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Switch to: References
Citations of:
Metatheory and Mathematical Practice in Frege
Philosophical Topics 25 (2):213-264 (1997)
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Why should one think Frege's definition of the ancestral correct? It can be proven to be extensionally correct, but the argument uses arithmetical induction, and that seems to undermine Frege's claim to have justified induction in purely logical terms. I discuss such circularity objections and then offer a new definition of the ancestral intended to be intensionally correct; its extensional correctness then follows without proof. This new definition can be proven equivalent to Frege's without any use of arithmetical induction. This (...) |
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According to Frege, judgement is the ‘logically primitive activity’. So what is judgement? In his mature work, he characterizes judging as ‘acknowledging the truth’ (‘Anerkennen der Wahrheit’). Frege’s remarks about judging as acknowledging the truth of a thought require further elaboration and development. I will argue that the development that best suits his argumentative purposes takes acknowledging the truth of a thought to be a non-propositional attitude like seeing an object; it is a mental relation between a thinker, a thought, (...) |
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It is widely believed that some puzzling and provocative remarks that Frege makes in his late writings indicate he rejected independence arguments in geometry, particularly arguments for the independence of the parallels axiom. I show that this is mistaken: Frege distinguished two approaches to independence arguments and his puzzling remarks apply only to one of them. Not only did Frege not reject independence arguments across the board, but also he had an interesting positive proposal about the logical structure of correct (...) |
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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 (...) |
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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. |
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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, (...) |
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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 (...) |
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We shift attention from the development of model theory for demarcated languages to the development of this theory for fragments of a language. Although it is often assumed that model theory for demarcated languages is not compatible with a universalist conception of logic, no one has denied that model theory for fragments of a language can be compatible with that conception. It thus seems unwarranted to ignore the universalist tradition in the search for the origins and development of model theory. (...) |
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It is often argued that by assuming the existence of a universal language, one prohibits oneself from conducting semantical investigations. It could thus be thought that Tarski’s stance towards a universal language in his fruitful Wahrheitsbegriff differs essentially from Carnap’s in the latter’s less successful Untersuchungen zur allgemeinen Axiomatik. Yet this is not the case. Rather, these two works differ in whether or not the studied fragments of the universal language are languages themselves, i.e., whether or not they are closed (...) |
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Since its publication in 1967, van Heijenoort’s paper, “Logic as Calculus and Logic as Language” has become a classic in the historiography of modern logic. According to van Heijenoort, the contrast between the two conceptions of logic provides the key to many philosophical issues underlying the entire classical period of modern logic, the period from Frege’s Begriffsschrift (1879) to the work of Herbrand, Gödel and Tarski in the late 1920s and early 1930s. The present paper is a critical reflection on (...) |
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In this paper I suggest an answer to the question of what Frege means when he says that his logical system, the Begrijfsschrift, is like the language Leibniz sketched, a lingua characteristica, and not merely a logical calculus. According to the nineteenth century studies, Leibniz's lingua characteristica was supposed to be a language with which the truths of science and the constitution of its concepts could be accurately expressed. I argue that this is exactly what the Begriffsschrift is: it is (...) |
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In this paper I suggest an answer to the question of what Frege means when he says that his logical system, the Begriffsschrift, is like the language Leibniz sketched, a lingua characteristica, and not merely a logical calculus. According to the nineteenth century studies, Leibniz’s lingua characteristica was supposed to be a language with which the truths of science and the constitution of its concepts could be accurately expressed. I argue that this is exactly what the Begriffsschrift is: it is (...) |
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Frege claims that the laws of logic are characterized by their “generality,” but it is hard to see how this could identify a special feature of those laws. I argue that we must understand this talk of generality in normative terms, but that what Frege says provides a normative demarcation of the logical laws only once we connect it with his thinking about truth and science. He means to be identifying the laws of logic as those that appear in every (...) |
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In the last two decades, there has been increasing interest in a re-evaluation of Frege’s stance towards consistency- and independence proofs. Papers by several authors deal with Frege’s views on these topics. In this note, I want to discuss one particular problem, which seems to be a main reason for Frege’s reluctant attitude towards his own proposed method of proving the independence of axioms, namely his view that thoughts, that is, intensional entities are the objects of metatheoretical investigations. This stands (...) |
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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 (...) |
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In recent years, several scholars have been investigating Frege’s mathematical background, especially in geometry, in order to put his general views on mathematics and logic into proper perspective. In this article I want to continue this line of research and study Frege’s views on geometry in their own right by focussing on his views on a field which occupied center stage in nineteenth century geometry, namely, projective geometry. |
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In a series of articles dating from 1903 to 1906, Frege criticizes Hilbert’s methodology of proving the independence and consistency of various fragments of Euclidean geometry in his Foundations of Geometry. In the final part of the last article, Frege makes his own proposal as to how the independence of genuine axioms should be proved. Frege contends that independence proofs require the development of a ‘new science’ with its own basic truths. This paper aims to provide a reconstruction of this (...) |
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Over the past decades, Patricia Blanchette has developed a sophisticated account of Frege's conception of logic and his views on logical consequence. One of the central components of her interpretation is the idea that Frege's conception of logical consequence is ‘semantically laden’ and not purely formal. The aim of the present paper is to provide precise explications of this as well as related ideas that inform her account, and to discuss their significance for the philosophy of logic in general and (...) |
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Recent historical studies have investigated the first proponents of methodological structuralism in late nineteenth-century mathematics. In this paper, I shall attempt to answer the question of whether Peano can be counted amongst the early structuralists. I shall focus on Peano’s understanding of the primitive notions and axioms of geometry and arithmetic. First, I shall argue that the undefinability of the primitive notions of geometry and arithmetic led Peano to the study of the relational features of the systems of objects that (...) No categories |
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This paper explores the relationship between Hume's Prinicple and Basic Law V, investigating the question whether we really do need to suppose that, already in Die Grundlagen, Frege intended that HP should be justified by its derivation from Law V. |
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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. |
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An overview of what Frege accomplishes in Part II of Grundgesetze, which contains proofs of axioms for arithmetic and several additional results concerning the finite, the infinite, and the relationship between these notions. One might think of this paper as an extremely compressed form of Part II of my book Reading Frege's Grundgesetze. $104.96 new (collection) View on Amazon.com Direct download (2 more)  1 citation |
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Ludwig Wittgenstein once wrote to G.E. Moore that he had stirred up a philosophical wasps’ nest with his paradox, associated with the sentence “I believe it’s raining and it’s not raining”. The problem is that it would be odd for a speaker to assert this thought about herself, although it could be true about her, and although the sentence is well-formed and not contradictory. -/- Making use of the notion of a sentence having sense in a context of significant use (...) |
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