often called a "platonist". In connection with his philosophy we can talk about platonism concerning three kinds of entities: numbers, or logical objects more generally; concepts, or functions more generally; thoughts, or senses more generally. I will only be concerned about the first of these three kinds here, in particular about the natural numbers. I will also focus mostly on Frege's corresponding remarks in The Foundations of Arithmetic (1884), supplemented by a few asides on Basic Laws of (...) Arithmetic (1893/1903) and "Thoughts" (1918). My goal is to clarify in which sense the Frege of Foundations and Basic Laws is a platonist concerning the natural numbers.1.. (shrink)

The paper critically scrutinizes the widespread idea that Russell subscribes to a "Universalist Conception of Logic." Various glosses on this somewhat under-explained slogan are considered, and their fit with Russell's texts and logical practice examined. The results of this investigation are, for the most part, unfavorable to the Universalist interpretation.

This paper challenges a standard interpretation according to which Frege’s conception of logic (early and late) is at odds with the contemporary one, because on the latter’s view logic is formal, while on Frege’s view it is not, given that logic’s subject matter is reality’s most general features. I argue that Frege – in Begriffsschrift – retained the idea that logic is formal; Frege sees logic as providing the ‘logical cement’ that ties up together the contentful concepts of specific sciences, (...) not the most general truths. Finally, I discuss how Frege conceives of the application of Begriffsschrift, and of its status as a ‘lingua characteristica’. (shrink)

This two-part paper reviews a scholarly debate on an alleged tension in Frege ’s philosophy of logic. In Section 1 of Part I, I discuss Frege ’s view that logic is concerned with establishing norms for correct thinking and is therefore a normative science. In Section 2, I explore a different understanding of the role of logic that Frege seems to advance: logic is constitutive of the very possibility of thought, because it sets forth necessary conditions for thought. Hence, the (...) tension the view according to which logic is normative for thought seems to be incompatible with the idea that abiding by the laws of logic forms a precondition for thought. In Section 1 of Part II, I survey a number of interpretations of Frege ’s conception of logic that deal with this question. I show that they are for the most part either normative readings or constitutive readings. Finally, in Section 2, I adjudicate the debate and aim at reconciling the normative and the constitutive strands in Frege ’s conception of logic. (shrink)

This two-part paper reviews a scholarly debate on an alleged tension in Frege's philosophy of logic. In Section 1 of Part I, I discuss Frege's view that logic is concerned with establishing norms for correct thinking and is therefore a normative science. In Section 2, I explore a different understanding of the role of logic that Frege seems to advance: logic is constitutive of the very possibility of thought, because it sets forth necessary conditions for thought. Hence, the tension the (...) view according to which logic is normative for thought seems to be incompatible with the idea that abiding by the laws of logic forms a precondition for thought. In Section 1 of Part II, I survey a number of interpretations of Frege's conception of logic that deal with this question. I show that they are for the most part either normative readings (emphasising the former understanding of the nature of logic) or constitutive readings (emphasising the latter). Finally, in Section 2, I adjudicate the debate and aim at reconciling the normative and the constitutive strands in Frege's conception of logic. (shrink)

According to the species of neo-logicism advanced by Hale and Wright, mathematical knowledge is essentially logical knowledge. Their view is found to be best understood as a set of related though independent theses: (1) neo-fregeanism-a general conception of the relation between language and reality; (2) the method of abstraction-a particular method for introducing concepts into language; (3) the scope of logic-second-order logic is logic. The criticisms of Boolos, Dummett, Field and Quine (amongst others) of these theses are explicated and assessed. (...) The issues discussed include reductionism, rejectionism, the Julius Caesar problem, the Bad Company objections, and the charge that second-order logic is set theory in disguise. (shrink)

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

Frege's concern in GGI §10 is neither with the epistemological issue of how we come to know about value-ranges, nor with the semantic-metaphysical issue of whether we have said enough about such objects in order to ensure that any kind of reference to them is possible. The problem which occupies Frege in GGI §10 is the general problem according to which we ‘cannot yet decide’, for any arbitrary function, what value ‘’ has if ‘ℵ’ is a canonical value-range name. This (...) is a problem with the ‘reference’ of value-range names, but only in the weak sense that, if we do not exercise care, value-range terms might become ‘bedeutungslos’ for purely formal reasons. Frege addresses the general problem only for the primitive function- and object-names he has already introduced into his concept-script. I argue that this methodology was perfectly intentional: his intention for GG in general, on display in GGI §10, is to check, for each primitive function- and object-name, as it is introduced into concept-script, whether it interacts with the other primitive names which have already been introduced in such a way that these atomic combinations of primitive names do not become bedeutungslos. If there is a risk of producing a bedeutungslos combination, Frege will make an arbitrary stipulation to ensure that logical hygiene is maintained. I argue that this interpretation does not violate some of the other principal commitments of GG. (shrink)

This paper addresses an issue with the sign ‘⊢’ in Frege’s mature version of Begriffsschrift, i.e., the version in ‘Function and Concept’ and Grundgesetze. The sign is a performative for asserting in that writing down ‘⊢p’ is equivalent to asserting that p. Frege further says that writing ‘ p’ is also equivalent to identifying the reference of ‘p’ with the truth-value True. It looks as if he holds that asserting that p consists in identifying the True with the reference of (...) ‘p’. Frege’s commitment to it, however, seems to encounter a number of tensions. This paper aims to show that these tensions can be avoided by endorsing a non-assertive conception of identification under which making an identification is not making an identity assertion. The suggested reading leads to an entirely di erent understanding of the compositionality of the sign ‘⊢’ as well as Frege’s conception of assertion and judgment. (shrink)

It is argued that the finitist interpretation of wittgenstein fails to take seriously his claim that philosophy is a descriptive activity. Wittgenstein's concentration on relatively simple mathematical examples is not to be explained in terms of finitism, But rather in terms of the fact that with them the central philosophical task of a clear 'ubersicht' of its subject matter is more tractable than with more complex mathematics. Other aspects of wittgenstein's philosophy of mathematics are touched on: his view that mathematical (...) propositions are 'grammatical propositions' (and so, Strictly speaking, Not genuine propositions at all) and his view that in mathematics 'everything is algorithm, Nothing meaning'. His views on consistency and his anti-Foundationalism are linked with this central thesis. (shrink)

Frege's strict alignment between his syntactic and ontological categories is not, as is commonly assumed, some kind of a philosophical thesis. There is no thesis that proper names refer only to objects, say, or that what refers to an object is a proper name. Rather, the alignment of categories is internal to Frege's conception of what syntax and ontology are. To understand this, we need to recognise the pride of place Frege assigns within his theorising to the notion of truth. (...) For both language and the world, the Fregean categories are logical categories, categories, that is, of truth. The elaboration of this point makes clear the incoherence of supposing that they might not align. (shrink)

Commentators regularly attribute to Frege realist, idealist, and quietist responses to metaphysical questions concerning the abstract objects he calls ‘thoughts’. But despite decades of effort, the evidence offered on behalf of these attributions remains unconvincing. I argue that Frege deliberately avoids commitment to any of these positions, as part of a metaphysical separatist policy motivated by the fact that logic is epistemologically autonomous from metaphysics. Frege’s views and arguments prove relevant to current attempts to argue for epistemological autonomy, particularly that (...) of ethics. (shrink)

In a famous passage (A68/B93), Kant writes that “the understanding can make no other use of […] concepts than that of judging by means of them.” Kant's thought is often called the thesis of the priority of judgments over concepts. We find a similar sounding priority thesis in Frege: “it is one of the most important differences between my mode of interpretation and the Boolean mode […] that I do not proceed from concepts, but from judgments.” Many interpreters have thought (...) that Frege's priority principle is close to (or at least derivable from) Kant's. I argue that it is not. Nevertheless, there was a gradual historical development that began with Kant's priority thesis and culminated in Frege's new logic. (shrink)

In his 'Meaning and Truth-Conditions', Gary Kemp offers a reconstruction of Frege's infamous 'regress argument' which purports to rely only upon the premises that the meaning of a sentence is its truth-condition and that each sentence expresses a unique proposition. If cogent, the argument would show that only someone who accepts a form of semantic holism can use the notion of truth to explain that of meaning. I respond that Kemp relies heavily upon what he himself styles 'a literal, rather (...) wooden' understanding of truth-conditions. I explore alternatives, and say a few words about how Frege's regress argument might best be understood. (shrink)

An investigation of Frege’s various contributions to the study of language, focusing on three of his most famous doctrines: that concepts are unsaturated, that sentences refer to truth-values, and that sense must be distinguished from reference.

The aim of this paper is to give a detailed reconstruction of Frege's solution to his puzzle about the cognitive function of truth, which is this: On the one hand, the concept of truth seems to play an essential role in acquiring knowledge because the transition from the mere hypothetical assumption that p to the acknowledgement of its truth is a crucial step in acquiring the knowledge that p, while, on the other hand, this concept seems to be completely redundant (...) because the sense of the word 'true' does not make any essential contribution to the senses of the sentences in which it occurs. (shrink)

The aim of this article is to contribute to a better understanding of Frege’s views on semantics and metatheory by looking at his take on several themes in nineteenth century geometry that were significant for the development of modern model-theoretic semantics. I will focus on three issues in which a central semantic idea, the idea of reinterpreting non-logical terms, gradually came to play a substantial role: the introduction of elements at infinity in projective geometry; the study of transfer principles, especially (...) the principle of duality; and the use of counterexamples in independence arguments. Based on a discussion of these issues and how nineteenth century geometers reflected about them, I will then look into Frege’s take on these matters. I conclude with a discussion of Frege’s views and what they entail for the debate about his stance towards semantics and metatheory more generally. (shrink)

Las interpretaciones tradicionales de Frege tienden a explicar su antipsicologismo como indicando que ninguna consideración subjetivista debe interferir en la elaboración de un lenguaje lógico; dichas interpretaciones, sin embargo, tienen problemas para explicar el énfasis dado por el mismo Frege a conceptos como el de fuerza asertiva. Este artículo pretende solucionar esta extrañeza investigando, a partir de su concepto de ciencia, algunas de las ideas epistemológicas de Frege. Esto nos llevará a concluir que una concepción de la verdad como fenómeno (...) esencialmente público puede resolver este y otros problemas interpretativos. (shrink)

While the topic of assertion has recently received a fresh wave of interest from Peirce scholars, to this point no systematic account of Peirce’s view of assertion has been attempted. We think that this is a lacuna that ought to be filled. Doing so will help make better sense of Peirce’s pragmatism; further, what is hidden amongst various fragments is a robust pragmatist theory of assertion with unique characteristics that may have significant contemporary value. Here we aim to uncover this (...) theory, and to show that assertion for Peirce is not a mere corollary of pragmatic conceptions of truth, judgement, and belief, but is rather a central aspect of his philosophy. (shrink)

In this essay, I distinguish two ways of depsychologizing psychology: ‘anti-psychologism’ and ‘non-psychologism’. Both positions are responses to the Fregean sharp distinction between the logical and the psychological. But where anti-psychologism, which I find in John McDowell, attempts to overcome the sharp distinction by arguing that psychological states and their expressions are apt to be articulated into judgments, Stanley Cavell's non-psychologism, a powerful and neglected alternative, wants to overcome the sharp distinction by abandoning judgment as the paradigm expression of thought (...) and communication. (shrink)

External obstacles to properly understanding Wittgenstein’s philosophy of mathematics are not lacking, either. For one thing, there is the piecemeal way that his mathematical manuscripts have been made available. The editors of Remarks on the Foundations of Mathematics write that “the time has not yet come to print the whole of Wittgenstein’s MSS on these... topics”. One wonders what sorts of reasons there could be for that editorial choice.

This paper, written in Romanian, compares fictionalism, nominalism, and neo-Meinongianism as responses to the problem of objectivity in mathematics, and then motivates a fictionalist view of objectivity as invariance.

A general survey of Frege's views on truth, the paper explores the problems in response to which Frege's distinctive view that sentences refer to truth-values develops. It also discusses his view that truth-values are objects and the so-called regress argument for the indefinability of truth. Finally, we consider, very briefly, the question whether Frege was a deflationist.

This paper argues that Frege is not the metaphysical platonist about mathematics that he is standardly taken to be. It is shown that Frege’s project has two distinct stages: the identification of what is true of our ordinary notions, and then the provision of a systematic account that shares the identified features. Neither of these stages involves much metaphysics. The paper criticizes in detail Dummett’s interpretation of §§55-61 of Grundlagen. These sections fall under the heading ‘Every number is a self-subsistent (...) object’ and are described by Dummett as containing the worst arguments put forward by Frege. It is argued that essentially all of Dummett’s interpretive points are mistaken. Finally, I show that Frege’s claims about the independence of mathematics from humans and their activities does not commit him to any particularly metaphysical position either. (shrink)