Gottlob Frege

Frege maintains that there are not two distinct acts, assertion and denial; rather, denying p is one and the same as asserting not-p. Wittgenstein appears not to recognise this identity in Frege, attributing to him the contrary view that a proposition may have one of two verbs, "is true" or "is false". This paper explains Wittgenstein’s attribution as a consequence of Frege’s treatment of content as theoretically prior to the act of judgment. Where content is prior to judgment, the denial (...) of p—what is rejected in asserting p—is distinct from the assertion of not-p. Wittgenstein’s own embrace of Frege’s identity is then considered, an embrace explained in part by his repudiation of Frege’s theoretical order. (shrink)

El presente texto es una guía para una primera lectura de los Los fundamentos de la aritmética de Gottlob Frege para estudiantes del grado de Filosofía. -/- No pretende hacer ninguna aportación a la investigación sobre Frege sino ofrecer los instrumentos para hacer una primera lectura mediante la recopilación y la ordenación de los textos relevantes de los estudiosos de Frege, especialmente de la literatura en inglés. En la mayor parte de los casos, las referencias a otros autores (Autorfecha) preceden (...) a una traducción de los escritos referidos así como una unificación de la terminología, el recorte de lo no relevante y el añadido de las explicaciones pertinentes para hacer el texto accesible a un estudiante del grado de Filosofía. En el caso de que la cita justifique una afirmación que no se desarrolla, se verá precedida por la palabra "ver" dentro del paréntesis de la referencia. El texto usado para las citas de Los fundamentos de la aritmética es la traducción de Ulises Moulines dado que el que esto escribe no sabe alemán. La guía no supone más matemáticas que las que se estudian en los cursos anteriores a la universidad. Sin embargo, es recomendable haber hecho un curso elemental de lógica matemática y teoría de conjuntos. La razón fundamental es que la formalización de ciertas proposiciones permite aclarar extremos que se hacen muy enojosos sin esos instrumentos. (shrink)

This article argues that history played a larger role in the thought of Gottlob Frege than has usually been acknowledged. Frege’s logical writings frequently employed statements about the past as examples that included references to historical persons. Frege also described history as a science and argued that historical propositions could support valid inferences and reliably identify historical persons and events. But Frege’s eternalist theory of reference, designed primarily for formal concepts and objects, struggled to accommodate such propositions. Identifying an objective (...) referent for the subjectivity of historical actors was particularly problematic. The article suggests that Frege’s writings are interesting for the philosopher of history for at least two reasons: first, his work is clarificatory when considering the key features that historical propositions must have to count as objective knowledge, and second, it foreshadowed the issues with historicity that analytical philosophy experienced in the twentieth century. It concludes that the problems Frege raised exposed the need for a new concept of inter-subjectivity to replace his own Platonic foundation for objectivity. (shrink)

In this paper we continue the investigation carried out in Albuquerque et al. (2018) on assertional logics and their relation with the Frege hierarchy, through the notions of relative point-regularity and relative congruence orderability. We provide new characterizations for the classes of logics within the Frege hierarchy under the underlying assumption of assertionality. In particular, an assertional logic S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}$$\end{document} is fully Fregean if and only if the class AlgS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} (...) \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsf{Alg}}{\mathcal {S}}$$\end{document} is congruence orderable. Moreover, an assertional logic S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}$$\end{document} is protoalgebraic if and only the class AlgS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsf{Alg}}{\mathcal {S}}$$\end{document} is point-regular. Finally, we introduce a new notion of relative strong congruence orderability and prove that the class AlgS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textsf{Alg}}{\mathcal {S}}$$\end{document} satisfies this property if and only if the intrinsic variety V(S)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {V}({\mathcal {S}})$$\end{document} is congruence orderable. As a consequence, we prove a sufficient condition for the variety problem in AAL. (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)

We compare Fregean theorizing about sets with the theorizing of an ontologically non-committal, natural-deduction based, inferentialist. The latter uses free Core logic, and confers meanings on logico-mathematical expressions by means of rules for introducing them in conclusions and eliminating them from major premises. Those expressions (such as the set-abstraction operator) that form singular terms have their rules framed so as to deal with canonical identity statements as their conclusions or major premises. We extend this treatment to pasigraphs as well, in (...) the case of set theory. These are defined expressions (such as ‘subset of’, or ‘power set of’) that are treated as basic in the lingua franca of informal set theory. Employing pasigraphs in accordance with their own natural-deduction rules enables one to ‘atomicize’ rigorous mathematical reasoning. (shrink)

Following logicism, Frege famously held that logic is a science on its own. Particularly, he held the informativity thesis, viz., that logic is a science because it is deductively informative. This paper aims to understand Frege’s informativity thesis and its connection with the conception of logic as science. For such, it focuses on some features of Frege’s philosophy that are key for understanding this connection, particularly his conception of analyticity, the role of judgments in inferential reasoning, and the use of (...) decomposition of functions as a means for concept-formation, which Frege names as fruitful definitions. Keywords: Gottlob Frege; Informativity of Logic; Judgements; Fruitful Definitions. (shrink)

Frege's attempt to provide a foundation for the possibility of language and communication, like Kant's attempt to provide a foundation for the possibility of knowledge, fails to provide us with something absolute and foundational in a fixed sense. However, both these philosophers succeed in showing something about necessity that can be preserved independently of their absolutisms. Part III of this paper will provide reasons for accepting this thesis, while Parts I and II will provide an expository background on Frege's view (...) in preparation for supporting the thesis. (shrink)

info:eu-repo/semantics/publishedVersion.

Neo-Fregean logicists claim that Hume’s Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A long-standing 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)

It is widely recognized that Frege's systematic conception of science has a major impact on his work. I argue that central to this conception and its impact is Frege's Simplicity Requirement that a scientific system must have as few primitive truths as possible. Frege states this requirement often, justifies it in several ways, and appeals to it to motivate important aspects of his broader views. Acknowledging its central role illuminates several aspects of his work in new ways, including his treatment (...) of truth values, logical concepts, and the dependence relations that induce a ‘natural order’ of truths. (shrink)

Part one of this two-part paper looked at the modern semantic developments underlying Gilson’s innovative and highly influential semantic theory in Being and Some Philosophers (BSP)—the existential neutrality of the copula, the distinction between predication and some positing or “thetic” function of judgment, and the distinction between predication and assertion. The present part of this paper offers a rereading of Gilson’s work in light of this modern backdrop. It argues that Gilson’s BSP, rather than being a purely historical exegesis of (...) the logical writings of a thirteenth-century friar, is a work of original philosophizing inspired by Aquinas, but principally engaged with modern debates. In particular, it advances a Brentano- inspired reading of Aquinas in place of Maritain’s Pfänder-inspired reading of him. Rereading Gilson in his historical setting clarifies the meaning and implication of many phrases and theses that have become commonplaces in philosophical discourse, in part because of Gilson’s work. (shrink)

Gilson’s Being and Some Philosophers (BSP) has been widely influential well beyond Thomistic circles, but its modern historical sources and logical consequences call for further investigation. The first part of this two-part article explores three modern semantic assumptions or principles without which BSP’s innovated theory of existential judgment cannot be fully appreciated—the existential neutrality of the copula ubiquitous among modern logicians; Kant’s introduction of a positing or “thetic” function of judgment, the understanding of which evolved in nineteenth-century logic; and the (...) distinction between predication and assertion, generally accepted by late nineteenth century logicians. Part two of this paper offers a rereading of Gilson’s BSP as an implicit critique of and alternative to Maritain’s synthesis of Aquinas with these modern developments. (shrink)

This paper aims to construct a picture of knowledge out of Frege’s comments on truth, judgment, assertion, and knowledge. Frege takes truth to be an object, and the act of judgment to be the act of non-judgmental identification of truth qua an object with the reference of a sentence. For him, the propositional knowledge that p is the non-propositional knowledge of the identity between truth and |p|. Propositional knowledge thusly understood is produced by our knowledge of truth qua an object, (...) which is constituted by our abilities to identify truth as such. The Fregean picture of knowledge provides a tight connection between knowledge-how, objectual knowledge, and propositional knowledge. (shrink)

The given article provides some views on the development of the problems of formalization and representation of a notion of «sense», which is being conceptualized by an outstanding German philosopher Gottlob Frege, and what’s mainly being analyzed and elaborated in his work “On Sense and Reference”. The key critical theories of understanding the notion of «sense» according to Frege are characterized in the conceptual aspect, that is, a paradigmatic concept that does not receive unambiguous fixation in further developments of logicians (...) and philosophers in the analytic philosophy, but it is assumed as a functional multifaceted concept. In particular, we find examples of conceptualization in the classics of such analytical philosophers as L. Wittgenstein, B. Russell, R. Carnap, and A. Church. Some of the modern works on this topic are analyzed, as well as some classical works in which, opinionate, the directions of the sense conceptualization in the Frege’s works were mostly reflected. We come to the conclusion about the presence of four basic conceptualizations, among which there are two groups: the exclusively logical ones, and the semiotics. Hence, we diagnose the need to deep down the study of the socio-cultural context of works of the analytical philosophy due to the topic. The possibility of the entry of analytical methods into the interdisciplinary study of the problem of sense is fixed there. Method of conceptualization itself is considered and demonstrated by examples, which, on the one hand, are necessary for more detailed study of the development of ideas of analytical philosophy, and, on the other hand, as a source of rejection of irrelevant to the classical philosophy ideas and views. The proximity of the conceptualized semantic refinement on the paradigm method is investigated, from which it is concluded that conceptualization acts as a certain climate for new areas: formalization and representation. A separate consideration of these parts makes obvious the relationship between concepts of meaning and process of its gradual separation in the works of those analytical philosophers who has been developing this topic. (shrink)

The following dissertation explains the psychologism debate as it played out in the 19thand early 20th Centuries and then shows how Martin Heidegger radicalized the debate by undermining its key themes and assumptions. First, I explain each side of the psychologism debate, starting with the psychologicists. I explore the philosophies of Jakob Friedrich Fries and John Stuart Mill in order to encapsulate the full spectrum of psychologism in the 19th Century, from Neo-Kantian to British Empiricist. The investigation will show a (...) set of common themes within psychologism, such as the grounding of logic in the constitution of the human subject, a reliance on introspection, and the prioritization of epistemology over metaphysics. I then turn to the logicists, focusing on Gottlob Frege and Edmund Husserl. I start by explaining two influential arguments against psychologism: that it treats logical objects as mental entities and that it devolves into relativism. I analyze how both Frege and Husserl developed alternate explanations for the foundations of logic, where Frege grounded logic in the relationships of objects in an ideal third realm and Husserl founded them on the relationships between the ideal objects of intentional experience. I end the section with a brief explanation of logicism’s relationship to the themes highlighted in the prior part, namely that logicism founded logic on the object side of the subject-object dichotomy but agreed with the prioritization of epistemology. The final part of the dissertation will be Heidegger’s critique and radicalization of the psychologism debate and his own understanding of logic. It starts with his critique of Husserl, v followed by a Heideggerian critique of Frege, and finally his critique of psychologism. I show that Heidegger undermined the debate by arguing that theoretical modes of engagement with the world were derivative of our more fundamental ontological structure, that the subject-object dichotomy is illegitimate, and that the psychologism debate relies on a flawed understanding of truth. Afterward, I explain Heidegger’s philosophy and construct a Heideggerian understanding of logic based on his understanding of truth and the logos. (shrink)

In the Tractatus, Wittgenstein conceives of tautology as 'saying nothing'. More precisely, he holds -- or so this essay contends -- that it says nothing in virtue of possessing a zero quantity of sense. Insofar as it is the limit of a series of propositions of diminishing quantity of sense, tautology resembles a degenerate conic section. But it also resembles the result of a summing together of equal and opposite linear vector quantities. Both of these models shape Wittgenstein's conception of (...) tautology in the Tractatus, though they are not fully reconciled. Many of the puzzling features of the Tractatus's view of logic arise from this failure to affect the needed reconciliation. These points are argued for by working through the development of Wittgenstein's views on these matters in the Wartime Notebooks and other pre-Tractatus writings. (shrink)

In this paper, Dominique Pradelle focuses on the act of symbolic institution that establishes the alphabet of signs: Is it a matter of the free establishment of an operative meaning or is it governed by necessary laws? In the latter case, where did they come from? The author shows that these laws come from the upper layers of meaning and ideal objects—which has the essential consequence, in transcendental phenomenology, of removing the Husserlian concept of transcendental constitution from the paradigm of (...) a free production of the meaning and of the objects. (shrink)

Le problème de Frege-Geach est un problème qui se pose pour les théories selon lesquelles les jugements normatifs n’ont pas de contenu cognitif, mais expriment plutôt des états mentaux non cognitifs. Dans cet article, je présente le problème de Frege-Geach ; j’examine certaines stratégies existantes pour l’aborder dans sa forme traditionnelle ; et je me demande enfin si un problème de Frege-Geach se pose pour les raisons, et si l’usage des raisons peut mener à une solution. J’esquisse une réponse positive (...) à cette question. (shrink)

This volume does what it says on the tin: it collects Eva Picardi’s (1984-2017) essays on themes in Frege’s philosophy. The first part, ‘Frege in Context: Logic.

Criticizing psychologism about logic in the Foreword of Grundgesetze, Frege examines an answer to the question of how we can justify our acknowledgment of logical axioms as true—the logical laws that cannot be proved from other laws. The answer he entertains states that we cannot reject logical axioms if we do not want to give up our judgment altogether. Suspending his judgment about this answer, Frege points out that it is still compatible with his anti-psychologist conception of logic. There are (...) two issues with this paragraph. First, it is difficult to read this paragraph consistently. Second, it is not clear why Frege brings up this particular view regarding the justification for logical axioms in the middle of his criticism of psychologism. This paper develops a consistent interpretation of this paragraph. The view on the justification for logical axioms, which Frege is dealing with, is neo-Kantian. In the paragraph, Frege is criticizing a psychologistic misunderstanding of the neo-Kantian justification for logical axioms. The way Frege explains away this misunderstanding shows that he takes the neo-Kantian justification for logical axioms to be a non-evidential epistemic warrant if it is an epistemic warrant. It is this notion of non-evidential epistemic warrant that Frege suspends his judgment about. Thus, the paragraph shows that Frege has a reason to have reservations about neo-Kantianism. Furthermore, it turns out that the sole issue between Frege and psychological logicians is whether truth is objective, which provides a different way to explain away a much-discussed tension between Begriffsschrift and Frege's anti-psychologism. (shrink)

The letters that were found after years are the evidence of hard intellectual work that had been conducted under very infavourable conditions. They refer to the time when Wittgenstein was writing "Tractatus" while Frege was working on his articles "Thought" and "Negation". Correspondence between Frege and Wittgenstein prove the common will of communication and mutual understanding. Thus remains the question why those two close and well known to each other thinkers have not realized that willingness.

Die geschichte der macht.--Ewige menschheit und wandel der reiche.--Ahnung der form.--Blick auf das werden.

Moral expressivism suggests that 1) moral sentences lack truth conditions and 2) our purpose in asserting moral sentences is to express non-cognitive attitudes such as desires, approval, or disapproval. Moral expressivism meets a fundamental challenge, known as the Frege-Geach problem. Sentences that express moral judgments can form part of semantically complex sentences. “P” (a moral sentence) contradicts “~P”, and “Q” follows logically, by modus ponens, from (1) “P” and (2) “if P, then Q”. Geach argued that noncognitivists are committed to (...) denying that moral predicates mean the same thing in embedded contexts as they do in unembedded sentences (atomic sentences). If “P” does not mean the same as the antecedent of (2), the argument would be invalid. The problem is that the above-mentioned argument is obviously valid. Blackburn has argued that the complex sentence expresses a ‘higher-order’ attitude toward the attitudes expressed by the smaller sentences which make it up. If we accept the premises of a valid argument but deny its conclusion our attitudes clash in the same way that they do if we both believe that P and ~P. Blackburn's meta-attitudes approach faces several problems. Someone who endorses the premises but denies the conclusion of the valid argument commits himself to a moral inconsistency, not a logical one. In addition, uttering both ‘P’ and ‘~P’ seem to be inconsistent but expressivism cannot explain the inconsistency between these two obviously inconsistent sentences. Blackburn's strategy of inventing a new attitude, such as tolerance, is also unable to solve this problem. Introduction Moral expressivism suggests that 1) moral sentences lack truth conditions (negative claim) and 2) our purpose in asserting moral sentences is to express non-cognitive attitudes such as desires, approval, or disapproval (positive claim). Moral expressivism meets a fundamental challenge known as the Frege-Geach problem (Geach, 1960, 1965). In the following, the Frege-Geach problem and Simon Blackburn’s response to it will be examined. Blackburn has developed several different answers to the Frege-Geach Problem In this article, only the answer which appeals to the higher-order attitudes (Blackburn, 1984) will be examined, and it will be shown that this expressivist recipe can not solve the Frege-Geach problem. The Frege-Geach problem The Frege-Geach problem is the idea that moral predicate functions as a ‘logical’ one so that sentences containing this predicate enter into logical relations with other sentences. Sentences that express moral judgments can form part of semantically complex sentences. “P” (a moral sentence) contradicts “~P”, and “Q” follows logically, by modus ponens, from (1) “P” and (2) “if P, then Q”. Geach argued that noncognitivists are committed to denying that moral predicates mean the same thing in embedded contexts as they do in unembedded sentences (atomic sentences). They cannot explain that the meaning of complex sentences is in terms of the meanings of their parts. They cannot explain why modus ponens arguments are always valid. Arguments of the form of modus ponens, according to expressivism, commit a fallacy of equivocation; because it appears to imply that “P” does have different meanings in premise (1) and in the antecedent of premise (2). “P” is asserted in premise (1) and remains unasserted in premise (2). The utterance of (2) does not seem to express approval or disapproval of “P”. If “P” does not mean the same as the antecedent of (2), the argument would be invalid. The problem is that the abovementioned argument is obviously valid. Geach (Geach, 1960, p. 223) calls this point about assertion the Frege point who was the first logician who made the point in his distinction between the sense and the reference of a sentence. “A thought may have just the same content whether you assent to its truth or not; a proposition may occur in discourse now asserted, now unasserted, and yet be recognizably the same proposition” (Geach, 1965, p. 449). The higher-order attitudes Blackburn denies that any of his claims entail valid moral arguments are invalid. He employs an account of what we are doing when we use ethical sentences in terms of expressing meta-attitudes about a moral sensibility; The complex sentence expresses a ‘higher-order’ attitude toward the attitudes expressed by the smaller sentences which make it up. There are logical relationships that exist among attitudes as there are among beliefs. If we accept the premises of a valid argument but deny its conclusion, our attitudes clash in the same way they do if we both believe that P and ~P. So we can explain the logical validity of the moral argument via the attitudes one can hold without clash; anyone approving (1) “P” and (2) “if P, then Q” must hold the consequential approval of “Q”. If he does not, his attitudes clash. The inconsistency originates from the clash of attitudes (failing to do something which one has committed oneself to do). Blackburn says that such a clash would involve a “fractured sensibility which cannot itself be an object of approval” because “such a sensibility cannot fulfill the practical purposes for which we evaluate things” (Blackburn, 1984, p. 195). Logical conflict and moral conflict There is a most important problem with this strategy for Blackburn’s expressivism. The clash of attitudes can only show moral conflict and moral conflict does not necessarily lead to logical inconsistency. What needs to be explained in the moral argument is a logical inconsistency. Moreover, as Mark van Roojen (1996, pp. 21-320) has pointed out, higher-order-attitudes logic commits us to believe in logical inconsistency where there is no such inconsistency. He considers the origin of this problem to be the confusion between logical inconsistency and practical inconsistency (van Roojen, 1996, p. 332). The negation problem The other problem for Blackburn’s expressivism was originally raised by Unwin (Nicholas Unwin, 1999, 2001). This problem is known as the ‘negation problem’. As Schroeder points out (Schroeder, 2010, p. 134), the negation problem is the same as the Frege-Geach problem, which has received more attention in recent years than its conditional form, which is presented in the modus ponens argument. The most important semantic property of the negation operator in descriptive sentences is that it makes the negated sentence inconsistent with the main sentence, in the sense that for any sentence ‘P’, ‘P’ is inconsistent with ‘~P’. According to expressivism, “murdering is wrong” (P) is used to express the attitude of disapproval towards the act of murdering. But what is expressed by “murdering is not wrong” (~P)? To utter both ‘P’ and ‘~P’ seems to be inconsistent but expressivism cannot explain the inconsistency between these two obviously inconsistent sentences; because it cannot tell us which state of mind is expressed in the moral sentences like “murdering is not wrong.” To avoid the problem, Blackburn (Blackburn, 1988, p. 189) introduces a new attitude, such as tolerance, and says that “murdering is wrong” expresses disapproval of murdering, and “murdering is not wrong” expresses tolerance of murdering. “Tolerance toward P (T!p) is equivalent to not hooraying ~p, that is, not booing p.” (Blackburn, 1988, p. 189). But there is a real difference between not accepting something and actually accepting its negation (Unwin, 1999, p. 341). Conclusion According to Blackburn, We are expressing our attitudes about a moral sensibility when we use ethical sentences. The question now is whether Blackburn can explain the validity of the inference from the premises of the argument to its conclusion through this strategy. This article has shown that Blackburn’s meta-attitudes approach faces several problems. Someone who endorses the premises but denies the conclusion of the valid argument commits himself to a moral inconsistency, not a logical one. In addition, expressivism has insufficient structure to account for the various ways in which a moral sentence can be negated because it cannot tell us which state of mind is expressed in moral sentences like “murdering is not wrong.” Blackburn’s strategy of inventing a new attitude, such as tolerance, is also unable to solve this problem. (shrink)

Wittgenstein afirma no Tractatus que a teoria das classes é supérflua na Matemática e que isso está relacionado ao fato de que a generalidade exigida pela Matemática não é “acidental” (TLP 6.031). O objetivo deste texto é elucidar essa afirmação chamando a atenção para o que, seguindo Gregory Landini, tomaremos como uma forma de Logicismo compartilhada por Frege e Russell. Esta forma de Logicismo tem dois princípios básicos, a saber: o uso de uma teoria lógica cujas variáveis estruturadas incorporam o (...) que hoje chamamos de teoria dos tipos simples e a análise de atribuições numéricas em termos de afirmações sobre conceitos ou atributos que empregam conceitos de ordem superior exatamente análogos aos chamados quantificadores ‘numericamente definidos’. Argumentamos que é esse arcabouço teórico que Wittgenstein está rejeitando em 6.031 e não apenas o uso da Teoria dos Conjuntos na Matemática. Também é defendido que a noção de classe ainda tem um papel essencial a desempenhar no Tractatus.AbstractWittgenstein claims in the Tractatus that the theory of classes is superfluous in Mathematics and that this is related to the fact that the generality required by Mathematics is not “accidental” (TLP 6.031). My aim in this paper is to elucidate this claim by calling attention to what, following Gregory Landini, I refer to as a form of Logicism shared by Frege and Russell. This form of Logicism has two main tenets, namely:the use of a logical theory whose structured variables embody what we nowadays call the simple theory of types and the analysis of number ascriptions in terms of assertions about concepts or attributes which employ higher-order concepts exactly analogous to so-called ‘numerically definite’ quantifiers. It is argued that it is this shared theoretical framework that Wittgenstein is rejecting in 6.031 and not just the use of Set Theory in Mathematics. It is also argued that the notion of class still has an essential role to play in the Tractatus. (shrink)

Nosso objetivo é responder a uma questão proposta por Richard Heck no artigo “Formal Arithmetic Before Grundgesetze”. Heck indaga a respeito das razões pelas quais Frege levou quase oito anos para honrar suas promessas de concluir seu grandioso projeto de fundamentar a matemática na lógica. Embora Heck tenha fornecido algumas respostas, pensamos que uma discussão filosófica mais adequada ainda pode ser oferecida. Este artigo tentará preencher essa lacuna apresentando o que entendemos ser o problema central enfrentado por Frege em _Die (...) Grundlagen der Arithmetik_: a falta de um critério unificado para fixar o significado das proposições de identidade da matemática. Acreditamos que o filósofo alemão finalmente tenha decidido preencher essa lacuna, fornecendo uma nova interpretação para o conceito de “extensão”, uma que acrescentasse alguns refinamentos importantes a sua concepção anterior. O novo conceito assim construído permitiu a Frege unificar seu tratamento das proposições de identidade, incluindo em seu sistema um critério universal para decidir a verdade de qualquer proposição de identidade apoiado por sua famosa lei básica V. Assim, nossa alegação será que a resistência de Frege e as dúvidas sobre a inclusão do axioma V como uma lei lógica em seu sistema foram a causa primária desse atraso. (shrink)

Questo volume ripropone sei articoli fondamentali di Gottlob Frege nella versione italiana corredata dalla guida alla lettura di Carlo Lazzerini, che ha fatto di questa raccolta un’introduzione particolarmente felice alla logica moderna: un testo classico molto intenso viene proposto in modo tanto semplice da poter essere la porta di ingresso nella logica e nella filosofia della logica del Novecento anche per lettori che non ne hanno nessuna nozione e che ne iniziano da qui lo studio. Gli articoli di Frege, composti (...) in epoche diverse e caratterizzati da un’evoluzione verso una maggiore sistematicità, formano nel complesso un testo organico e seminale per lo sviluppo della logica matematica, della semantica e della filosofia del linguaggio del Novecento, come si constata anche ad una prima lettura, quando si tenga conto dell’influenza da un lato della geniale intuizione di Frege della distinzione nel significato dei nomi, dei concetti e delle proposizioni fra «senso» e «denotazione», e dall’altro della sua scoperta del valore funzionale del concetto. La lettura di Frege è un’occasione d’incontro tra cultura filosofica e scientifica. Se nell’antichità la logica (come, ad esempio, anche la fisica e la matematica) era intesa come disciplina filosofica, col volgere del tempo la logica è diventata sempre più una scienza autonoma, con un oggetto ben definito e finalità pratiche preminenti. Anzi, proprio la «logica matematica» promossa da Frege ha dato maggiore impulso a questo distacco, configurandosi sempre più come una tecnica precisa, un calcolo, come la forma elementare della matematica o come un ramo particolare della matematica stessa. Eppure, che gli scritti di Frege abbiano un valore filosofico, si avverte anche ad una lettura superficiale, nonostante l’impressione di tecnicismo che gli argomenti da lui preferiti possono dare. La ricerca matematica di Frege, il suo interesse per il problema dei «fondamenti» ed il suo inevitabile sconfinare di continuo nella logica, non sono mai promossi né da esigenze di applicazione, né di costruzione «all’in su» della scienza, ma sempre da un’esigenza di perfezionamento teoretico, verso il profondo. Dalla matematica ai fondamenti e dai fondamenti alla logica: se questo è l’itinerario della ricerca di Frege la sua qualità filosofica non è data dalla direzione dell’itinerario stesso, dal fatto che approda alla logica, ma dal modo della ricerca, che gli fa trascurare le possibilità costruttive della scienza (e da ciò viene la legittima diffidenza dei matematici suoi contemporanei), per tener dietro alle prospettive teoretiche che si aprono, una dopo l’altra, in direzioni del tutto imprevedibili. (shrink)

This chapter clarifies that it was the works Giuseppe Peano and his school that first led Russell to embrace symbolic logic as a tool for understanding the foundations of mathematics, not those of Frege, who undertook a similar project starting earlier on. It also discusses Russell’s reaction to Peano’s logic and its influence on his own. However, the chapter also seeks to clarify how and in what ways Frege was influential on Russell’s views regarding such topics as classes, functions, meaning (...) and denotation, etc., and summarizes the correspondence between Frege and Russell and the light it sheds on the philosophical logic of both. (shrink)

Russell corresponded with Sir Michael Dummett (1925–2011) between 1953 and 1963 while the latter was working on a book on Frege, eventually published as Frege: Philosophy of Language (1973). In their letters they discuss Russell’s correspondence with Frege, translating it into English, as well as Frege’s attempted solution to Russell’s paradox in the appendix to vol. 2 of his Grundgesetze der Arithmetik. After Dummett visited the University of Münster to view Frege’s Nachlaß, he sent reports back to Russell concerning both (...) the philosophical materials Frege left behind, as well as information from Frege’s journal revealing his anti-semitic political opinions. Their interaction contains interpretive conjectures and insights on Dummett’s side, and some dark humor on Russell’s. (shrink)

This chapter contains sections titled: Introduction Frege on Function, Concept and Object Sinn (Sense) and Bedeutung (Reference) Identity Statements and Bearerless Names: Russell's View of Names as Associated with Descriptions Names and Communication Russell's Theory of Descriptions Indirect Discourse Conclusion.

Of all philosophers, it is Frege whom Wittgenstein held in greatest esteem. The aim of philosophy, Wittgenstein wrote in the Tractatus, “is the logical clarification of thoughts”, a characterization that might well be taken to be true of Frege's philosophy. The clarity that Wittgenstein saw as an important philosophical virtue is arguably nowhere better illustrated than in Frege's writings, even if one disagrees with the substantive philosophical claims that Frege makes. Peter Geach reports a remark that Wittgenstein made to him (...) when they were discussing Frege's essay “On Concept and Object”. Wittgenstein may have envied Frege's style, but he nevertheless felt it had a strong effect on his own writing. Frege is explicitly cited as an influence on the Tractatus, but although he is rarely mentioned by name in his later writings, his views continued to be a major source of inspiration to the very end of Wittgenstein's life. (shrink)

This chapter deals with the following questions: What does Michael Dummett demand of a “systematic” theory of meaning, and what understanding of Frege's “level of sense” leads him to conclude that, if Wittgenstein is correct in denying that there is such a level, then no systematic theory of meaning is possible? For Dummett, an understanding of the meaning side of language is not “systematic” if it must hold that a sentence is understood only because it has been previously learned as (...) a whole. To reconstruct how Dummett arrives at Frege's level of “sense” as crucial for a systematic understanding of complexity, we can begin with a hypothetical concrete situation and proceed step by step to rule out what is not meant by “meaning” when we speak of it as being systematically developed. (shrink)