Frege: Works

In the explanations of logical laws and inference rules of the mature version of Begriffsschrift in Grundgesetze, Frege uses the predicate “… is the True.” Scholars like Greimann maintain that this predicate is a metalinguistic truth-predicate for Frege. This paper examines an argument for this claim that is based on the “nominal reading” of Frege’s conception of sentences—the claim that for Frege a sentence “ $ p $ ” is equivalent to a nonsentential phrase like “the truth-value of the thought (...) that $ p $.” In particular, this paper attempts to establish two points concerning this argument based on the nominal reading. First, the argument implies a claim about the nature of assertion which Frege repeatedly denies in his mature works. Secondly, the nominal reading on which the argument depends is false. A sentence “ $ p $ ” is not equivalent to a nonsentential phrase like “the truth-value of the thought that $ p $ ” for Frege. Our discussion will lead to an important lesson about Frege’s conception of sentences and of assertion. (shrink)

In the explanations of logical laws and inference rules of the mature version of Begriffsschrift in Grundgesetze, Frege uses the predicate “… is the True.” Scholars like Greimann maintain that this predicate is a metalinguistic truth-predicate for Frege. This paper examines an argument for this claim that is based on the “nominal reading” of Frege’s conception of sentences—the claim that for Frege a sentence “ $ p $ ” is equivalent to a nonsentential phrase like “the truth-value of the thought (...) that $ p $.” In particular, this paper attempts to establish two points concerning this argument based on the nominal reading. First, the argument implies a claim about the nature of assertion which Frege repeatedly denies in his mature works. Secondly, the nominal reading on which the argument depends is false. A sentence “ $ p $ ” is not equivalent to a nonsentential phrase like “the truth-value of the thought that $ p $ ” for Frege. Our discussion will lead to an important lesson about Frege’s conception of sentences and of assertion. (shrink)

Gottlob Frege is considered a founder of analytic philosophy and mathematical logic, but the traditions that claim Frege as a forebear never embraced his Begriffsschrift, or “conceptual notation”—the invention he considered his most important accomplishment. Frege believed that his notation rendered logic visually observable. Rejecting the linearity of written language, he claimed Begriffsschrift exhibited a structure endogenous to logic itself. But Frege struggled to convince others to use his notation, as his frustrated pedagogical efforts at the University of Jena illustrate. (...) Teaching Begriffsschrift meant using words to explain it; rather than replacing spoken language, notation became its obverse in a bifurcated style of argument that separated deduction from commentary. Both registers of this discourse, however, remained within Frege’s monologue, imposing a consequential passivity on his students. In keeping with Frege’s visual understanding of notation, they learned by silently observing it, though never in isolation: notation and language were always mixed together. (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)

Neste artigo daremos principal atenção aos dois critérios de identidade de conteúdo conceitual que estão presentes nos §§ 3 e 8 da Conceitografia de Gottlob Frege. Nosso propósito é analisar as características destes critérios da notação conceitual de Frege porque pretendemos delimitar a discussão em torno dos problemas relacionados às noções de identidade intensional e extensional. Dessa forma, pretendemos: analisar os critérios de identidade de conteúdo conceitual presentes nos §§ 3 e 8 da Conceitografia com o objetivo de mostrar que (...) Frege apresentou uma caracterização híbrida da noção de conteúdo conceitual e fazer considerações a respeito de relações que podemos estabelecer entre os critérios intensionais e extensionais de Frege e Richard Kirkham presentes no livro Teorias da Verdade: Uma Introdução Crítica. Com tais comparações pretendemos averiguar: se o critério de identidade do § 8 da Conceitografia é idêntico ao critério extensional de equivalência material de Kirkham e se o critério de identidade do § 3 da Conceitografia é mais forte, mais fraco ou idêntico aos critérios intensionais de equivalência essencial e de equivalência de sinonímia de Kirkham. (shrink)

Well over a century after its introduction, Frege's two-dimensional Begriffsschrift notation is still considered mainly a curiosity that stands out more for its clumsiness than anything else. This paper focuses mainly on the propositional fragment of the Begriffsschrift, because it embodies the characteristic features that distinguish it from other expressively equivalent notations. In the first part, I argue for the perspicuity and readability of the Begriffsschrift by discussing several idiosyncrasies of the notation, which allow an easy conversion of logically equivalent (...) formulas, and presenting the notation's close connection to syntax trees. In the second part, Frege's considerations regarding the design principles underlying the Begriffsschrift are presented. Frege was quite explicit about these in his replies to early criticisms and unfavorable comparisons with Boole's notation for propositional logic. This discussion reveals that the Begriffsschrift is in fact a well thought-out and carefully crafted notation that intentionally exploits the possibilities afforded by the two-dimensional medium of writing like none other. (shrink)

It is widely taken that the first-order part of Frege's Begriffsschrift is complete. However, there does not seem to have been a formal verification of this received claim. The general concern is that Frege's system is one axiom short in the first-order predicate calculus comparing to, by now, the standard first-order theory. Yet Frege has one extra inference rule in his system. Then the question is whether Frege's first-order calculus is still deductively sufficient as far as the first-order completeness is (...) concerned. In this short note we confirm that the missing axiom is derivable from his stated axioms and inference rules, and hence the logic system in the Begriffsschrift is indeed first-order complete. (shrink)

This volume contains English translations of Frege's early writings in logic and philosophy and of relevant reviews by other leading logicians. Professor Bynum has contributed a biographical essay, introduction, and extensive bibliography.

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 (...) a language, since, unlike calculi, its sentential expressions express truths, and it is a characteristic language, since the meaning of its complex expressions depend only on the meanings of their constituents and on the way they are put together. In fact it is in itself already a science composed in accordance with the Classical Model of Science. What makes the Begriffsschrift so special is that Frege is able to accomplish these goals with using only grammatical or syncategorematic terms and so has a medium with which he can try to show analyticity of the theorems of arithmetic. (shrink)

This volume contains English translations of Frege's early writings in logic and philosophy and of relevant reviews by other leading logicians. Professor Bynum has contributed a biographical essay, introduction, and extensive bibliography.

The dissertation seeks to gain a comprehensive understanding of Frege's watershed work of logic, the 1879 Begriffsschrift . The treatise is the first since Aristotle's instauration of the discipline to have succeeded in completely recasting logic. Very generally stated, the dissertation aims at making explicit the conceptual origins of modern logic. To do that will require that each of the achievements of Concept-script be examined from one or more vantage points: either within the perspective of Frege's larger and guiding ambition, (...) that of providing a logical justification of arithmetic; or with respect to the horizon of modernity; or in relation to later developments in logical theory and the philosophy of language, developments stimulated, albeit not exclusively, by Frege's very work. The vantage point adopted in each case will depend on the nature of the topic and the extent to which it requires to be made explicit. One perspective will have preponderance, that of Frege's relationship to his early modern ancestors, a group that includes philosophers, scientists, and mathematicians. Since modernity is constituted in a nearly constant critical, if not polemical, relationship to the Ancients, they will also at times be drawn into the discussion. The elucidation of Frege's breakthrough work will be guided by the following thesis: Frege's attempt to justify arithmetic logically and the fine work of clarification he produced in actualizing this project; his discovery of function-theoretic logic and his fashioning of a concept-script ; and his critique of natural language and the attendant semantics developed over the course of that critique constitute an unfolding and an extension of the primacy of universal method over ontology that is the signature of modern inquiry. (shrink)

This dissertation is a largely historical study in the theory of concepts. My discussion centers on the notion of a "complex concept" or, more illuminatingly, on varieties of logical complexity that attach specifically to concepts. My main contention is that one of the principal innovations associable with the logical doctrine of Frege's Begriffsschrift is the introduction of a novel conception of complex concepts. The innovation of Frege's position can only be appreciated by seeing it against the background of the logical (...) tradition from which it departed---a tradition that I characterize with reference to sources both from the early modern period and from the context of nineteenth-century algebraic logic. Broadly speaking, the purveyors of the earlier tradition held that logical complexity was, if not strictly identifiable with the complexity of concepts, at least parasitic upon a prior notion of the complexity of concepts. For this reason, explanations of predication and inference were characteristically based upon prior and independent accounts of concept-formation. It was just this order of priority that Frege opposed in his early elucidations of the function-argument syntax of Begriffsschrift. Frege departed from the earlier logical tradition by propounding the thesis that judgements are logically prior to concepts. I maintain that the central implication of this thesis is that an adequately general account of the complexity of concepts presupposes a prior and independent account of propositional structure. The main goal of the dissertation is to put Frege's views on concept-formation in perspective both by giving an account of the earlier approaches that Frege opposed and by relating Frege's views to elements of his broader philosophy of logic and language. (shrink)

Complete French translation of Gottlob Frege's Begriffsschrift.

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 (...) a language, since, unlike calculi, its sentential expressions express truths, and it is a characteristic language, since the meaning of its complex expressions depend only on the meanings of their constituents and on the way they are put together. In fact it is in itself already a science composed in accordance with the Classical Model of Science. What makes the Begriffsschrift so special is that Frege is able to accomplish these goals with using only grammatical or syncategorematic terms and so has a medium with which he can try to show analyticity of the theorems of arithmetic. (shrink)

The analysis of identity as coreference is strongly associated with Frege ; it is the view in Begriffsschrift, and, some have argued, henceforth throughout his work. This thesis is incorrect: Frege never held that identity is coreference. The case is made not by interpretation of “proof-quotes”, but rather by exploring how Frege actually deploys the concept. Two cases are considered. The first, from Grundgesetze, are the definitions of the core concepts, zero and truth; the second, from Begriffsschrift, is the validity (...) of Leibniz's Law. In both cases, if identity is coreference, results ensue that would be unacceptable to Frege. (shrink)

In Part III of his 1879 logic Frege proves a theorem in the theory of sequences on the basis of four definitions. He claims in Grundlagen that this proof, despite being strictly deductive, constitutes a real extension of our knowledge, that it is ampliative rather than merely explicative. Frege furthermore connects this idea of ampliative deductive proof to what he thinks of as a fruitful definition, one that draws new lines. My aim is to show that we can make good (...) sense of these claims if we read Frege’s notation diagrammatically, in particular, if we take that notation to have been designed to enable one to exhibit the (inferentially articulated) contents of concepts in a way that allows one to reason deductively on the basis of those contents. (shrink)

It is a familiar fact that different systems of notation can function in radically different ways. Consider, to take a very simple example, the difference between the sign-designs ‘twenty-three’, ‘XXIII’, and ‘23’. The first is an expression of English tracing the sounds a speaker makes in uttering the words ‘twenty’ and ‘three’. The second is a Roman numeral that uses signs for collections of things—‘X’ for ten things and ‘I’ for one thing—to present by addition the idea of ten and (...) ten and one and one and one, that is, twenty-three, things. Instead of tracing the sounds a speaker makes in speaking in some natural language, the Roman numeral ‘XXIII’ directly represents a collection of things. The Arabic numeral ‘23’ is different again. Like the Roman numeral ‘XXIII’ it in some way represents a number directly; unlike the Roman numeral it is not immediately additive. The numeral ‘23’ is not to be read as designating two and three things. The Arabic numeration system functions as a notational system in a way that is different both from the notational system of a written natural language such as English and from that of the Roman numeration system. (shrink)

If the begriffsschrift from Frege does represent the logical form of natural language it either lacks a logical form itself, or its logical form is different to that of natural language. But Frege insists that his notation has a logical form. So the second disjunct holds. This suggests that Frege's notation will generate consequences different to those that can be derived with natural language, with its different logical form. For anyone looking for "a means of avoiding misunderstandings", as Frege does, (...) this would be unpalatable, to say the least. (shrink)

Frege uses Greek letters in two different ways in his Begriffsschrift. One way is the familiar use of bound variables, in conjunction with variable-binding operators, to mark and close argument-places. The other, which is quite unfamiliar, employs letters to mark places for operators to reach into, without thereby closing these places. Frege thereby invents a powerful and compact notation for functional operations which can be recommended even today. His regrettable double use of Greek letters obscured his invention, and this, together (...) with the fact that in the Grundgesetze he no longer has need of function-valued functions, explains why the device was overlooked and has not passed into general use. (shrink)

Before Frege, the term Begriffsschrift'was used to indicate a language the expressions of which adequately represent the structure of the judgements or concepts which they signify, and a language the written signs of which designate ideas rather than sounds. In 1879 Frege follows . Later he adopts —and with it the Aristotelian theory of language in which it is embedded.

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 (...) a language, since, unlike calculi, its sentential expressions express truths, and it is a characteristic language, since the meaning of its complex expressions depend only on the meanings of their constituents and on the way they are put together. In fact it is in itself already a science composed in accordance with the Classical Model of Science. What makes the Begrijfsschrift so special is that Frege is able to accomplish these goals with using only grammatical or syncategorematic terms and so has a medium with which he can try to show analyticity of the theorems of arithmetic. (shrink)

The formal structure of Frege’s ‘concept script’ has been widely adopted in logic text books since his time, even though its rather elaborate symbols have been abandoned for more convenient ones. But there are major difficulties with its formalisation of pronouns, predicates, and propositions, which infect the whole of the tradition which has followed Frege. It is shown first in this paper that these difficulties are what has led to many of the most notable paradoxes associated with this tradition; the (...) paper then goes on to indicate the lines on which formal logic—and also the lambda calculus and set theory—needs to be restructured, to remove the difficulties. Throughout the study of what have come to be known as first-, second-, and higher-order languages, what has been primarily overlooked is that these languages are abstractions. Many well known paradoxes, we shall see, arose because of the elementary level of simplification which has been involved in the abstract languages studied. Straightforward resolutions of the paradoxes immediately appear merely through attention to languages of greater sophistication, notably natural language, of course. The basic problem has been exclusive attention to a theory in place of what it is a theory of, leading to a focus on mathematical manipulation, which ‘brackets off’ any natural language reading. (shrink)

Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought (1879), by G. Frege.--Some metamathematical results on completeness and consistency; On formally undecidable propositions of Principia mathematica and related systems I; and On completeness and consistency (1930b, 1931, and 1931a), by K. Gödel.--Bibliography (p. [111]-116).