Dans cet article, j’essaie de montrer que le dépassement et le rejet du dogmatisme sont un aspect décisif du changement dans la pensée de Wittgenstein qui a eu lieu au début des années 30, quand il commence à mettre en valeur l’autonomie de la grammaire du langage et à parler d’images grammaticales et de jeux de langage en tant qu’objets de comparaison. En examinant certains traits fondamentaux de ce changement, je mettrai en évidence l’impulsion et les idées décisives que Wittgenstein (...) a reçues des nouveaux développements dans les mathématiques et la science naturelle du début du 20e siècle.It will be argued that the overcoming and avoiding of dogmatism is a decisive feature of the change in Wittgenstein’s thinking that takes place in the beginning of the 1930’s when he starts to emphasise the autonomy of the grammar of language and to talk about grammatical pictures and language games as objects of comparison. By examining certain crucial features in this change in Wittgenstein’s thinking, it will be shown that he received decisive impulses and ideas from new developments in mathematics and natural science in the early 20th century. (shrink)

It is a remarkable fact that Hilbert's programmatic papers from the 1920s still shape, almost exclusively, the standard contemporary perspective of his views concerning (the foundations of) mathematics; even his own, quite different work on the foundations of geometry and arithmetic from the late 1890s is often understood from that vantage point. My essay pursues one main goal, namely, to contrast Hilbert's formal axiomatic method from the early 1920s with his existential axiomatic approach from the 1890s. Such a contrast illuminates (...) the circuitous beginnings of the finitist consistency program and connects the complex emergence of existential axiomatics with transformations in mathematics and philosophy during the 19th century; the sheer complexity and methodological difficulties of the latter development are partially reflected in the well known, but not well understood correspondence between Frege and Hilbert. Taking seriously the goal of formalizing mathematics in an effective logical framework leads also to contemporary tasks, not just historical and systematic insights; those are briefly described as “one direction” for fascinating work. (shrink)

The idea that logic and reasoning are somehow related goes back to antiquity. It clearly underlies much of the work in logic, as witnessed by the development of computability, and formal and mechanical deductive systems, for example. On the other hand, a platitude is that logic is the study of correct reasoning; and reasoning is cognitive if anything Is. Thus, the relationship between logic, computation, and correct reasoning makes an interesting and historically central case study for mechanism. The purpose of (...) this article is to begin the articulation of this relationship, pointing out its sources and its limitations. (shrink)

Moritz Pasch (1843ber neuere Geometrie (1882), in which he also clearly formulated the view that deductions must be independent from the meanings of the nonlogical terms involved. Pasch also presented in these lectures the main tenets of his philosophy of mathematics, which he continued to elaborate on throughout the rest of his life. This philosophy is quite unique in combining a deductivist methodology with a radically empiricist epistemology for mathematics. By taking into consideration publications from the entire span of Paschs (...) highly original, but virtually unknown, philosophy of mathematics is presented. (shrink)

We explore the technical details and historical evolution of Charles Peirce's articulation of a truth table in 1893, against the background of his investigation into the truth-functional analysis of propositions involving implication. In 1997, John Shosky discovered, on the verso of a page of the typed transcript of Bertrand Russell's 1912 lecture on ?The Philosophy of Logical Atomism? truth table matrices. The matrix for negation is Russell's, alongside of which is the matrix for material implication in the hand of Ludwig (...) Wittgenstein. It is shown that an unpublished manuscript identified as composed by Peirce in 1893 includes a truth table matrix that is equivalent to the matrix for material implication discovered by John Shosky. An unpublished manuscript by Peirce identified as having been composed in 1883?1884 in connection with the composition of Peirce's ?On the Algebra of Logic: A Contribution to the Philosophy of Notation? that appeared in the American Journal of Mathematics in 1885 includes an example of an indirect truth table for the conditional. (shrink)

Toisaalta ennennäkemätön äärettömien joukko-opillisten menetelmien hyödyntäminen sekä toisaalta epäilyt niiden hyväksyttävyydestä ja halu oikeuttaa niiden käyttö ovat ratkaisevasti muovanneet vuosisatamme matematiikkaa ja logiikkaa. Tämän kehityksen vaikutus nykyajan filosofiaan on myös ollut valtaisa; merkittävää osaa siitä ei voi edes ymmärtää tuntematta sen yhteyttä tähän matematiikan ja logiikan vallankumoukseen. Lähestymistapoja, jotka tavalla tai toisella hyväksyvät äärettömän matematiikan ja perinteisten logiikan sääntöjen (erityisesti kolmannen poissuljetun lain) soveltamisen myös sen piirissä, on tullut tavaksi kutsua klassiseksi matematiikaksi ja logiikaksi erotuksena nämä hylkäävistä radikaaleista intuitionistisista ja (...) konstruktivistisista opeista. (shrink)

Towards the end of his 1930 paper on boundary numbers and domains of sets Zermelo briefly discusses the questions of consistency and of the existence of an unbounded sequence of strongly inaccessible cardinals, deferring a detailed discussion to a later paper which never appeared. In a report to the Emergency Community of German Science from December 1930 about investigations in progress he mentions that some of the intended extensions of these topics had been worked out and were nearly ready for (...) publication. Using manuscripts from his Nachlass, we exhibit essential features of these extensions. (shrink)

The paper examines the roots of Husserlian phenomenology in Weierstrass's approach to analysis. After elaborating on Weierstrass's programme of arithmetization of analysis, the paper examines Husserl's Philosophy of Arithmetic as an attempt to provide foundations to analysis. The Philosophy of Arithmetic consists of two parts; the first discusses authentic arithmetic and the second symbolic arithmetic. Husserl's novelty is to use Brentanian descriptive analysis to clarify the fundamental concepts of arithmetic in the first part. In the second part, he founds the (...) symbolic extension of the authentically given arithmetic with stepwise symbolic operations. In the process of doing so, Husserl comes close to defining the modern concept of computability. The paper concludes with a brief comparison between Husserl and Frege. While Frege chose to subject arithmetic to logical analysis, Husserl wants to clarify arithmetic as it is given to us. Both engage in a kind of analysis, but while Frege analyses within Begriffsschrift, Husserl analyses our experiences. The difference in their methods of analysis is what ultimately grows into two separate schools in philosophy in the 20th century. (shrink)

1. Setting the stage Dinda Gorlée’s seminal On Translating Signs explores translation of texts under the guise of ‘semio-translation.’ In her previous book, Semiotics and the Problem of Translation, Gorlée originally coined the term semiotranslation, which subsequently gained some notoriety in translation theory circles. She sees no need to give up the term, in fact including it in hyphenated form in the subtitle of the book under review. For, semio-translation ‘retains an ongoing excitement and creates a niche in semiotic and (...) translationtheoretical scholarship’. The author perpetuates its use with the hope that ‘it will remain a vital and fruitful challenge to future researchers everywhere, especially those who pay no mind to the noise of passing scholarly genres and fashions’. Thus OTS can be effectively read as a continuation of SPT. (shrink)

Both Hilbert's axiomatics and Cassirer's philosophy of symbolic forms have their roots in Leibniz's idea of a 'universal characteristic,' and grow on Hertz's 'principles of mechanics,' and Dedekind's 'foundations of arithmetic'. As Cassirer recalls in the introduction to his Philosophy of Symbolic Forms, it was the discovery of the analysis of infinity that led Leibniz to focus on "the universal problem inherent in the function of symbolism, and to raise his 'universal characteristic' to a truly philosophical plane." In Leibniz's view, (...) the logic of 'things' cannot be separated from the logic of 'signs,' as "the sign is no mere accidental cloak of the idea, but its necessary and essential organ."Every 'law' of .. (shrink)

In “Unarticulated Comparison Classes” 2018 [2009], Richard Vallée adopts John Perry’s (2012 [2001]) reflexive-referential theory of meaning and content as well as his concept of unarticulated constituents (Perry 1986) to deal with certain context-sensitive elements of the truth-conditions of statements containing relative gradable predicates. I am sympathetic both with the general framework and with the assumption that unarticulated constituents are involved in the truth-conditions of bare positives such as “Monica is tall.” I do not share, however, Vallée’s main conclusions on (...) the examples he provides as pre-theoretical evidence. This leads me to disagree with some details of his proposal for the semantics and pragmatics of relative gradable adjectives. (shrink)

In this paper I consider three mathematicians who allowed some role for menial processes in the foundations of their logical or mathematical theories. Boole regarded his Boolean algebra as a theory of mental acts; Cantor permitted processes of abstraction to play a role in his set theory; Brouwer took perception in time as a cornerstone of his intuitionist mathematics. Three appendices consider related topics.

Post's Nachlass has recently been made available to the public in an archive in the U.S.A. After a short summary of his life and career, this article indicates the character and content of the manuscripts, and their significance is assessed. Two short passages are transcribed; and. as a separate item, a paper of the 1930s on the paradoxes is reproduced.

There has been considerable discussion in the literature of one kind of identity problem that mathematical structuralism faces: the automorphism problem, in which the structure is unable to individuate the mathematical entities in its domain. Shapiro (Philos Math 16(3):285–309, 2008) has partly responded to these concerns. But I argue here that the theory faces an even more serious kind of identity problem, which the theory can’t overcome staying within its remit. I give two examples to make the point.

It is widely assumed that Russell's problems with the unity of the proposition were recurring and insoluble within the framework of the logical theory of his Principles of Mathematics. By contrast, Frege's functional analysis of thoughts (grounded in a type-theoretic distinction between concepts and objects) is commonly assumed to provide a solution to the problem or, at least, a means of avoiding the difficulty altogether. The Fregean solution is unavailable to Russell because of his commitment to the thesis that there (...) is only one ultimate ontological category. This, combined with Russell's reification of propositions, ensures that he must hold concepts and objects to be of the same logical and ontological type. In this paper I argue that, while Frege's treatment of the unity of the proposition has immediate advantages over Russell's, a deeper consideration of the philosophical underpinnings and metaphysical consequences of the two approaches reveals that Frege's supposed solution is, in fact, far from satisfactory. Russell's repudiation of the Fregean position in the Principles is, I contend, convincing and Russell's own position, despite its problems, conforms to a greater extent than Frege's with common sense and, furthermore, with certain ideas which are central to our understanding of the origins of the analytical tradition. (shrink)

It can be argued that only the equational theories of some sub-elementary function algebras are finitistic or intuitive according to a certain interpretation of Hilbert's conception of intuition. The purpose of this paper is to investigate the relation of those restricted forms of equational reasoning to classical quantifier logic in arithmetic. The conclusion reached is that Edward Nelson's ‘predicative arithmetic’ program, which makes essential use of classical quantifier logic, cannot be justified finitistically and thus requires a different philosophical foundation, possibly (...) as a restricted form of logicism. (shrink)

I attribute an 'intensional reading' of the second incompleteness theorem to its author, Kurt G del. My argument builds partially on an analysis of intensional and extensional conceptions of meta-mathematics and partially on the context in which G del drew two familiar inferences from his theorem. Those inferences, and in particular the way that they appear in G del's writing, are so dubious on the extensional conception that one must doubt that G del could have understood his theorem extensionally. However, (...) on the intensional conception, the inferences are straightforward. For that reason I conclude that G del had an intensional understanding of his theorem. Since this conclusion is in tension with the generally accepted view of G del's understanding of mathematical truth, I explain how to reconcile that view with the intensional reading of the theorem that I attribute to G del. The result is a more detailed account of G del's conception of meta-mathematics than is currently available. (shrink)

The papers where Gerhard Gentzen introduced natural deduction and sequent calculi suggest that his conception of logic differs substantially from the now dominant views introduced by Hilbert, Gödel, Tarski, and others. Specifically, (1) the definitive features of natural deduction calculi allowed Gentzen to assert that his classical system nk is complete based purely on the sort of evidence that Hilbert called ?experimental?, and (2) the structure of the sequent calculi li and lk allowed Gentzen to conceptualize completeness as a question (...) about the relationships among a system's individual rules (as opposed to the relationship between a system as a whole and its ?semantics?). Gentzen's conception of logic is compelling in its own right. It is also of historical interest, because it allows for a better understanding of the invention of natural deduction and sequent calculi. (shrink)

This paper discusses the neo-logicist approach to the foundations of mathematics by highlighting an issue that arises from looking at the Bad Company objection from an epistemological perspective. For the most part, our issue is independent of the details of any resolution of the Bad Company objection and, as we will show, it concerns other foundational approaches in the philosophy of mathematics. In the first two sections, we give a brief overview of the "Scottish" neo-logicist school, present a generic form (...) of the Bad Company objection and introduce an epistemic issue connected to this general problem that will be the focus of the rest of the paper. In the third section, we present an alternative approach within philosophy of mathematics, a view that emerges from Hilbert's Grundlagen der Geometrie (1899, Leipzig: Teubner; Foundations of geometry (trans.: Townsend, E.). La Salle, Illinois: Open Court, 1959.). We will argue that Bad Company-style worries, and our concomitant epistemic issue, also affects this conception and other foundationalist approaches. In the following sections, we then offer various ways to address our epistemic concern, arguing, in the end, that none resolves the issue. The final section offers our own resolution which, however, runs against the foundationalist spirit of the Scottish neo-logicist program. (shrink)

This work presents the concepts of translation and conservative translation between logics. By using algebraic semantics we introduce several conservative translations involving the classical propositional calculus and the many-valued calculi of Post and Lukasiewicz.

The paper focuses on Gottlob Frege’s so called Context Principle (CP hereafter), which counts as one of the most controversial points of his philosophy. Due to its importance and centrality in Frege’s thought, a detailed discussion of the principle requires a detailed analysis of almost all aspects of his philosophy. Obviously, such a task cannot be successfully accomplished here. Thus I limit myself to address only two questions concerning the CP: what role does the principle play (in Grundlagen) and how (...) can we interpret it. Addressing the first problem is required in order to address the second. Most authors interpreted CP from the perspective of Frege’s later distinction between sense and reference, which I will call the ‘semantic interpretation’. Although I accept this perspective as valuable and important, I will initially inverse the action and I will try to approach CP, and generally Grundlagen, in a more natural way, contextually, namely setting them in the initial logicist plan of the Begriffschrift. Finally, I will try to provide an interpretation concerning the alleged conflict between CP and Frege’s compositionality thesis such that they could coherently stay together. (shrink)

The middle chapters of Soames’s The World Philosophy Made are briefly summarized and examined. There are some local slips, but globally the work displays an impressive knowledge of and a distinctive viewpoint on a wide range of important intellectual disciplines and their original roots in and continuing connections with philosophy.

What particular privilege has this little Agitation of the Brain which we call Thought, that we must make it the Model of the whole Universe? (Hume, Dialogues Concerning Natural Religion, 1976, p. 168)******...at once it struck me, what quality went to form a Man (sic) of Achievement especially in Literature and which Shakespeare possessed so enormously—I mean Negative Capability, that is when someone is capable of being in uncertainties, Mysteries, doubts without any irritable reaching after fact and reason. (Keats 1959, (...) p. 261)******Die menschliche Vernunft hat das besondere Schicksal in ihrer Gattung ihrer Erkenntnisse: dass sie durch Fragen belästigt wird, die sie nicht abweisen kann; denn sie sind ihr durch die Natur der Vernunft selbst aufgegeben, die sie aber auch nicht beantworten kann; denn sie übersteigen alles Vermögen der menschlichen Vernunft. (Opening lines of the Kritik der reinen Vernunft (1956), A VII; the italics in the text are mine)Human reason has the particular fa .. (shrink)

The aim of this paper is to clarify why propositional logic is Post complete and its weak completeness was almost unnoticed by Hilbert and Bernays, while first-order logic is Post incomplete and its weak completeness was seen as an open problem by Hilbert and Ackermman. Thus, I will compare propositional and first-order logic in the Prinzipien der Mathematik, Bernays’s second Habilitationsschrift and the Grundzüge der Theoretischen Logik. The so called “arithmetical interpretation”, the conjunctive and disjunctive normal forms and the soundness (...) of the propositional rules of inference deserve special emphasis. (shrink)

This paper is concerned with the way different axiom systems for set theory can be justified by appeal to such intuitions as limitation of size, predicativity, stratification, etc. While none of the different conceptions historically resulting from the impetus to provide a solution to the paradoxes turns out to rest on an intuition providing an unshakeable foundation,'each supplies a picture of the set-theoretic universe that is both useful and internally well motivated. The same is true of more recently proposed axiom (...) systems for non-well-founded universes, and an attempt is made to motivate such axiom systems on the basis of an old and respected ‘algebraic’ intuition. (shrink)

The origin of paraconsistent logic is closely related with the argument, ‘from the assertion of two mutually contradictory statements any other statement can be deduced’; this can be referred to as ex contradictione sequitur quodlibet (ECSQ). Despite its medieval origin, only by the 1930s did it become the main reason for the unfeasibility of having contradictions in a deductive system. The purpose of this article is to study what happened earlier: from Principia Mathematica to that time, when it became well (...) established. The two main historical claims that I am going to advance are the following: (1) the first explicit use of ECSQ as the main argument for supporting the necessity of excluding any contradiction from deductive systems is to be found in the first edition of the book Grundzüge der Theoretischen Logik (Hilbert, D. and Ackermann, W. 1928. Grundzüge der Theoretischen Logik. Berlin: Julius Springer Verlag); (2) Łukasiewicz's position regarding the logical constraints against contradictions varies considerably from his studies on the principle of (non-) contradiction in Aristotle, published in 1910 and what is stated in his `authorized lectured notes' on mathematical logic that appeared in 1929. The two texts are: 1) a paper in German (Łukasiewicz, J. 1910. `Über den Satz des Widerspruchs bei Aristotles'. Bulletin International de l'Académie des sciences de Cracovie, Classe d'Histoire et de Philosophie, pp. 15–38) [English translation: Łukasiewicz, J. 1971. ‘On the principle of contradiction in Aristotle’, Review of Metaphysics, XXIV, 485–509]; and 2) a book in Polish. Łukasiewicz, J. 1910. O zasadzie sprzecznosci u Aristotelesa Studium krytyczne, Warsaw: Panstwowe Wydawnictwo Naukowe [German translation: Łukasiewicz, J. 1993. Über den Satz des Widerspruchs bei Aristotles. Hildesheim: Georg Olms Verlag]. The lecture notes were then published as a book (Łukasiewicz, J. 1958. Elementy Logiki Matematycznej. Warszawa: Panstwowe Wydawnictwo Naukowe [PWN] and then translated into English (Łukasiewicz, J. 1963. Elements of Mathematical Logic. Oxford, New York: Pergamon Press/the Macmillan Company). The second half of this article will concentrate on Łukasiewicz's position on ECSQ. This will lead me to propose that to regard him as a forerunner of paraconsistent logic by virtue of those early writings is accurate only if his book published in Polish is considered but not if the analysis is restricted to the paper originally published in German (as has been the case for the principal reconstructions of the history of paraconsistent logic). Furthermore, I will stress that in the 1929 book he presented one formalization of ECSQ as an axiom for sentential calculus and, also, he used ECSQ to defend the necessity of consistency, apparently independently of Hilbert and Ackermann's book. At the end, I will suggest that the aim of twentieth century usage of ECSQ was to change from the centuries-long philosophical discussion about contradictions to a more ‘technical’ one. But with paraconsistent logic viewed as a technical solution to this restriction, then, the philosophical problem revives but having now at one's disposal an improved understanding of it. Finally, Łukasiewicz's two different positions about ECSQ open an interesting question about the history of paraconsistent logic: do we have to attempt a consistent reconstruction of it, or are we prepared to admit inconsistencies within it? (shrink)