Gottlob Frege

Philosophy undergraduate course completion work published in 2015. This work examines Frege's concept of logical law and its relationship to other normative and descriptive approaches in the history of philosophy, as well as epistemological conceptions of the a priori aspects of mathematical knowledge.

Hybrid expressivists claim to solve the Frege-Geach problem by offloading the explanation of the logico-semantic properties of moral sentences onto beliefs that are components of hybrid states they express. We argue that this strategy is undermined by one of hybrid expressivism’s own commitments: that the truth of the belief-component is neither necessary nor sufficient for the truth of the hybrid state it composes. We articulate a new approach. Instead of explaining head-on what it is for, say, a pair of moral (...) sentences to be inconsistent, expressivists should “sidestep” and explain what it is _to think that_ a pair of moral sentences is inconsistent. To think so is to think they cannot both be true – a modal notion. Since expressivists have given accounts of such modals, we illustrate how sentences like ‘“lying is wrong” and “lying is not wrong” are inconsistent’ express sensible – and rationally compelling – states of mind. (shrink)

Tento článek se zabývá dvěma aritmetickými systémy - konkrétně systémem, který představil R. Dedekind a systémem, který vytvořil G. Frege - a paradoxy, které se zde vyskytují - tedy Burali-Fortiho paradoxem (což je vůbec první fomrulace moderního paradoxu), Cantorovým paradoxem a Russellovým paradoxem. Hlavním cílem je ukázat, co mají tyto paradoxy společného a zdůvodnit, že ačkoli se tyto paradoxy vyskytují v různých systémech, mají společné znaky. Na základě studia uvedených systémů, paradoxů i různých řešení těchto paradoxů, autorka dospívá k tvrzení, (...) že zkoumané paradoxy vznikají na stejném základě, a to na základě problému nehierarchizovanosti.V dodatku tohoto článku navíc čtenář nalezne dva zajímavé historické exkurzy. První z nich předkládá v historicky autentické podobě odvození Burali-Fortiho paradoxu Druhý exkurz je věnován Russellovu paradoxu a představuje jej nejen v podobě, jak jej vyjádřil B. Russell, ale také ve formulaci G. Frega. (shrink)

It is in heavy dispute in the literature whether Frege’s conception of logic is universalism and excludes the possibility of metalogical investigations. In the book, the author argues that there is indeed tension between universalism and non-universalism in it, which finds expression in Grundgesetze. It dismisses the notion of reinterpretation playing a crucial role in the post-Tarskian metalogical investigations as illegitimate; for a language in which a theory, whose subject matter is thoughts rather than formulae per se, is formulated must (...) not be reinterpreted in that fully interpreted formulae express different thoughts once reinterpreted. However, it does not preclude such metalogical investigations that do not make use of the notion: they are not only legitimate, understandable to it but also indispensable to Frege’s logicist project. (shrink)

This book is about Gottlob Frege. The guiding thought is that Frege left philosophy a legacy which has been largely ignored, not least of all by his admirers. In order of logical priority, Frege's first concern was to locate the law-like behaviour of truths and falsehoods merely by virtue of their being such. The just-mentioned legacy lies in his first step towards that goal. It consists in winnowing the 'logical' from the 'psychological', the business of being true as such from (...) that of holding, or holding forth as true-and to keep these separate. A first lesson: what belongs to what is thus abstracted cannot be read directly back into what it was abstracted from. This is what is most widely ignored. The book is divided in three parts. The first presents Frege's general picture of the business of being true-of what belongs to the abstraction. The second is primarily concerned with steps Frege takes between 1891 and 1895, to pave the way for what became, after logic itself, his central project, that whose attempted carrying out is contained in Grundgesetze I. The third part concerns views of logic, truth, the inexorableness of logic, which Frege eventually came to hold, and what it might be to study 'The Mind' as opposed to minds. (shrink)

Robert Hanna's twelfth book, The Fate of Analysis, is a comprehensive revisionist study of Analytic philosophy from the early 1880s to the present, with special attention paid to Wittgenstein's work and the parallels and overlaps between the Analytic and Phenomenological traditions. By means of a synoptic overview of European and Anglo-American philosophy since the 1880s-including accessible, clear, and critical descriptions of the works and influence of, among others, Gottlob Frege, G.E. Moore, Bertrand Russell, Alexius Meinong, Franz Brentano, Edmund Husserl, The (...) Vienna Circle, W.V.O. Quine, Saul Kripke, Wilfrid Sellars, John McDowell, and Robert Brandom, and, particularly, Ludwig Wittgenstein-The Fate of Analysis critically examines and evaluates modern philosophy over the last 140 years. In addition to its critical analyses of the Analytic tradition and of professional academic philosophy more generally, The Fate of Analysis also presents a thought-provoking, forward-looking, and positive picture of the philosophy of the future from a radical Kantian point of view. (shrink)

Blackburn has outlined a formal account for moral expressivism, and we argued that the moral Frege-Geach problem can be solved formally by appending two rules for the boo-operator which are missing from his account. We then extended Blackburn’s formal account to generate a similar solution to the problem in modal context and showed that the validity of the modal argument can be preserved too in modal expressivism. However, the higher-order element endorsed by Blackburn does not seem necessary for solving the (...) Frege-Geach problem. Nor is his extension from moral expressivism to modal expressivism tenable, since the latter violates its own ontological constraint. A general moral is drawn on the basis on three observations made in evaluating Blackburn’s expressivism. (shrink)

Wittgenstein seeks to throw light on our concept of understanding by looking at how misunderstandings arise and what kinds of failure they involve. He discerns a peculiar sort of misunderstanding in the writings of the social anthropologist James Frazer. In Frazer’s hands, the anthropological project of enabling us to understand human behavior seems to yield the result that there are certain forms of human behavior that simply cannot be understood. The source of Frazer’s misunderstanding, according to Wittgenstein, is that he (...) places narrow requirements on what could count as meaningful, prior to and independently of his encounter with the subjects of his interpretation. Frazer, like some of the philosophers Wittgenstein addresses in his other works, succumbs to nonsense in his very attempt to draw the limits of sense. My aim in this chapter is to clarify the connections between Wittgenstein’s criticism of Frazer and his criticism of his fellow philosophers, in particular of Frege. The materials I draw on stem from various periods in Wittgenstein’s career, and they reveal, in my mind, an important continuity in Wittgenstein’s thought: addressing misunderstanding, in Wittgenstein’s view, is fundamentally an ethical problem, not a theoretical one. (shrink)

Neo-Fregean logicists claim that Hume's Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A longstanding problem for neo-Fregean logicism is that HP is not deductively conservative over pure axiomatic second-order logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck's Two-sorted Frege Arithmetic (2FA), a variation on HP which has been thought to be deductively conservative over second-order logic. We show that it isn't. (...) In fact, 2FA is not conservative over $n$-th order logic, for all $n \geq 2$. It follows that in the usual one-sorted setting, HP is not deductively Field-conservative over second- or higher-order logic. (shrink)

This paper continues my discussion with Michael Dummett on Frege’s senses, published in The Philosophy of Michael Dummett and further developed in Diametros. In his reply to my original paper, Dummett came to agree with me that senses are neither objects nor functions, since they have a categorially different kind of linguistico-metaphysical function to perform. He then asks how we might quantify over senses, if they are neither objects nor functions. He discusses two main options, and finds one unviable and (...) the other “very un-Fregean.” I then offer a Fregean or neo-Fregean option in my rejoinder. And I still hold that this way out will do the job, or is at least plausible enough that the burden of persuasion is on those who disagree. But I hope to show in this paper that on a more complete examination of Frege, there are at least twenty Fregean or neo-Fregean ways out, with the one I proposed being option. (shrink)

Frege argues that number is so unlike the things we accept as properties of external objects that it cannot be such a property. In particular, (1) number is arbitrary in a way that qualities are not, and (2) number is not predicated of its subjects in the way that qualities are. Most Aristotle scholars suppose either that Frege has refuted Aristotle's number theory or that Aristotle avoids Frege's objections by not making numbers properties of external objects. This has led some (...) to conclude that Aristotle's accounts of arithmetical and geometrical objects differ substantially. I close this supposed gap by showing that Aristotle's arithmetical objects, like geometrical objects, are just certain sensible things qua certain properties they in fact possess. Specifically, numbers are pluralities qua quantitative or relational properties like ten units or ten. I show that this view is resistant to the Fregean concerns about arbitrariness and numerical predication. (shrink)

In [Boccuni 2010], a predicative fragment of Frege’s blv augmented with Boolos’ unrestricted plural quantification is shown to interpret pa2. The main disadvantage of that axiomatisation is that it does not recover Frege Arithmetic fa because of the restrictions imposed on the axioms. The aim of the present article is to show how [Boccuni 2010] can be consistently extended so as to interpret fa and consequently pa2 in a way that parallels Frege’s. In that way, the presented system will be (...) compared with the system pe in [Ferreira 2018] and some relevant differences between the two will be highlighted. (shrink)

In contemporary historical studies Peano is usually linked to the logical tradition pioneered by Frege. In this paper I question this association. Specifically, I claim that Frege and Peano developed significantly different conceptions of a logical calculus. First, I clam that while Frege put the systematisation of the notion of inference at the forefront of his construction of an axiomatic logical system, Peano modelled his early logical systems as mathematical calculi and did not really attempt to justify reasoning. Second, I (...) argue that in later works on logic Peano advanced towards a deductive approach that was closer to Frege’s standpoint. (shrink)

By means of a reductio argument, Wittgenstein’s Tractatus calls into question the very idea that we can represent logical form. My paper addresses three interrelated questions: first, what conception of logical form is at issue in this argument? Second, whose conception of logic is this argument intended to undermine? And third, what could count as an adequate response to it? I show that the argument construes logical form as the universal, underlying correlation of any representation and the reality it represents. (...) I further show that the argument seeks to undermine core commitments of Frege’s and Russell’s. But the reductio, as I read it, is not intended to establish the falsity of any of their specific assumptions. Rather, its aim is to make manifest the indeterminacies that underlie the language in which these assumptions are framed, and establish the need for a transformation of that language. So understood, Wittgenstein’s argument exemplifies his idea that philosophy is not a theory, but an activity of elucidation. The interpretation I propose bears on one of the central debates in the literature, namely how we should understand Wittgenstein’s contention that his elucidations succeed despite being nonsensical. (shrink)

In [Boccuni 2010], a predicative fragment of Frege’s blv augmented with Boolos’ unrestricted plural quantification is shown to interpret pa2. The main disadvantage of that axiomatisation is that it does not recover Frege Arithmetic fa because of the restrictions imposed on the axioms. The aim of the present article is to show how [Boccuni 2010] can be consistently extended so as to interpret fa and consequently pa2 in a way that parallels Frege’s. In that way, the presented system will be (...) compared with the system pe in [Ferreira 2018] and some relevant differences between the two will be highlighted. (shrink)

Danielle Macbeth's purpose in Macbeth 2005 is threefold. Her monograph proposes "to provide a logical justification for all aspects of Frege's peculiar notation, to motivate and explain the developments in Frege's views over the course of his intellectual life, and to explicate his most developed, critically reflective conception of his Begriffschrift, his formula language of pure thought" (p. vii). I shall focus here on a few selected aspects of the first and third points and leave on the side the discussion (...) of the historical development of Frege's views on logic, semantics and mathematics. The book's underlying claim is that although Frege's logical language, "can of course be read as a language of quantificational logic, [it] can also be read very differently" (p. vii), so much so, as a matter of fact, that although Frege's logic is usually thought of as a "notational variant of standard quantificational logic", it is nevertheless "unknown to us" (p. 1). This naturally prompts three questions : "What is Frege's logic?", "What is quantificational logic?", and given the way we understand formal languages for first and higher order predicate logic, most notably the notions of argument 2 place (for a predicate symbol), of scope and of binding (of variables - either individual or predicate of first or higher order): "How come Frege's logic says nothing about such notions, or something so different from what we say?". (shrink)

En el apartado Freges Versuch, incluido en Filosofía de la aritmética, Husserl abiertamente señala que en los Fundamentos de la aritmética de G. Frege no existe un análisis lógico adecuado del concepto de número en términos de equinumerosidad. Según Husserl, la caracterización de Frege del concepto de número cardinal, en estrecha conexión con la noción de correspondencia uno- a-uno, es errónea. El objetivo principal de este artículo es mostrar que esta interpretación de Husserl sobre la obra de Frege, específicamente en (...) este apartado, contiene una serie de errores y confusiones graves. Palabras clave: Husserl, Frege, psicologismo, Filosofía de la aritmética, equinumerosidad, fenomenología, correspondencia uno-a-uno. (shrink)

Resumen En esta discusión expongo algunas objeciones a las siguientes tesis de Velázquez 2020: 1) que tanto Descartes como Frege sostienen que las entidades aritméticas son irreductibles a procesos empíricos; 2) que, en el caso de Descartes, dichas entidades son “perennes, inherentes a la propia constitución y funcionamiento de la mente” y 3) que Frege impugnó la filosofía matemática de Mill por psicologista. Sostengo que la segunda tesis no es, per se, controversial, pero que sí lo es en el contexto (...) del artículo de Velázquez.In this discussion I present objections to the following theses in Velázquez 2020: 1) that both Descartes and Frege maintain that arithmetic entities are irreducible to empirical processes; 2) that, in the case of Descartes, these entities are “perennial, inherent to the very constitution and functioning of the mind”, and 3) that Frege contested Mill’s mathematical philosophy for being psychologistic. I argue that the second thesis is not, per se, controversial, but that it is controversial in the context of Velázquez’s article. (shrink)

I take issue with Glezakos’s explanation of why Frege’s puzzle is un-puzzling. On her view, Frege’s statement – how can sentences of the form a=a and a=b, if true, differ in cognitive value if they express the same semantic content/are made true by the same object’s self-identity? – should not be considered any puzzling either because it is on the whole circular, or because, neutrally stated, it cannot even be set up. I argue against this that if, as she takes (...) it, Frege’s statement is “problematic” it is not for the reasons she gives, but because it rests upon a couple of questionable assumptions: i) there is no ambiguity as to which aspect of the sign (in its relation to the referent) is relevant to a neutral statement of the puzzle ii) it is of the (sentence) forms themselves one may sensibly say they differ in cognitive value. (shrink)

Logical Investigations (notably, The Thought) is one of the works wherein Frege voices his hostility to psychologism in logic, science, and semantics. Such hostility lies, arguably, behind his threefold conceptual distinction between thought (Gedanke), thinking (Denken), and ideas (Vorstellungen). In this essay I investigate, to begin with, Frege’s motivations for drawing the distinction and keeping thinking episodes (one of the meanings of ‘thoughts’ in English) out of the picture. It turns out, or so I argue, that psychologism threatens, on Frege’s (...) view, to destroy the objectivity requirement by which thoughts are defined as answerable contents. I then draw a connection between this feature in Frege’s view and an on-going debate in philosophy of mind and language over the nature of world-directed thoughts and show that the feature lies at the heart of the singularist stance. Finally, I show that, despite Frege’s own hostility to all that bears the mark of the subjective in logic, science, and semantics, a psychological argument can be mounted in support of singularism (i.e. the view that some of our thoughts are about particulars qua particulars) using his disparaged notion of idea (Vorstellung) and that, appearances notwithstanding, the argument is compatible with the objectivity requirement on answerable thought-contents. (shrink)

My aim in this paper is to show in what sense one might characterize ‘analytic philosophy’. In its first part I present some meta-philosophical ideas about the topic and in its following and more substantial parts I develop, as an example of what is being said, one and only one philosophical problem in the Philosophy of Language. The philosophical problem I consider, the so called ‘identity problem’ or ‘Frege Puzzle’ was first advanced by Frege and was treated and developed by (...) several philosophers during the 20th Century. I consider some of the main solutions for the puzzle trying to show by philosophical considerations what I had previously said by meta-philosophical ideas about analytic philosophy. (shrink)

We note that a plural version of logicism about arithmetic is suggested by the standard reading of Hume's Principle in terms of `the number of Fs/Gs'. We lay out the resources needed to prove a version of Frege's principle in plural, rather than second-order, logic. We sketch a proof of the theorem and comment philosophically on the result, which sits well with a metaphysics of natural numbers as plural properties.

Fregesche Variationen [Variations of Frege], a Festschrift edited by Matthias Wille, assembles twelve papers in honour of Christian Thiel on the occasion of his 80th birthday in 2017. The papers ex...