Frege: Logic and Philosophy of Logic, Misc

This article derives from a project attempting to show that Western formal logic, from Aristotle onward, has both been partially constituted by, and partially constitutive of, what has become known as racism. In the present article, I will first discuss, in light of Frege’s honorary role as founder of the philosophy of mathematics, Reuben Hersh’s What is Mathematics, Really? Second, I will explore how the infamous section of Frege’s 1924 diary (specifically the entries from March 10 to April 9) supports (...) Hersh's claim regarding the link between political conservatism and the (historically and currently) dominant school of the philosophy of mathematics (to which Frege undeniably belongs). Third, I will examine Frege’s attempt at a more reader-friendly introduction to his philosophy of mathematics, The Foundations of Arithmetic. And finally, I will briefly analyze Frege’s Begriffsschrift to see how questions of race arise even at the heights of his logical abstraction. (shrink)

There are two dominant approaches to quantification: the Fregean and the Tarskian. While the Tarskian approach is standard and familiar, deep conceptual objections have been pressed against its employment of variables as genuine syntactic and semantic units. Because they do not explicitly rely on variables, Fregean approaches are held to avoid these worries. The apparent result is that the Fregean can deliver something that the Tarskian is unable to, namely a compositional semantic treatment of quantification centered on truth and reference. (...) We argue that the Fregean approach faces the same choice: abandon compositionality or abandon the centrality of truth and reference to semantic theory. Indeed, we argue that developing a fully compositional semantics in the tradition of Frege leads to a typographic variant of the most radical of Tarskian views: variabilism, the view that names should be modeled as Tarskian variables. We conclude with the consequences of this result for Frege's distinction between sense and reference. (shrink)

In this essay, I argue that Frege plagiarized the Stoics --and I mean exactly that-- on a large scale in his work on the philosophy of logic and language as written mainly between 1890 and his death in 1925 (much of which published posthumously) and possibly earlier. I use ‘plagiarize' (or 'plagiarise’) merely as a descriptive term. The essay is not concerned with finger pointing or casting moral judgement. The point is rather to demonstrate carefully by means of detailed evidence (...) that there are numerous (over a hundred) and extensive parallels both in formulation and --more importantly-- in content between the Stoics and Frege, parallels so plentiful that one would be hard pressed to brush them off as coincidence. These parallels include several that appear to occur in no other modern works that were published before Frege’s own and were accessible to him. Additionally, a cluster of corroborating historical data is adduced to support the suggestion, showing how easy it would have to been for Frege to plagiarize the Stoics. This (first) part of the essay is easy to read and vaguely entertaining, or so I hope. (shrink)

In this paper I argue that the two-dimensional character of Frege’s Begriffsschrift plays an epistemological role in his argument for the analyticity of arithmetic. First, I motivate the claim that its two-dimensional character needs a historical explanation. Then, to set the stage, I discuss Frege’s notion of a Begriffsschrift and Kant’s epistemology of mathematics as synthetic a priori and partly grounded in intuition, canvassing Frege’s sharp disagreement on these points. Finally, I argue that the two-dimensional character of Frege’s notations play (...) the epistemological role of facilitating our grasp of logical truths independently of intuition. The rest of this paper critically evaluates Frege’s view and discusses Macbeth’s account. (shrink)

Even though Frege is a major figure in the history of analytic philosophy, it is not surprising that there are still issues surrounding his views, interpreting them, and labeling them. Frege’s view on numbers is typically termed as ‘Platonistic’ or at least a type of Platonism (Reck 2005). Still, the term ‘Platonism’ has views and assumptions ascribed to it that may be misleading and leads to mischaracterizations of Frege’s outlook on numbers and ideas. So, clarification of the term ‘Platonism’ is (...) required to portray Frege’s views more accurately (Reck 2005). This clarification gives us a better picture of what Frege is interested in and what he does not emphasize. Moreover, in such a clarifying process, we find that Frege draws significant influence from Rudolf Hermann Lotze, who is frequently called a Neo-Kantian (Vagnetti 2018). In Lotze’s major work, Logik, Lotze has a central focus on validity, in its most general form as he used it, that investigates various related topics, i.e., concepts, language, etc. (Vagnetti 2018; Lotze 1888). Furthermore, we observe that Frege’s work is so similar to Lotze, that it seems questionable to call his outlook ‘Platonism’. Therefore, attributing ‘Platonism’ to Frege may be a slight misnomer. This paper’s entirety is mostly a synthesis of a variety of articles related to Frege, Lotze, and their respective outlooks and the original works of Frege and Lotze that I use to support the view that the term ‘Platonism’ is a slight issue when predicated to Frege. As such, I include an overview of Frege’s treatment in other work that highlights the usage of the term ‘Platonism’ and how broad its uses tend to be utilized (Balaguer 2006; Burge 1992). In sum, it is observed that the general label ‘Platonism’ becomes less appropriate when we consider Lotze in the picture and contrast Lotze alongside Mr. Frege. Overall, this paper is just an explanatory one of Mr. Frege, the Platonist, and the issues of applying the term ‘Platonism’ onto him as his views are seemingly more of a segue from Lotze. Keywords: Frege, Lotze, number, objectification, Platonism, validity. (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)

Of Zeno’s book of forty paradoxes, it was the first that attracted Socrates’ attention. This is the paradox of the like and the unlike. On contemporary assessments, this paradox is largely considered to be Zeno’s weakest surviving paradox. All of these assessments, however, rely heavily on reconstructions of the paradox. It is only relative to these reconstructions that there is nothing paradoxical involved, or that there is some rather obvious mistake being made. This paper puts forward and defends a novel (...) interpretation of this paradox, according to which the concept of a unit plays a central role. There is every reason to think the paradox turns on the concept of a unit: after the presentation of the paradox the text of the Parmenides immediately turns to a discussion of units, and the concept of a unit is also central to the Greek conception of a plurality. If this interpretation is correct then the paradox that Zeno presented was the same as one discussed and solved in Frege’s Grundlagen. (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)

We propose a reconstruction of the constellation of problems and philosophical positions on the nature and number of the primitives of logic in four authors of the nineteenth century logical scene: Peano, Padoa, Frege and Peirce. We argue that the proposed reconstruction forces us to recognize that it is in at least four different senses that a notation can be said to be simpler than another, and we trace the origins of these four senses in the writings of these authors. (...) We conclude that Frege, and even more so Peirce, developed new notations not to make drawing logical conclusions easier but in order to answer the needs of logical analysis. (shrink)

This paper addresses Frege's discussion of illogical thought in the introduction to Basic Laws of Arithmetic. After a brief introduction, I discuss Frege's claims that logic is normative vis-à-vis thought, and not descriptive, and his opposition to the idea that logical laws express psychological necessities. I argue that these two strands of Frege's polemic against psychologism constitute two motivating factors behind his allowing for the possibility of illogical thought. I then explore a line of thought—originally advanced by Joan Weiner—according to (...) which Frege should have rejected illogical thought as not constituting a genuine possibility. I argue that, once developed, this line of thought constitutes an important correction (moreover, one that is consistent with his two aforementioned anti-psychologistic strands) to Frege's own response to the possibility of illogical thought. -/- . (shrink)

Various critiques of important analytic thinkers made by Alain Badiou in the late 1960s have been largely overlooked by continental philosophers and entirely overlooked by analytic philosophers. This paper looks in detail at Badiou’s 1969 essay ‟Mark and Lack,” providing an exposition and clarification of his direct and sustained critique of Gottlob Frege’s supposed ideological philosophical commitments. Badiou’s intellectual context is analyzed in some detail, not only explaining his theoretical debt to his then-master Louis Althusser, but also clarifying his understandings (...) of the notions of ‟the scientific” and ‟the ideological” in light of the projects of Gaston Bachelard and Georges Canguilhem. A philosophical exposition of Badiou’s point-by-point critique of Frege’s conception of logic follows. Finally, the paper concludes with an analysis of the more general relevance of Badiou’s half-century-old critiques in light of developments in contemporary analytic metaphysics, especially those indebted to W. V. O. Quine and Donald Davidson. In essence, Davidson’s Fregean reconfiguration of Tarski’s work on truth places contemporary analytic metaphysics within the scope of what Badiou directly criticizes. It is suggested that Badiou’s critique find a place in discussions of analytic metaphysics. (shrink)

Hintikka’s semantic approach to meaning, a development of Wittgenstein’s view of meaning as use, is the general theme of this chapter. We will focus on the analysis of quantified sentences and on the scope of the principle of compositionality and compare Hintikka’s take on these issues with that of Frege. The aim of this paper is to show that Hintikka’s analysis of quantified expressions as choice functions, in spite of its obvious dissimilarities with respect to the higher-order approach, is actually (...) very close to the Fregean stance on compositionality and context dependence. In particular, we will defend that the Fregean approach to quantifiers is unavoidably linked to the idea that quantified expressions are context-dependent, and therefore should not be conceived under the traditional inside-out model for analysis. (shrink)

Frege wanted to show that arithmetical truths are logical by proving them from purely logical basic laws. But how do we know that these basic laws are logical? Frege uses generality and undeniability to make a prima facie case for logicality—if a truth is general and undeniable, then it’s likely logical. I argue that Frege could, did, and had to make a deeper case for why we’re right in recognizing his basic laws as logical. Implicit in his work is a (...) view of logical laws as epistemically analytic—his arguments for his basic laws serve to elicit a reflective awareness of the fact that understanding them is sufficient for recognizing them to be true. This view both fits with Frege’s comments concerning the connection between logic, truth, and normativity, and serves to explain why and in what sense he took logic to be general and conceptually undeniable. In my view, semantics must play a distinctive role in any rational reconstruction of Frege’s case for logicality—the aforementioned “reflective awareness” must be an explicit appreciation of how the truth of formulas expressing Frege’s laws follows quickly from his stipulations governing terms which figure in those formulas. Opposing this view is the elucidatory interpretation of Thomas Ricketts, Warren Goldfarb, and Joan Weiner, which holds that Frege’s arguments for his basic laws can’t be taken at face value, and must serve the merely elucidatory purpose of easing us into the language. Another reading is the correctness interpretation of Richard Heck and Jason Stanley, which holds that Frege’s primary purpose in his arguments is justifying the claim that Frege’s axioms, qua formulas, are true. I argue against both of these interpretations, and in doing so clarify the role and limits of semantics in Frege’s enterprise. (shrink)

Para Frege las relaciones lógicas se dan entre contenidos judicables, entre pensamientos. Aquellas relaciones son inferenciales. Los pensamientos se definen a través de sus relaciones inferenciales con otros. De acuerdo con esto es discutible afirmar, como lo hizo Schröder, que el proyecto lógico de Frege es como el proyecto lógico de Boole. También es cuestionable afirmar, como lo hizo Dummett, que la relación inferencial no es siempre central en el proyecto fregeano. En este texto defenderé una lectura del proyecto lógico (...) de Frege, en contra de las posturas de Schröder y Dummett. (shrink)

The focus of the article is the self-predication principle, according to which the/a such-and-such is such-and-such. We consider contemporary approaches (Frege, Russell, Meinong) to the self-predication principle, as well as fourteenth-century approaches (Burley, Ockham, Buridan). In crucial ways, the Ockham-Buridan view prefigures Russell’s view, and Burley’s view shows a striking resemblance to Meinong’s view. In short the Russell-Ockham-Buridan view holds: no existence, no truth. The Burley-Meinong view holds, in short: intelligibility suffices for truth. Both views approach self-predication in a uniform (...) way. We were unable to find a medieval philosopher who, like Frege, approaches self-predication in a non-uniform way. We do not want to dispute that there are also considerable differences between the contemporary and the fourteenth-century approach to self-predication. Importantly, fourteenth-century accounts of self-predication rely strongly on the identity theory of predication, and this theory is not endorsed by contemporary philosophers of language. Nevertheless, some basic tenets seem to us to be clearly shared between fourteenth-century and contemporary thinkers. (shrink)

In this paper I take the opportunity of the recent publication of Pieranna Garavaso’s and Nicla Vassallo’s Frege on Thinking and Its Epistemic Significance (with whose main tenets this paper is in constant dialogue) to provide an overview of some important components of Frege’s conception of logic. Section 1 discusses Frege’s view that the task of logic is to provide justification for what we think, and in sections 2 and 3 this idea is shown to play a central role in (...) his anti-psychologistic view that logic is normative for thinking. Section 4 offers a survey of Frege’s manifold conception of thinking. Finally, section 5 discusses the rela-tions between thinking and language in Frege’s philosophy. (shrink)

We set out the implication fragment of Frege’s Grundgesetze, clarifying the implication rules and showing that this system extends Absolute Implication, or the implication fragment of Intuitionist logic. We set out a sequent calculus which naturally captures Frege’s implication proofs, and draw particular attention to the Cut-like features of his Hypothetical Syllogism rule.

Apresentamos neste artigo explicitações histórico-conceituais sobre o surgimento da predicação lógica contemporânea. Quando se trata de predicação, remete-se de imediato à obra de Aristóteles, mas, com as transformações trazidas pela Lógica Contemporânea, o estudo da predicação deixa o plano do estudo lógico-gramatical para o estudo do plano da análise lógicomatemática. Veremos, nesse sentido, a importância dos trabalhos de Peirce, Frege e Russell para o surgimento da predicação lógica contemporânea. Embora Peirce tenha sido o precursor da introdução do conceito de função (...) proposicional na História da Lógica, ganha destaque, contemporaneamente, o modelo de interpretação da predicação inicialmente proposto por Frege. (shrink)

Frege overlooked the role of quantifiers as expressing, by their formal dependence on each other, the actual dependences between variables bound to them. The resulting flaw in Frege’s and other logicians’ logic began to be corrected only in IF logic. The dependence relations are codified in the Skolem functions that correspond to existential-force quantifiers. Their existence is the natural truth condition. Such functions are not adequately handled in first-order predicate logic. In any adequate logic, a fixed mode of identification is (...) presupposed. The frameworks of identification can be perspectival or public. Kripke makes the same mistake about quantifiers as Frege and in addition assumes that only perspectival indentification is needed in the last analysis. He also overlooks dependence relations between modal operators and quantifiers. (shrink)

Gottlob Frege (1848-1925) invented the discipline of mathematical logic. In this short work, it is clearly stated what Frege did and did not accomplish.

Along with offering an historically-oriented interpretive reconstruction of the syntax of PM ( rst ed.), I argue for a certain understanding of its use of propositional function abstracts formed by placing a circum ex on a variable. I argue that this notation is used in PM only when de nitions are stated schematically in the metalanguage, and in argument-position when higher-type variables are involved. My aim throughout is to explain how the usage of function abstracts as “terms” (loosely speaking) is (...) not inconsistent with a philosophy of types that does not think of propositional functions as mind- and language-independent objects, and adopts a nominalist/substitutional semantics instead. I contrast PM’s approach here both to function abstraction found in the typed λ-calculus, and also to Frege’s notation for functions of various levels that forgoes abstracts altogether, between which it is a kind of intermediary. (shrink)

Frege’s system of first-order logic is presented in a contemporary framework. The system described is distinguished by economy of expression and an unusual syntax.

Drawing on the original conception of Kant’s synthetic a priori and the relevant related developments in philosophy, this book presents a reconstruction of the intellectual history of the conception of quantity and offers an entirely ...

O presente artigo busca debater a posição de Charles Sanders Peirce e dos primeiros estudantes peirceanos de Lógica (Christine Ladd e O. H. Mitchell nos Studies in Logic, 1883) dentro do debate inspirador da visão da linguagem dentro da Filosofia Analítica, conhecido como “Lingua Universalis contra Calculus Ratiocinator”, cujos primórdios podem ser traçados desde a filosofia de Gottfried Leibniz. Para isso, comparamos esse campo do pensamento peirceano com o debate crucial entre a conceitografia de Gottlob Frege (Begriffsschrift, 1879) e a (...) lógica algébrica de George Boole (An investigation of the Laws of Thought on which are founded the mathematical theories of Logic and Probabilities, 1854). O nosso objetivo principal é observar que esse momento da filosofia peirceana pode ser comparado com o pensamento wittgensteiniano, especialmente em sua nova vertente, chamada New Wittgenstein, que tenta superar a tradicional divisão entre Tractatus Logico-Philosophicus e Investigações Filosóficas. Pela comparação entre pensadores influenciados por C. S. Peirce e a filosofia de Ludwig Wittgenstein, pretendemos abrir caminhos na compreensão do conceito de jogo de linguagem, especialmente em sua gramática e em seus operadores simbólicos, através da observação das funções de verdade [truth functions] e as tabelas de verdade [truth tables] enquanto lógica algébrica. (shrink)

In this paper we summarize some results about sets in Frege structures. The resulting set theory is discussed with respect to its historical and philosophical significance. This includes the treatment of diagonalization in the presence of a universal set.

Gregory Landini offers a detailed historical account of Frege's notations and the philosophical views that led Frege from Begriffssscrhrift to his mature work Grundgesetze, addressing controversial issues that surround the notations.

Abstract I argue against the two most influential readings of Frege's methodology in the philosophy of logic. Dummett's ?semanticist? reading sees Frege as taking notions associated with semantical content?and in particular, the semantical notion of truth?as primitive and as intelligible independently of their connection to the activity of judgment, inference, and assertion. Against this, the ?pragmaticist? reading proposed by Brandom and Ricketts sees Frege as beginning instead from the independent and intuitive grasp that we allegedly have on the latter activity (...) and only then moving on to explain semantical notions in terms of the nature of such acts. Against both readings, I argue, first, that Frege gives clear indication that he takes semantical and pragmatical notions to be equally primitive, such that he would reject the idea that either sort of notion could function as the base for a non-circular explanation of the other. I argue, secondly, that Frege's own method for conveying the significance of these primitive notions?an activity that Frege calls ?elucidation??is, in fact, explicitly circular in nature. Because of this, I conclude that Frege should be read instead as conceiving of our grasp of the semantical and pragmatical dimensions of logic as far more of a holistic enterprise than either reading suggests. (shrink)

Frege's account of indirect proof has been thought to be problematic. This thought seems to rest on the supposition that some notion of logical consequence ? which Frege did not have ? is indispensable for a satisfactory account of indirect proof. It is not so. Frege's account is no less workable than the account predominant today. Indeed, Frege's account may be best understood as a restatement of the latter, although from a higher order point of view. I argue that this (...) ascent is motivated by Frege's conception of logic. (shrink)

In the opening sentence of his Methods of Logic, W. V. O. Quine writes, “Logic is an old subject, and since 1879 it has been a great one.”1 Quine is referring to the year in which Gottlob Frege presented his Begriffschrift, or “concept-script,” one of the first published accounts of a logical system or calculus with quantification and a function-argument analysis of propositions. There can be no doubt as to the importance of these introductions, and, indeed, Frege’s orientation and advances, (...) if not his particular system, have proven to be highly significant for much of mathematical logic and research pertaining to the foundations of mathematics. Nevertheless, Quine’s assessment, often repeated, obscures other .. (shrink)

This chapter begins with a discussion of Kant's theory of judgment-forms. It argues that it is not true in Kant's logic that assertoric or apodeictic judgments imply problematic ones, in the manner in which necessity and truth imply possibility in even the weakest systems of modern modal logic. The chapter then discusses theories of judgment-form after Kant, the theory of quantification, Frege's Begriffsschrift, C. I. Lewis and the beginnings of modern modal logic, the proof-theoretic approach to modal logic, possible world (...) semantics, correspondence theory, and modality and quantification. (shrink)

We show that the untyped λ -calculus can be extended with Frege's interpretation of propositional notions, provided we restrict β -conversion to positive expressions. The system of illative λ -calculus so obtained admits a natural Scott-style semantics.

1 Logic in philosophy The century that was Logic has played an important role in modern philosophy, especially, in alliances with philosophical schools such as the Vienna Circle, neopositivism, or formal language variants of analytical philosophy. The original impact was via the work of Frege, Russell, and other pioneers, backed up by the prestige of research into the foundations of mathematics, which was fast bringing to light those amazing insights that still impress us to-day. The Golden Age of the 1930s (...) deeply affected philosophy, and heartened the minority of philosophers with a formalanalytical bent. As Brand Blanshard writes in Reason and Analysis (1964) – I quote from memory here, to avoid the usual disappointment when re-reading an original text. (shrink)

This paper is concerned with locating the specific assumption that led Frege into Russell's Paradox. His understanding of reflexive pronouns was weak, for one thing, but also, by assimilating concepts to functions he was misled into thinking one could invariably replace a two-place relation with a one-place property. /// Este trabajo se ocupa de localizar el supuesto específico que llevó a Frege a la Paradoja de Russell. Por una parte, su comprensión de los pronombres reflexivos era débil pero, por otra, (...) al asimilar los conceptos a las funciones pensó equivocadamente que siempre podíamos reemplazar una relación diádica con una propiedad monádica. (shrink)

Der Begriff der logischen Form charakterisiert einerseits einen der Gegenstande und andererseits eine der Methoden der analytischen Philosophie. Diese zweifache Rolle verdankt der Formbegriff den Formulierungen Russells, der in Auseinandersetzung mit seinen Zeitgenossen verschiedene Aspekte des Inhalts dieses Begriffs definiert. Die vorliegende Untersuchung rekonstruiert die Theorie Russells als einen Begriffszusammenhang und stellt die Ursprunge und die Besonderheiten der Entwicklung des Begriffs der logischen Form dar, welche die moderne Herangehensweise an die logisch-philosophische Problematik entscheidend pragen und somit die Gestalt der analytischen (...) Tradition bestimmen.". (shrink)