In this manuscript, published here for the first time, Tarski explores the concept of logical notion. He draws on Klein's Erlanger Programm to locate the logical notions of ordinary geometry as those invariant under all transformations of space. Generalizing, he explicates the concept of logical notion of an arbitrary discipline.

After sketching an argument for radical anti-realism that does not appeal to human limitations but polynomial-time computability in its definition of feasibility, I revisit an argument by Wittgenstein on the surveyability of proofs, and then examine the consequences of its application to the notion of canonical proof in contemporary proof-theoretical-semantics.

A realist theory of truth for a class of sentences holds that there are entities in virtue of which these sentences are true or false. We call such entities ‘truthmakers’ and contend that those for a wide range of sentences about the real world are moments (dependent particulars). Since moments are unfamiliar, we provide a definition and a brief philosophical history, anchoring them in our ontology by showing that they are objects of perception. The core of our theory is the (...) account of truthmaking for atomic sentences, in which we expose a pervasive ‘dogma of logical form’, which says that atomic sentences cannot have more than one truthmaker. In contrast to this, we uphold the mutual independence of logical and ontological complexity, and the authors outline formal principles of truthmaking taking account of both kinds of complexity. (shrink)

In Part III of his Remarks on the Foundations of Mathematics Wittgenstein deals with what he calls the surveyability of proofs. By this he means that mathematical proofs can be reproduced with certainty and in the manner in which we reproduce pictures. There are remarkable similarities between Wittgenstein's view of proofs and Hilbert's, but Wittgenstein, unlike Hilbert, uses his view mainly in critical intent. He tries to undermine foundational systems in mathematics, like logicist or set theoretic ones, by stressing the (...) unsurveyability of the proof-patterns occurring in them. Wittgenstein presents two main arguments against foundational endeavours of this sort. First, he shows that there are problems with the criteria of identity for the unsurveyable proof-patterns, and second, he points out that by making these patterns surveyable, we rely on concepts and procedures which go beyond the foundational frameworks. When we take these concepts and procedures seriously, mathematics does not appear as a uniform system, but as a mixture of different techniques. (shrink)

In this paper, I present a summary of the philosophical relationship betweenWittgenstein and Brouwer, taking as my point of departure Brouwer's lecture onMarch 10, 1928 in Vienna. I argue that Wittgenstein having at that stage not doneserious philosophical work for years, if one is to understand the impact of thatlecture on him, it is better to compare its content with the remarks on logics andmathematics in the Tractactus. I thus show that Wittgenstein's position, in theTractactus, was already quite close to (...) Brouwer's and that the points of divergence are the basis to Wittgenstein's later criticisms of intuitionism. Among the topics of comparison are the role of intuition in mathematics, rule following, choice sequences, the Law of Excluded Middle, and the primacy of arithmetic over logic. (shrink)

The philosophy of arithmetic of Wittgenstein's Tractatus is outlined and the central role played in it by the general notion of operation is pointed out. Following which, the language, the axioms and the rules of a formal theory of operations, extracted from the Tractatus, are presented and a theorem of interpretability of the equational fragment of Peano's Arithmetic into such a formal theory is proven.

Bolzano fut le premier philosophe à établir une distinction explicite entre les procédés déductifs qui nous permettent de parvenir à la certitude d’une vérité et ceux qui fournissent son fondement objectif. La conception que Bolzano se fait du rapport entre ce que nous appelons ici, d’une part, « conséquence subjective », à savoir la relation de raison à conséquence épistémique et, d’autre part, la « conséquence objective », c’est-à-dire la fondation , suggère toutefois que Bolzano défendait une conception « explicativiste (...) » de la preuve : les preuves par excellence sont celles qui reflètent l’ordre de la fondation objective. Dans cet article nous faisons état des problèmes liés à une telle conception et argumentons en faveur d’une démarcation plus stricte entre la préoccupation ontologique et la préoccupation épistémologique dans l’élaboration d’une théorie de la preuve.Bolzano was the first to establish an explicit distinction between the deductive methods that allow us to recognise the certainty of a given truth and those that provide its objective ground. His conception of the relation between what we, in this paper, call “subjective consequence”, i.e. the relation from epistemic reason to consequence and “objective consequence”, i.e. grounding however suggests that Bolzano advocated an “explicativist” conception of proof : proofs par excellence are those that reflect the objective order of grounding. In this paper, we expose the problems involved by such a conception and argue in favour of a more rigorous demarcation between the ontological and the epistemological concern in the elaboration of a theory of proof. (shrink)

An implication is a proposition, a consequence is a relation between propositions, and an inference is act of passage from certain premise-judgements to another conclusion-judgement: a proposition is true, a consequence holds, whereas an inference is valid. The paper examines interrelations, differences, refinements and linguistic renderings of these notions, as well as their history. The truth of propositions, respectively the holding of consequences, are treated constructively in terms of verification-objects. The validity of an inference is elucidated in terms of the (...) existence of a chain of immediately evident steps linking premises and conclusion. (shrink)

Consequence is at the heart of logic, and an account of consequence offers a vital tool in the evaluation of arguments. This text presents what the authors term as 'logical pluralism' arguing that the notion of logical consequence doesn't pin down one deductive consequence relation; it allows for many of them.

In his Grundgesetze, §32, Frege launched the idea that the meaning of a sentence is given by its truth condition, or, in his particular version, the condition under which it will be a name of the True. This, indeed, was only one of the many roles in which truth has to serve within the Fregean system. In particular, truth is an absolute notion in the sense that bivalence holds: every Gedanke is either true or false, in complete independence of any (...) conative activity, whether by God or man. Thus various epistemological notions, such as the correctness of an assertion made, or judgement passed, are reducible to this absolute notion of truth: an assertion made through the utterance of a declarative sentence is correct when the proposition expressed by the sentence in question is true. Given this absolute status of truth it is not surprising that Frege is of the opinion that truth is a sui generis notion which has to be left unanalyzed and, indeed, which is indefinable. (shrink)

Truth-maker analyses construe truth as existence of proof, a well-known example being that offered by Wittgenstein in theTractatus. The paper subsumes the intuitionistic view of truth as existence of proof under the general truth-maker scheme. Two generic constraints on truth-maker analysis are noted and positioned with respect to the writings of Michael Dummett and theTractatus. Examination of the writings of Brouwer, Heyting and Weyl indicates the specific notions of truth-maker and existence that are at issue in the intuitionistic truth-maker analysis, (...) namely that of proof in the sense of proof-object (Brouwer, Heyting) and existence in the nonpropositional sense of a judgement abstract (Weyl). Furthermore, possible anticipations in the writings of Schlick and Pfänder are noted. (shrink)

In the present paper I wish to regard constructivelogic as a self-contained system for the treatment ofepistemological issues; the explanations of theconstructivist logical notions are cast in anepistemological mold already from the outset. Thediscussion offered here intends to make explicit thisimplicit epistemic character of constructivism.Particular attention will be given to the intendedinterpretation laid down by Heyting. This interpretation, especially as refined in the type-theoretical work of Per Martin-Löf, puts thesystem on par with the early efforts of Frege andWhitehead-Russell. This quite (...) recent work, however,has proved valuable not only in the philosophy andfoundations of mathematics, but has also foundpractical application in computer science, where thelanguage of constructivism serves as an implementableprogramming language, and within the philosophy oflanguage.\footnote{Nordstr\"{o}m et al. give an overview of the work in computerscience, whereas Ranta provides an impressiveconstructivist alternative to Montague Grammar usingthe richer type structure of Martin-L\"{o}f in placeof the simple classical type theory of Church.} Mypresentation will be carried out through a contrastwith standard metamathematical work.\footnote{Troelstra and van Dalen give an encyclopedictreatment of the metamathematics of constructivism.}In the course of the development I have occasion tooffer some novel considerations on thenature of proof and inference. (shrink)

Reprint of paper first published in Philosophy and Phenomenological Research in 1984.

En logique mathématique, on doit distinguer entre une conception « axiomatique »de la logique, qui fut celle de Frege, Russell et Hilbert, et une conception plus « pragmatique »en termes d’actes de preuves, que l’on retrouve dans les systèmes de déduction naturelle de Gentzen. Des parallèles sont esquissés entre la conception de l’inférence et de la logique dans le Tractatus Logico-philosophicus de Wittgenstein et celle de Gentzen. Ce cadre permet en outre de jeter un regard neuf sur l’argument de Wittgenstein (...) sur « suivre une règle ». (shrink)

This is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. Reassessing the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as our understanding of mathematics, Michael Potter places arithmetic at the interface between experience, language, thought, and the world.

An implication is a proposition, a consequence is a relation between propositions, and an inference is act of passage from certain premise-judgements to another conclusion-judgement: a proposition is true, a consequence holds, whereas an inference is valid. The paper examines interrelations, differences, refinements and linguistic renderings of these notions, as well as their history. The truth of propositions, respectively the holding of consequences, are treated constructively in terms of verification-objects. The validity of an inference is elucidated in terms of the (...) existence of a chain of immediately evident steps linking premises and conclusion. (shrink)

En logique mathématique, on doit distinguer entre une conception « axiomatique »de la logique, qui fut celle de Frege, Russell et Hilbert, et une conception plus « pragmatique »en termes d’actes de preuves, que l’on retrouve dans les systèmes de déduction naturelle de Gentzen. Des parallèles sont esquissés entre la conception de l’inférence et de la logique dans le Tractatus Logico-philosophicus de Wittgenstein et celle de Gentzen. Ce cadre permet en outre de jeter un regard neuf sur l’argument de Wittgenstein (...) sur « suivre une règle ». (shrink)

We will discuss a mathematical proof found in Wittgenstein’s Nachlass, a constructive version of Euler’s proof of the infinity of prime numbers. Although it does not amount to much, this proof allows us to see that Wittgenstein had at least some mathematical skills. At the very last, the proof shows that Wittgenstein was concerned with mathematical practice and it also gives further evidence in support of the claim that, after all, he held a constructivist stance, at least during the transitional (...) period of his thought (1929-33). (shrink)