Göran Sundholm

Existence, Proof and Truth-Making : A Perspective on the Intuitionistic Conception of Truth

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.

Inference versus consequence revisited: inference, consequence, conditional, implication

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.

The proper explanation of intuitionistic logic: on Brouwer’s demonstration of the Bar Theorem

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.

Semantic Values for Natural Deduction Derivations

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 (logical) consequence among propositions and inference among judgements. Derivations in Gentzen’s (1934–5) format with derivable formulae dependent upon open assumptions, stand, it is suggested, for proof-objects (of propositions), whereas derivations in Gentzen’s (1936) sequential format are (blue-prints for) proof-acts.

Frege, August Bebel and the Return of Alsace-Lorraine: The dating of the distinction between Sinn and Bedeutung

A detailed chronology is offered for the writing of Freges 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 Bebels views on the return of Alsace-Lorraine. Likely sources are offered for Freges use of the terms Bestimmungsweise, Art des Gegebenseins and Sinn und Bedeutung.

Vestiges of Realism

In the Trinity Term of 1977 I had the privilege of hearing Michael Dummett lecture in the Old Library at All Souls on the justification and criticism of logical laws, and subsequently I also assisted Mark Helme in the production of some unauthorised, not altogether successful, notes on these lectures. I can vividly recall my surprise when Prof. Dummett stated, in connection with the idea with which he had up till then been mainly concerned, namely the idea that meaning is primarily determined in terms of the Gentzen introduction rules, that my old tutor, the Swedish logician Per Martin-Lof, in correspondence, held that “the meanings of the logical constants CAN IN NO WAY be given by the introduction rules, rather they are given by the elimination rules”.1 This was, as far as I knew from tutorials and conversations with Martin-Lof, not a position he would defend at the time and certainly not today. In fact, he has steadfastly held by the introduction-rule orthodoxy in all relevant publications from 1979 onwards.2 Taken jointly those publications constitute, as far as I know, the only sustained effort towards a realisation of a constructivist theory of meaning for a sizeable interpreted language serving the needs of pure mathematics on a scale comparable to that of Frege’s Grundgesetze. From this circumstance alone it seems clear that much of Dummettian interest can be found in the works of Martin-Lof.

Containment and Variation; Two Strands in the Development of Analyticity from Aristotle to Martin-Löf

My original training as a philosopher, at Uppsala and at Oxford, was ruggedly analytical. Also the notion of an analytic judgement, or ‘proposition’, or ‘sentence’, or ‘statement’, (one did not overly distinguish these notions) was repeatedly treated of by excellent teachers and colleagues. There were aficionados of Quine and experts on Kant among them, but no names, no pack-drill! If there was one central topic in traditional epistemology on which I felt philosophically at ease, it was that of analyticity. In the early 1980s, I entered for the first time a pluralist philosophical environment in the Philosophy Department of the Catholic University at Nijmegen, with ample representation in phenomenology, Hegelian idealism, and (neo)Thomism. To my considerable surprise, I discovered that it could be enjoyable as well as instructive talking to such rare birds in the philosophical aviary. A colleague drew my attention to Thomas Aquinas’ Five Ways, which I had never read, having adopted, from the exposition in Anders Wedberg’s History of Philosophy, the opinion that, like Kant’s transcendental deduction, Aquinas’ demonstrations were ‘worthless’. However, the Summa Theologica was readily available on open shelves in the library at Nijmegen, and my curiosity got the better of me. Upon consultation of its second question, my shock was great. In a discussion of whether the judgement Deus est admits of demonstration, Aquinas introduces the notion of a propositio per se nota, that is, an S is P judgement known in, or—perhaps better—from itself: The explanation offered is that the predicate P is included, or contained, in the notion (= concept) of the subject S. Needless to say, in view of my previous deep and thorough (as I misguidedly thought) exposure to analyticity, I had a powerful deja lu experience, pertaining to Kant, four centuries later. Clearly, I had been choused. What was the hidden tale behind this, and why had my eminent teachers not told me that the notion of an analytic judgement was known long before Kant?

Constructive Recursive Functions, Church’s Thesis, and Brouwer’s Theory of the Creating Subject: Afterthoughts on a Parisian Joint Session

The first half of the paper discusses recursive versus constructive functions and, following Heyting, stresses that from a constructive point the former cannot replace the latter. The second half of the paper treats of the Kreisel-Myhill theory CS for Brouwer’s Creating Subject, and its relation to BHK meaning-explanations and Kripke’s Schema. Kripke’s Schema is reformulated as a principle and shown to be classically valid. Assuming existence of a verification-object for this principle, a modification of a proof of conservativeness of Van Dalen’s, is shown to give a relative BHK meaning explanation for the Kreisel-Myhill connective. The result offers an explanation of why Kripke’s Schema can be used as a replacement of the Theory of Creating Subject when formulating Brouwerian counter-examples. It also shows that the Theory of Creating Subject is classically valid.

Demonstrations Versus Proofs, Being an Afterword to Constructions, Proofs, and the Meaning of the Logical Constants

The spring of 1980 I spent as visiting lecturer at Utrecht. The volume of Heyting’s Collected Papers had not yet been put together, and his philosophical papers could not be found at Oxford. Accordingly, I availed myself of the opportunities offered by Dutch libraries and read the relevant papers. A couple of years earlier, I had learned about Constructive Type Theory from Per Martin-Lof, and Michael Beeson, who had just written a paper on a theory of constructions, was an eager sparring partner in almost daily discussions at Utrecht. The outcome of these ponderings was this chapter on which you are reading now as an afterword. It was ready toward the end of the summer 1981, and my Oxford Professor Dana Scott suggested to me that I should submit it to Richmond Thomason, the editor of the Journal of Philosophical Logic, at a meeting of authors for the Handbook of Philosophical Logic at Bad Homburg. I did so and the paper was readily accepted; however, a special issue on intuitionism was being prepared, and Thomason suggested that I might want to wait in order to have it appear in that issue. Thus, the paper appeared only in 1983 but had circulated rather widely in the intervening time.

The Vocabulary of Epistemology, with Observations on Some Surprising Shortcomings of the English Language

After some observations on the conference performance of Kevin Mulligan, the chapter notes shortcomings in the English epistemic vocabulary concerning the crucial terms knowledge, science, evidence, certainty, proof, demonstration, and proposition.