The aim of this paper is to present a constructive solution to Frege's puzzle (largely limited to the mathematical context) based on type theory. Two ways in which an equality statement may be said to have cognitive significance are distinguished. One concerns the mode of presentation of the equality, the other its mode of proof. Frege's distinction between sense and reference, which emphasizes the former aspect, cannot adequately explain the cognitive significance of equality statements unless a clear identity criterion for (...) senses is provided. It is argued that providing a solution based on proofs is more satisfactory from the standpoint of constructive semantics. (shrink)

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)

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)

The notion of cognitive act is of importance for an epistemology that is apt for constructive type theory, and for epistemology in general. Instead of taking knowledge attributions as the primary use of the verb 'to know' that needs to be given an account of, and understanding a first-person knowledge claim as a special case of knowledge attribution, the account of knowledge that is given here understands first-person knowledge claims as the primary use of the verb 'to know'. This means (...) that a cognitive act is an act that counts as cognitive from a first-person point of view. The method of linguistic phenomenology is used to explain or elucidate our epistemic notions. One of the advantages of the theory is that an answer can be given to some of the problems in modern epistemology, such as the Gettier problem. (shrink)

We now move to the demonstration of the left-to-right direction of the equivalence result. Let us assume that there is a winning $$\mathbf {P}$$ P -strategy in the dialogical game for $$\varphi $$ φ.

The term verificationism is used in two different ways: the first is in relation to the verification principle of meaning, which we usually and rightly associate with the logical empiricists, although, as we now know, it derives in reality from Wittgenstein, and the second is in relation to the theory of meaning for intuitionistic logic that has been developed, beginning of course with Brouwer, Heyting and Kolmogorov in the twenties and early thirties, but in much more detail lately, particularly in (...) connection with intuitionistic type theory. It is therefore very natural to ask how these two forms of verificationism are related to one another: was the verificationism that we had in the thirties a kind of forerunner of what we have now, or was it something entirely different? I would like to discuss this question by considering a very particular problem, which was at the heart of Schlick’s interests, namely, the problem whether there might exist undecidable propositions or, if you prefer, unsolvable problems or unanswerable questions: it is merely a matter of wording which of these terms you choose. As I said, it is a problem which was at the heart of Schlick’s interests: it is explicitly discussed already in his early, programmatic paper Die Wende in der Philosophie in the first volume of Erkenntnis from 1930, and there is a short late paper, which has precisely Unanswerable Questions? as its title, from 1935, and he discussed it on several occasions in between also. (shrink)

Homotopy Type Theory is a proposed new language and foundation for mathematics, combining algebraic topology with logic. An important rule for the treatment of identity in HoTT is path induction, which is commonly explained by appeal to the homotopy interpretation of the theory's types, tokens, and identities as spaces, points, and paths. However, if HoTT is to be an autonomous foundation then such an interpretation cannot play a fundamental role. In this paper we give a derivation of path induction, motivated (...) from pre-mathematical considerations, without recourse to homotopy theory. (shrink)

On the basis of Martin-Löf’s meaning explanations for his type theory a detailed justification is offered of the rule of identity elimination. Brief discussions are thereafter offered of how the univalence axiom fares with respect to these meaning explanations and of some recent work on identity in type theory by Ladyman and Presnell.

The standard natural deduction rules for the identity predicate have seemed to some not to be harmonious. Stephen Read has suggested an alternative introduction rule that restores harmony but presupposes second-order logic. Here it will be shown that the standard rules are in fact harmonious. To this end, natural deduction will be enriched with a theory of definitional identity. This leads to a novel conception of canonical derivation, on the basis of which the identity elimination rule can be justified in (...) a proof-theoretical manner. (shrink)

The eta rule for a set A says that an arbitrary element of A is judgementally identical to an element of constructor form. Eta rules are not part of what may be called canonical Martin-Löf type theory. They are, however, justified by the meaning explanations, and a higher-order eta rule is part of that type theory. The main aim of this paper is to clarify this somewhat puzzling situation. It will be argued that lower-order eta rules do not, whereas the (...) higher-order eta rule does, accord with the understanding of judgemental identity as definitional identity. A subsidiary aim is to clarify precisely what an eta rule is. This will involve showing how such rules relate to various other notions of type theory, proof theory, and category theory. (shrink)

ZusammenfassungP. Bernays hat darauf hingewiesen, dass man, um die Widerspruchs freiheit der klassischen Zahlentheorie zu beweisen, den Hilbertschen flniter Standpunkt dadurch erweitern muss, dass man neben den auf Symbole sich beziehenden kombinatorischen Begriffen gewisse abstrakte Begriffe zulässt, Die abstrakten Begriffe, die bisher für diesen Zweck verwendet wurden, sinc die der konstruktiven Ordinalzahltheorie und die der intuitionistischer. Logik. Es wird gezeigt, dass man statt deesen den Begriff einer berechenbaren Funktion endlichen einfachen Typs über den natürlichen Zahler benutzen kann, wobei keine anderen (...) Konstruktionsverfahren für solche Funktionen nötig sind, als einfache Rekursion nach einer Zahlvariablen und Einsetzung von Funktionen ineinander .P. Bernays has pointed out that, in order to prove the consistency of classical number theory, it is necessary to extend Hilbert's finitary stand‐point by admitting certain abstract concepts in addition to the combinatorial concepts referring to symbols. The abstract concepts that so far have been used for this purpose are those of the constructive theory of ordinals and those of intuitionistic logic. It is shown that the concept of a computable function of finite simple type over the integers can be used instead, where no other procedures of constructing such functions are necessary except simple recursion by an integral variable and substitution of functions in each other. (shrink)

Semantic interpretations of both natural and formal languages are usually taken to involve the specification of a domain of entities with respect to which the sentences of the language are to be evaluated. A question that has received much attention of late is whether there is unrestricted quantification, quantification over a domain comprising absolutely everything there is. Is there a discourse or inquiry that has absolute generality? After framing the debate, this article provides an overview of the main arguments for (...) and against the possibility of unrestricted quantification, highlighting some of the broader implications of the debate. (shrink)

We define a generic notion of cut that applies to many first-order theories. We prove a generic cut elimination theorem showing that the cut elimination property holds for all theories having a so-called pre-model. As a corollary, we retrieve cut elimination for several axiomatic theories, including Church's simple type theory.

Some thirty years ago, two proposals were made concerning criteria for identity of proofs. Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal form in natural deduction. Lambek worked on a normalization proposal analogous to Prawitz's, based on reduction to cut-free form in sequent systems, but he also suggested understanding identity of proofs in terms of an equivalence relation based on generality, two derivations having the same generality if after generalizing maximally (...) the rules involved in them they yield the same premises and conclusions up to a renaming of variables. These two proposals proved to be extensionally equivalent only for limited fragments of logic. The normalization proposal stands behind very successful applications of the typed lambda calculus and of category theory in the proof theory of intuitionistic logic. In classical logic, however, it did not fare well. The generality proposal was rather neglected in logic, though related matters were much studied in pure category theory in connection with coherence problems, and there are also links to low-dimensional topology and linear algebra. This proposal seems more promising than the other one for the general proof theory of classical logic. (shrink)

The recent discovery of an interpretation of constructive type theory into abstract homotopy theory suggests a new approach to the foundations of mathematics with intrinsic geometric content and a computational implementation. Voevodsky has proposed such a program, including a new axiom with both geometric and logical significance: the Univalence Axiom. It captures the familiar aspect of informal mathematical practice according to which one can identify isomorphic objects. While it is incompatible with conventional foundations, it is a powerful addition to homotopy (...) type theory. It also gives the new system of foundations a distinctly structural character. (shrink)