This essay consists of two parts. In the first part, I focus my attention on the remarks that Frege makes on consistency when he sets about criticizing the method of creating new numbers through definition or abstraction. This gives me the opportunity to comment also a little on H. Hankel, J. Thomae—Frege’s main targets when he comes to criticize “formal theories of arithmetic” in Die Grundlagen der Arithmetik (1884) and the second volume of Grundgesetze der Arithmetik (1903)—G. Cantor, L. E. (...) J. Brouwer and D. Hilbert (1899). Part 2 is mainly devoted to Hilbert’s proof theory of the 1920s (1922–1931). I begin with an account of his early attempt to prove directly, and thus not by reduction or by constructing a model, the consistency of (a fragment of) arithmetic. In subsequent sections, I give a kind of overview of Hilbert’s metamathematics of the 1920s and try to shed light on a number of difficulties to which it gives rise. One serious difficulty that I discuss is the fact, widely ignored in the pertinent literature on Hilbert’s programme, that his language of finitist metamathematics fails to supply the conceptual resources for formulating a consistency statement qua unbounded quantification. Along the way, I shall comment on W. W. Tait’s objection to an interpretation of Hilbert’s finitism by Niebergall and Schirn, on G. Gentzen’s allegedly finitist consistency proof for Peano Arithmetic as well as his ideas on the provability and unprovability of initial cases of transfinite induction in pure number theory. Another topic I deal with is what has come to be known as partial realizations of Hilbert’s programme, chiefly advocated by S. G. Simpson. Towards the end of this essay, I take a critical look at Wittgenstein’s views about (in)consistency and consistency proofs in the period 1929–1933. I argue that both his insouciant attitude towards the emergence of a contradiction in a calculus and his outright repudiation of metamathematical consistency proofs are unwarranted. In particular, I argue that Wittgenstein falls short of making a convincing case against Hilbert’s programme. I conclude with some philosophical remarks on consistency proofs and soundness and raise a question concerning the consistency of analysis. (shrink)

Book reviewed in this article: GILEAD BAR‐ELLI, The Sense of Reference. Internationality in Frege, W. de Gruyter, BerliaNew York MICHAEL BEANEY, Frege: Making Sense, Duckworth, London.

In this paper I examine the question of logic’s normative status in the light of Carnap’s Principle of Tolerance. I begin by contrasting Carnap’s conception of the normativity of logic with that of his teacher, Frege. I identify two core features of Frege’s position: first, the normative force of the logical laws is grounded in their descriptive adequacy; second, norms implied by logic are constitutive for thinking as such. While Carnap breaks with Frege’s absolutism about logic and hence with the (...) notion that any system of logic should have a privileged claim to correctness, I argue that there is a sense in which Carnap’s framework-relative conception of logical norms has a constitutive role to play: though they are not constitutive for the conceptual activity for thinking, they do nevertheless set the ground rules that make certain forms of scientific inquiry possible in the first place. I conclude that Carnap’s principle of tolerance is tamer than one might have thought and that, despite remaining differences, Frege’s and Carnap’s conceptions of logic have more in common than one might have thought. (shrink)

Commentators regularly attribute to Frege realist, idealist, and quietist responses to metaphysical questions concerning the abstract objects he calls ‘thoughts’. But despite decades of effort, the evidence offered on behalf of these attributions remains unconvincing. I argue that Frege deliberately avoids commitment to any of these positions, as part of a metaphysical separatist policy motivated by the fact that logic is epistemologically autonomous from metaphysics. Frege’s views and arguments prove relevant to current attempts to argue for epistemological autonomy, particularly that (...) of ethics. (shrink)

The aim of this paper is to give a detailed reconstruction of Frege's solution to his puzzle about the cognitive function of truth, which is this: On the one hand, the concept of truth seems to play an essential role in acquiring knowledge because the transition from the mere hypothetical assumption that p to the acknowledgement of its truth is a crucial step in acquiring the knowledge that p, while, on the other hand, this concept seems to be completely redundant (...) because the sense of the word 'true' does not make any essential contribution to the senses of the sentences in which it occurs. (shrink)

Em sua obra, Robert Brandom apresenta uma teoria expressivista da lógica. Segundo o autor, a lógica tem o papel expressivo de tornar explı́citas as relações inferenciais implı́citas em nossas práticas linguı́sticas. Estas relações inferenciais conferem conteúdo aos conceitos. Neste artigo, exploraremos a tese de que o expressivismo lógico de Brandom é uma teoria antirrealista acerca da lógica. Realistas lógicos costumam defender a existência independente de fatos lógicos em relação a nossas práticas cognitivas e linguı́sticas, de modo que a lógica, uma (...) vez que seria capaz de representar estes fatos, descreveria verdadeiramente a estrutura da realidade. Por contraste, antirrealistas lógicos não se comprometem com a existência independente de fatos lógicos para entenderem a natureza da lógica. Em contraste com uma visão realista, mostramos que o antirrealismo de Brandom tem como consequências uma ontologia simples e econômica e a fundação de uma alternativa pragmatista para se examinar a normatividade da lógica, o fenômeno da pluralidade de lógicas rivais e a natureza de nossa racionalidade. (shrink)

Bogusław Wolniewicz – polski tłumacz Pism semantycznych Gottloba Fregego – przełożył termin Bedeutung na język polski jako „znaczenie”, zauważając w przypisie, że wątpliwości, jakie może budzić w tym przypadku użycie tego słowa, budzi również użycie w języku niemieckim przez Fregego słowa Bedeutung. Jednocześnie Wolniewicz, w tym samym przypisie, twierdzi, że „to, co Frege nazywa znaczeniem znaku, pokrywa się z tym, co w dzisiejszej terminologii semantycznej zwykło się nazywać »denotacją« albo »denotatem«” (Frege, 2014a, s. 62, przypis 27). W niniejszym artykule postaram (...) się przedstawić rozumienie Fregowskiej kategorii znaczenia (Bedeutung), które oddawałoby sposób, w jaki posługiwał się nim niemiecki filozof, tj. zarówno w odniesieniu do nazw, zdań, jak i predykatów. W części pierwszej przedstawię pokrótce wzajemny stosunek podziałów na przedmioty i funkcje oraz na sens i znaczenie wyrażeń. W drugiej zarysuję sposób, w jaki Frege podchodził do problemów językowych i filozoficznych. W części trzeciej dokonam szczegółowej interpretacji kategorii znaczenia dla nazw, zdań pojedynczych i złożonych, a także predykatów. Część ostatnia dotyczy kwestii Fregowskiego realizmu, która jest kluczowa z punktu widzenia późniejszych krytyk forsowanej przeze mnie interpretacji Bedeutung. (shrink)

In his logical foundation of arithmetic, Frege faced the problem that the semantic interpretation of his system does not determine the reference of the abstract terms completely. The contextual definition of number, for instance, does not decide whether the number 5 is identical to Julius Caesar. In a late writing, Quine claimed that the indeterminacy of reference established by Frege’s Caesar problem is a special case of the indeterminacy established by his proxy-function argument. The present paper aims to show that (...) Frege’s Caesar problem does not really support the conclusions that Quine draws from the proxy-function argument. On the contrary, it reveals that Quine’s argument is a non sequitur: it does not establish that there are alternative interpretations of our terms that are equally correct, but only that these terms are ambiguous. The latter kind of referential indeterminacy implies that almost all sentences of our overall theory of the world are either false or neither true nor false, because they contain definite descriptions whose uniqueness presupposition is not fulfilled. The proxy-function argument must therefore be regarded as a reductio ad absurdum of Quine’s behaviorist premise that the reference of terms is determined only by our linguistic behavior. (shrink)

In (1959), Carnap famously attacked Heidegger for having constructed an insane metaphysics based on a misconception of both the logical form and the semantics of ordinary language. In what follows, it will be argued that, once one appropriately (i.e., in a Russellian fashion) reads Heidegger’s famous sentence that should paradigmatically exemplify such a misconception, i.e., “the nothing nothings”, there is nothing either logically or semantically wrong with it. The real controversy as to how that sentence has to be evaluated—not as (...) to its meaning but as to its truth—lies at the metaphysico- ontological level. For in order for the sentence to be true one has to endorse an ontology of impossibilia and Leibniz’s principle of the identity of indiscernibles. (shrink)