Inference versus consequence , an invited lecture at the LOGICA 1997 conference at Castle Liblice, was part of a series of articles for which I did research during a Stockholm sabbatical in the autumn of 1995. The article seems to have been fairly effective in getting its point across and addresses a topic highly germane to the Uppsala workshop. Owing to its appearance in the LOGICA Yearbook 1997 , Filosofia Publishers, Prague, 1998, it has been rather inaccessible. Accordingly it is (...) republished here with only bibliographical changes and an afterword. (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)

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)

We present a new English translation of L.E.J. Brouwer's paper ‘De onbetrouwbaarheid der logische principes’ of 1908, together with a philosophical and historical introduction. In this paper Brouwer for the first time objected to the idea that the Principle of the Excluded Middle is valid. We discuss the circumstances under which the manuscript was submitted and accepted, Brouwer's ideas on the principle of the excluded middle, its consistency and partial validity, and his argument against the possibility of absolutely undecidable propositions. (...) We note that principled objections to the general excluded middle similar to Brouwer's had been advanced in print by Jules Molk two years before. Finally, we discuss the influence on George Griss' negationless mathematics. (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)

Brouwer's demonstration of his Bar Theorem gives rise to provocative questions regarding the proper explanation of the logical connectives within intuitionistic and constructivist frameworks, respectively, and, more generally, regarding the role of logic within intuitionism. It is the purpose of the present note to discuss a number of these issues, both from an historical, as well as a systematic point of view.

A detailed chronology is offered for the writing of Frege's central philosophical essays from the early 1890s. Particular attention is given to (the distinction between) Sinn and Bedeutung. Suggestions are made as to the origin of the examples concerning the Morning Star/Evening Star and August Bebel's views on the return of Alsace-Lorraine. Likely sources are offered for Frege's use of the terms Bestimmungsweise, Art des Gegebenseins and Sinn und Bedeutung.

Drawing upon Martin-Löf’s semantic framework for his constructive type theory, semantic values are assigned also to natural-deduction derivations, while observing the crucial distinction between consequence among propositions and inference among judgements. Derivations in Gentzen’s format with derivable formulae dependent upon open assumptions, stand, it is suggested, for proof-objects, whereas derivations in Gentzen’s sequential format are proof-acts.

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)

The paper offers an interpretation of thesis 6.01. The treatment touches upon variables, identity, elementary propositions, internal relations. Klammerausdrücke, and operations. Wittenstein's notations are found not to cover the particular form of definition by induction that is used at 6 and 6.01. It is concluded that Wittgenstein's ability to design of a formal system of logic does not match his outstanding logico-philosophical insight.

This chapter tells how, within a century, the notions of judgment and inference were driven out of logical theory and replaced by propositions and consequence. Systematic considerations guide the treatment. The history is unashamedly Whiggish: the current position is shown as the outcome, or even culmination, of a historical development.

The two different layers of logical theory—epistemological and ontological—are considered and explained. Special attention is given to epistemic assumptions of the kind that a judgement is granted as known, and their role in validating rules of inference, namely to aid the inferential preservation of epistemic matters from premise judgements to conclusion judgement, while ordinary Natural Deduction assumptions serve to establish the holding of consequence from antecedent propositions to succedent proposition.

The paper offers an interpretation of thesis 6.01. The treatment touches upon variables, identity, elementary propositions, internal relations. Klammerausdrücke, and operations. Wittenstein's notations are found not to cover the particular form of definition by induction that is used at 6 and 6.01. It is concluded that Wittgenstein's ability to design of a formal system of logic does not match his outstanding logico-philosophical insight.

The paper offers an interpretation of thesis 6.01. The treatment touches upon variables, identity, elementary propositions, internal relations. Klammerausdrücke, and operations. Wittenstein's notations are found not to cover the particular form of definition by induction that is used at 6 and 6.01. It is concluded that Wittgenstein's ability to design of a formal system of logic does not match his outstanding logico-philosophical insight.

The possibility of error is related to the existence a norm. Connections are spelled out to the notion of infallibility and to that of a modifying predicate, to traditional truth theories in connection with “truth of things”, as well as the primacy of the negative cases, for instance “ false friend”.

Tradition is classical. Surely, nothing could be more pleonastic than that? The logical tradition, certainly, was squarely classical from Bolzano to Carnap, with, say, Frege, Moore, Russell and the Wittgenstein of the Tractatus as intermediaries. Propositions are construed as being in themselves true-or-false. Indeed, in this tradition, a declarative sentence S expresses a proposition by being true-or-false. So the meaningfulness of a sentence consists in its being true-or-false. But S is true-orfalse, or so they say, only when S is true, (...) or when S is false. On the classical account the presumption of bivalence is built into the very notion of meaningfulness: there is no difference between asserting that A is a proposition and asserting that A is true-or-false. The matter came to the fore in the foundations of set theory. In his first attempt at giving an application criterion for sets Cantor noted:Eine Mannigfaltigkeit von Elementen, die irgendwelcher Begriffssphäre angehören, nenne ich wohldefiniert, wenn auf Grund ihrer Definition und infolge des logischen Prinzips vom ausgeschlossenen Dritten es als intern bestimmt angesehen werden muss, sowohl ob irgendein derselben Begriffssphäre angehöriges Objekt zu der gedachten Mannigfaltigkeit gehört oder nicht, wie auch, ob zwei zur Mannigfaltigkeit gehörige Objekte trotz formaler Unterschiede in der Art des Gegebenseins einander gleich sind oder nicht. (shrink)

A running commentary is offered on the first half of Frege’s Grundlagen der Arithmetik, §17, and suggests that Frege anticipated the method of demonstration used by Paul Bernays for the Deduction Theorem.

Bolzano reduced inferential validity of the inference (from premise judgements to conclusion judgment) to the holding of logical consequence between the propositions (in themselves) that serve as contents of the respective judgements. This explicit reduction of inferential validity among judgements to logical consequence among propositions (or, alternatively, to logical truth of certain implicational propositions) has been largely taken over by current logical theory, say, by Wittgenstein’s Tractatus, by Hilbert and Ackermann, by Quine, and by Tarski also. Frege, though, stands out (...) among those who did not adopt such an account. Under the Bolzano reduction also some blind inferences, with no epistemic support, are deemed valid, which is unacceptable. The Completeness Theorem reduces inferential validity to the logical truth of a certain well-formed formula, whence the unacceptable blindness phenomena are retained. Accordingly, the interest of the Completeness Theorem for logic, construed as the theory of inference, is nugatory. (shrink)

Essays on Wittgensteinian Themes Dedicated to Brian McGuinness Joachim Schulte, Göran Sundholm. PREFACE For thirty-five years the international community of philosophers have known Brian McGuinness as a major authority on the...

The paper presents a framework of distinctions for the philosophy of logic in which the interrelations between some central logical notions, such as statement, judgement, judgement, proposition, consequence, and inference are spelled out.

Göran Sundholm; Brouwer's Anticipation of the Principle of Charity*, Proceedings of the Aristotelian Society, Volume 85, Issue 1, 1 June 1985, Pages 263–276, ht.

The paper presents a framework of distinctions for the philosophy of logic in which the interrelations between some central logical notions, such as statement, judgement, judgement, proposition, consequence, and inference are spelled out.