Ansten Klev

The notions of category and type are here studied through the lens of logical syntax: Aristotle's as well as Kant's categories through the traditional form of proposition `S is P', and modern doctrines of type through the Fregean form of proposition `F(a)', function applied to argument. Topics covered include the conception of categories as highest genera; the parts of speech and their relation to categories; the attempt to derive categories from more fundamental notions; the notion of a range of significance; the notion of a type assignment; sortal concepts and the notions of identity and generality; and the distinction between formal and material categories.

This paper explores the view that numbers are symbolically constituted, that numbers just are meaningful symbols. Such a view is what results if we take the conception of number spelled out by Husserl in the second part of his Philosophy of Arithmetic to be self-standing rather than supported by the conception of numbers as abstracted from sets. It will be argued that this latter conception is problematic in itself and, moreover, that it cannot be regarded as providing a foundation for the former.

The eta rule for a set A says that an arbitrary element of A is judge-mentally 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.

What is the name of the sinus function? Mathematical symbolism offers two obvious alternatives: sin(x), including the (independent) variable x; or sin, where no x occurs. It will be argued here that a good mathematical symbolism should give room for both alternatives and, moreover, that each alternative corresponds to a distinct notion of function.

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.

This paper is to appear in the anthology "Reflections on the Foundations of Mathematics". It discusses some of the ways in which Martin-Löf type theory differs from set theory. The discussion concentrates on conceptual, rather than technical, differences. It revolves around four topics: sets versus types; syntax; functions ; and identity. The difference between sets and types is spelt out as the difference between unified pluralities and kinds, or sorts. A detailed comparison is then offered of the syntax of the two languages. Emphasis is put on the distinction between proposition and judgement, drawn by type theory, but not by set theory. Unlike set theory, type theory treats the notion of function as primitive. It is shown that certain inconveniences pertaining to function application that afflicts the set-theoretical account of functions are thus avoided. Finally, the distinction, drawn in type theory, between judgemental and propositional identity is discussed. It is argued that the criterion of identity for a domain cannot be formulated in terms of propositional identity. It follows that the axiom of extensionality cannot be taken as a statement of the criterion of identity for sets.

Dedekind's Theorem 66 states that there exists an infinite set. Its proof invokes such apparently non-mathematical notions as the thought-world and the self. This paper discusses the content and context of Dedekind's proof. It is suggested that Dedekind took the notion of thought-world from Lotze. The influence of Kant and Bolzano on the proof is also discussed, and the reception of the proof in the mathematical and philosophical literature is covered in detail.

It is argued here that criteria of identity do not have the form of predicate-logical formulae. The conclusion is drawn that Hume's Principle cannot serve as a criterion of identity for the concept of cardinal number. The way criteria of identity are formulated in Martin-Löf's type theory is presented as an alternative that is not affected by the argument.
This paper is forthcoming in the Logica Yearbook 2017.

I offer an analysis of the sentence 'the concept horse is a concept'. It will be argued that the grammatical subject of this sentence, `the concept horse', indeed refers to a concept, and not to an object, as Frege once held. The argument is based on a criterion of proper-namehood according to which an expression is a proper name if it is so rendered in Frege's ideography. The predicate `is a concept', on the other hand, should not be thought of as referring to a function. It will be argued that the analysis of sentences of the form `C is a concept' requires the introduction of a new form of statement. Such statements are not to be thought of as having function--argument form, but rather the structure subject--copula--predicate. The paper has been accepted for publication in the Review of Symbolic Logic.

Lecture notes from Husserl's logic lectures published during the last 20 years offer a much better insight into his doctrine of the forms of meaning than does the fourth Logical Investigation or any other work published during Husserl's lifetime. This paper provides a detailed reconstruction, based on all the sources now available, of Husserl's system of logical grammar. After having explained the notion of meaning that Husserl assumes in his later logic lectures as well as the notion of form of meaning as it features in 'doctrine of the forms of meaning', I present a system of rules that describes all the various forms of meaning that Husserl singles out in his lectures. The paper is accepted for publication in History and Philosophy of Logic.

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.

Unified science is a recurring theme in Carnap's work from the time of the Aufbau until the end of the 1930's. The theme is not constant, but knows several variations. I shall extract three quite precise formulations of the thesis of unified science from Carnap's work during this period: from the Aufbau, from Carnap's so-called syntactic period, and from Testability and Meaning and related papers. My main objective is to explain these formulations and to discuss their relation, both to each other and to other aspects of Carnap's work.