Frege: Logic and Philosophy of Logic, Misc

1. Frege’s mistake Frege is justifiably considered the most important thinker in the development of our contemporary “modern” logic. One corollary to this historical role of Frege’s is that his mistakes are found in a magnified form in the subsequent development of logic. This paper examines one such mistake and its later history. Diagnosing this history also reveals ways of overcoming some of the limitations that Frege’s mistake has unwittingly imposed on current forms of modern logic. Frege’s mistake concerns the (...) semantics (meaning) of quantifiers. The mistake is to assume that this semantics is exhausted by the quantifiers’ (quantified variables’) ranging over a class of values. These values are the members of the domain (universe of discourse) of the language to which the quantifiers belong. The entire job description of the quantifiers is to indicate whither or not at least one member of the domain has a certain (possible complex) predicate (existential quantifier) and to indicate whether all of them have one (universal quantifier). In other words, quantifiers are higher order predicates indicating whether or not a given lower-order predicate is nonempty or exceptionless. This is in fact precisely how Frege proposes to treat quantifiers in his logical theory. (See Frege 1984, pp. 153-154, pp. 26-27 of the original.) This is obviously part of the semantical task of quantifiers. However, it is not the only one. Quantifiers have another function in language. There is a task that any language must be capable of fulfilling if it is to serve as a language of science and for that matter as a language suitable for innumerable purposes in everyday life. This task is to.. (shrink)

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 ...

According to Peter Aczel, the inconsistency of Frege’s system in Grundgesetze is due, not to the introduction of sets, as is usually thought, but to the introduction of the Horizontal. His argument is that the principles governing sets are intuitively correct and therefore consistent, while the scheme introducing the Horizontal amounts to an internal definition of truth conflicting with Tarski’s classic result on the undefinability of truth in the object language. The aim of this paper is to show that the (...) Horizontal is innocent: Aczel’s diagnosis is based on a mistaken view of the structure underlying Frege’s ideal language. (shrink)

Die auf G. Frege zurückgehende logische Urteilslehre, die die universalen Aussagen im Sinne existenzfreier und die partikulären im Sinne existenzmitbehauptender Urteile deutet, hat ihren Ursprung in der nicht-mathematischen Logik des 19. Jahrhunderts. Bei J.F. Herbart findet sich die hypothetische Konzeption der Allaussage, die eine bedeutsame, Fregesche Gedankengänge antizipierende Verfeinerung durch Chr. Sigwart erfährt. Die genaue Struktur der partikulären Aussage bleibt vorerst noch im Dunkel. Erst F. Brentano gelingt es, die universalen wie die partikulären Aussagen in ihrer Eigenart herauszustellen. In dieser (...) Entwicklung spielen die Problematik fiktiver Gegenstände und leerer Begriffe, die apriorische Gesetzlichkeit sowie der Einfachheitsbegriff eine zentrale Rolle. (shrink)

Frege and Heidegger appear to appear to have diametrically opposed attitudes towards logic. Frege thinks logic must govern any investigation whatsoever, whereas Heidegger (in "What is Metaphysics?") apparently wants to dismantle logic. But when they try to explicate the nature of judgment, a striking similarity emerges. For while their accounts of judgment are radically different, each finds his account to be, by his own lights, _inexpressible<D>. This paper shows how Heidegger and Frege arrive at their respective accounts of judgment, explains (...) why they regard the status of their accounts as problematic, and explores and evaluates their responses to their parallel predicaments. (shrink)

Frege's docent's dissertation Rechnungsmethoden, die sich auf eine Erweiterung des Grössenbegriffes gründen(1874) contains indications of a bold attempt to extend arithmetic. According to it, arithmetic means the science of magnitude, and magnitude must be understood structurally without intuitive support. The main thing is insight into the formal structure of the operation of ?addition?. It turns out that a general ?magnitude domain? coincides with a (commutative) group. This is an interesting connection with simultaneous developments in abstract algebra. As his main application, (...) Frege studies iterations of functions. He does not yet pose the question of existence proofs. Measurement of magnitudes is also connected to numbers, but the discussion is here ambiguous in a way which calls for the systematic account of numbers in Grundgesetze. (shrink)

Frege?s project has been characterized as an attempt to formulate a complete system of logic adequate to characterize mathematical theories such as arithmetic and set theory. As such, it was seen to fail by Gödel?s incompleteness theorem of 1931. It is argued, however, that this is to impose a later interpretation on the word ?complete? it is clear from Dedekind?s writings that at least as good as interpretation of completeness is categoricity. Whereas few interesting first-order mathematical theories are categorical or (...) complete, there are logical extensions of these theories into second-order and by the addition of generalized quantifiers which are categorical. Frege?s project really found success through Gödel?s completeness theorem of 1930 and the subsequent development of first- and higher-order model theory. (shrink)

We reconstruct Frege?s treatment of certain deducibility problems posed by Boole. It turns out that in his formalization and solution of Boole?s problems Frege anticipates the idea of propositional resolution.

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)

One of Frege’s basic philosophical convictions has been that “a large part of the philosopher’s work consists—or at least should consist—in a fight against language.”. Besides, it is partly to his having yielded too easily to the deceptive suggestions of natural language that he himself attibuted, at the very end of his life, the failure of his attempt at defining numbers in logicist terms Nevertheless, he remained to the end convinced that the hopes raised by logic, one of whose tasks (...) is “to free us of the chains of language”, can only rest, in the last analysis, upon the possibility to trust the bulk of language and to make use of “such an imperfect instrument” to fight from the inside the uncertain fight that the philosopher must embark upon in order to free himself from its dominion and neutralize or eliminate its most damaging imperfections. (shrink)

This paper argues that a particular point raised by Schröder – that Frege's logical notation fails to be modelled on arithmetical notation – is based on a misunderstanding, for the modelling was meant as conceptual, rather than notational.

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.

In an essential way I make use of Frege’s and Vasilev’s ideas. N. A. Vasilev distinguished two levels in a logic. The abstract logic depends on gnoeologic assumptions while the empirical part of logic depends on ontological ones. Vasilev did not change the external logic but he did change the internal one. His system can be viewed as a non-standard syllogistics based on classical propositional logic . Vasilev’s ideas become plain if we discern cleary acts of predication and acts of (...) assertion, the relation of mental contents with the way things are. The second sourse of the idea seems to Frege’s differentation of mental process and assertion, statement . As a symbol of assertion for Frege serves a vertical stroke |. A horizontal stroke following it is correlated with holding a proposition, it makes the sign determined everywhere. In the string of signs the one following the vertical stroke does not express a proposition, it describes a possible state of affairs. (shrink)

Does Frege have a metatheory for his logic? There is an obvious and uncontroversial sense in which he does. Frege introduces and discusses his new logic in natural language; he argues, in response to criticisms of Begriffsschrift, that his logic is superior to Boole's by discussing formal features of both systems. In so far as the enterprise of using natural language to introduce, discuss, and argue about features of a formal system is metatheoretic, there can be no doubt: Frege has (...) a metatheory. There is also an obvious and uncontroversial sense in which Frege does not have a metatheory for his logic. The model theoretic semantics with which we are familiar today are a post-Fregean development. The question I address in this paper is, does Frege have a metatheory in the following sense: do his justifications of his basic laws and rules of inference employ, or even require, ineliminable use of a truth predicate and metalinguistic variables? My answer is ‘no’ on both counts. I argue that Frege neither uses, nor has any need to use, a truth predicate or metalinguistic variables in his justifications of his basic laws and rules of inference. Quine's famous explanation of the need for semantic ascent simply does not apply to Frege's logic. The purpose of the discussions that are typically understood as constituting Frege's metatheory is, rather, elucidatory. And once we see what the aim of these particular elucidations is, we can explain Frege's otherwise puzzling eschewal of the truth predicate in his discussions of the justification of the laws and rules of inference. (shrink)

The early formal logicians (Frege, Russell, Peano et al.) were worried about differentiating logic from psychology. As a result, they interpreted logic in the most abstract way possible: as a theory about inference patterns whose terms lacked descriptive content. Such a theory was also acontextual. What they did not realize was that psychological concepts like expecting someone, doubting, pain etc. each had their own logic, a logic that had two features: it was contextually oriented and its concepts had a restricted (...) sensible application. This is still a recognizable sense of logic but broader in scope than the conceptions that Frege and Russell had in mind. (shrink)

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)

An account of Aristotle's syllogistic (including a full square of opposition and allowing for empty nouns) as an integral part of first-order predicate logic is lacking. Some say it is not possible. It is not found in the tradition stemming from ukasiewicz's attempt nor in less formal approaches such as Strawson's. The ukasiewicz tradition leaves Aristotle's syllogistic as an autonomous axiomatized system. In this paper Aristotle's syllogistic is presented within first-order predicate logic with special restricted quantifiers. The theory is not (...) motivated primarily by historical considerations but as an accurate account of categorical sentences along lines suggested by recent work on natural language quantifiers and themes from supposition theory. It provides logical forms which conform to grammatical ones and is intended as a rival to accounts of quantifiers in natural language that appeal to binary quantifiers, for example, Wiggins or to restricted quantifiers, for example, Neale. (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.

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.

Hintikka and Sandu had argued that 'Frege's failure to grasp the idea of the standard interpretation of higher-order logic turns his entire foundational project into a hopeless daydream' and that he is 'inextricably committed to a non-standard interpretation' of higher-order logic. We disagree.

This book is far more than an exposition of Frege's logical system and semantic concepts, although it is that. The author puts forward the challenging thesis that in trying to cope with Russell's paradox Frege deserted principles of his system which he had relied on throughout. Sternfeld attempts to show, by offering his own interpretation of Frege's logical theory, that if Frege had relied consistently on his previously formulated logical principles, Russell's paradox would have given him no trouble. Further, he (...) uses these arguments as a basis for defending the general thesis that paradoxes and other difficulties with various logical systems can only be discussed relative to the philosophical principles underlying the logical system and adopted independently of it. While these are the most challenging of the book's theses they are not its only topics. The author seems to have read everything by and about Frege and is in control of his material. There will undoubtedly be disagreements over his interpretation of Frege because Frege did not write to make expositors happy. But all students of Frege, as well as students of the philosophy of logic and philosophy of mathematics, will find this book rewarding.--R. H. K. (shrink)

Frege begins his exposition of the symbol system employed in his Begriffsschrift by introducing the sign ⟝, whereby, he says, “[a] judgment is always to be expressed”.[The judgment sign] stands to the left of the sign or complex of signs in which the content of the judgment is given. If we omit the little stroke at the left of the horizontal stroke, then the judgment is to be transformed into a mere complex of ideas; the author is not expressing his (...) recognition or nonrecognition of the truth of this. (shrink)

Gottlob Frege's development of his logicist program has had a major impact on the history of logic and on the philosophy of mathematics. The year 1902 marked an important turning point for Frege: it was then that Russell discovered and communicated the inconsistency which could be derived in Frege's system. This problem arose because of the introduction of classes into this system, especially since Frege did not have type distinctions among his classes. ;Yet Frege had very clear type distinctions for (...) his functions. I show how we can easily develop his theory of function levels to get, essentially, a simple theory of types for Fregean functions and concepts. Within this "no-class" development we can prove many of the basic theorems of arithmetic in a way which parallels Frege's own derivations. Numbers appear as second-level concepts in this alternative development; Frege had construed numbers as classes which were considered logical objects in his structure. ;I examine the extent to which this "no-class" construction agrees with Frege's basic principles. This takes on new interest since the recent publication of many of his later papers and correspondence : Frege's writings after 1902 give fresh insights into his original motivations and thought, and they give us a glimpse of his reactions to various proposed solutions of the paradox, many of which are connected with a "no-class" development. ;For Frege the introduction of classes was very much entwined with his goal of construing numbers as logical objects. We are to note the significance of the Fregean thesis that numbers are objects. We are to examine the logical and philosophical repercussions of defining numbers as objects, and of defining them as second-level concepts. I point out the ontological and epistemological aspects of Frege's views here, and I stress their relevance to contemporary discussions in the philosophy of mathematics. We look at the way a Fregean reduction, seen as an explication, can set out to exhibit logical relationships, logical linkages among the truths of arithmetic. The importance of Frege's logicist thesis in the continuing exploration into the nature of number and arithmetic is not to be overlooked. (shrink)

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)

Convinced that logic has a history and that its history always manages to surprise the philosophers, Claude Imbert has devoted much of her work to the study of the Stoic school and of the late-nineteenth-century German logician Gottlob Frege. In the fifth chapter of her book Pour une histoire de la logique, she examines the trajectory of Frege's awareness of what his new logic entails, in particular the way it subverts the project of Kant.

Frege's philosophical writings, including the “logistic project,” acquire a new insight by being confronted with Kant's criticism and Wittgenstein's logical and grammatical investigations. Between these two points a non-formalist history of logic is just taking shape, a history emphasizing the Greek and Kantian inheritance and its aftermath. It allows us to understand the radical change in rationality introduced by Gottlob Frege's syntax. This syntax put an end to Greek categorization and opened the way to the multiplicity of expressions producing their (...) own intelligibility. This article is based on more technical analyses of Frege which Claude Imbert has previously offered in other writings. (shrink)