This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.

Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λ-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege’s Grundgesetze der Arithmetik for which Russell applied his famous paradox 1 and this led him to introduce the first theory of types, the Ramified Type Theory (rtt). We present rtt formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from rtt leading to the simple theory of types stt. We present stt and Church’s own simply typed λ-calculus (λ→C 2) and we finish by comparing rtt, stt and λ→C. §1. Introduction. Nowadays, type theory has many applications and is used in many different disciplines. Even within logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. But, before 1903 when Russell first introduced

In 1922, Thoraf Skolem published a paper titled "Some remarks on Axiomatized Set Theory". The paper presents a new proof of... This dissertation focuses almost exclusively on the first half of this project -- i.e., the half which tries to expose an initial tension between Cantor's theorem and the Löwenheim-Skolem theorem. I argue that, even on quite naive understandings of set theory and model theory, there is no such tension. Hence, Skolem's Paradox is not a genuine paradox, and there is very little reason to worry about (or even to investigate) the more extreme consequences that are supposed to follow from this paradox. The heart of my...

English contains singular terms, quantifiers and predicates (e.g. ‘it’, ‘something ’ and ‘... is an elephant’). But it also contains plural terms, quantifiers and predicates (e.g. ‘they’, ‘some things ’ and ‘... are scattered on the floor’). 1 Philosophers have become increasingly interested in plurals over the past couple of decades. The purpose of this paper is to explain why plurals might be thought to have philosophical importance, and why they have led to philosophical debate. 1

This paper argues that the basic problems of nominalisation are those of set theory. We shall therefore overview the problems of set theory, the various solutions and assess the influence on nominalisation. We shall then discuss Aczel's Frege structures and compare them with Scott domains. Moreover, we shall set the ground for the second part which demonstrates that Frege structures are a suitable framework for dealing with nominalisation. Keywords: Frege structures, Nominalisation, Logic and Type freeness. 1 The Problems We shall examine the problem of the semantics of nominalised terms from two angles: the formal theory and the existence of models. 1.1 The problem of the formal theory Any theory of nominalisation should be accompanied by some ontological views on concepts --- for predicates and open well-formed formulae act semantically as concepts. This is vague, however, if only because where I use the word concept, someone else might use class, predicate, set, property or even...

In this paper we shall meet the application of Scott domains to nominalisation and explain its problem of predication. We claim that it is not possible to find a solution to such a problem within semantic domains without logic. Frege structures are more conclusive than a solution to domain equations and can be used as models for nominalisation. Hence we develop a type theory based on Frege structures and use it as a theory of nominalisation. Keywords: Frege structures, Nominalisation, Logic and Type freeness. 1 Frege structures, a formal introduction Having in part I informally introduced Frege structures, I shall here fill in all the technical details and show that Frege structures exist. Consider F 0 , F 1 ; : : : ; a family F of collections where F 0 is a collection of objects, and (8n ? 0)[F n is a collection of n-ary functions from F n 0 to F 0 ]. Definition 1.1 (An explicitly closed family) A family F as above is explicitly closed iff: For every expression e[x 1 ; : : : ; x n...

The type free -calculus is powerful enough to contain all the polymorphic and higher order nature of functional programming and furthermore types could be constructed inside it. However, mixing the type free -calculus with logic is not very straightforward (see [Aczel 80] and [Scott 75]). In this paper, a system that combines polymorphism and higher order functions with logic is presented. The system is suitable for both the functional and the logical paradigms of programming as from the functional paradigms point of view, the system enables one to have all the polymorphism and higher order that exist in functional languages and much more. In fact even the fixed point operator Y which is defined as f:(x:f(xx))(x:f(xx)) can be type checked to ((ff ! ff) ! ff)) where ff is a variable type. (x:xx)(x:xx) can be type checked too, something not allowed in functional languages. From the point of view of theorem proving, the system is expressive enough to allow self referential sentences and ...

This article, based on an invited lecture at the Logic Colloquium '93 in Keele, is a sequel to van Lambalgen [1992]. Apart from presenting new results, it differs from its predecessor in the following respects: (i) the presentation of the axioms is simplified, following some suggestions of Wojciech Buszkowski, (ii) the axioms have been strengthened, and (iii) the philosophical discussion has (hopefully) been improved. The article has appeared in W. Hodges et al (eds.), Logic: from Foundations to Applications (European Logic Colloquium), Oxford University Press 1996

Logic, due to the paradoxes, is absent from the type free -calculus. This makes such a calculus an unsuitable device for Natural Language Semantics. Moreover, the problems that arise from mixing the type free -calculus with logic lead to type theory and hence formalisations of Natural Language were carried out in a strictly typed framework. It was shown however, that strict type theory cannot capture the self-referential nature of language ([Parsons 79], [Chierchia, Turner 88] and [Kamareddine, Klein 93]) and hence other approaches were needed. For example, the approach carried out by Parsons is based on creating a notion of floating types which can be instantiated to particular instances of types whereas the approaches of Chierchia, Turner and Kamareddine, Klein are based on a type free framework. In this paper, we will embed the typing system of [Parsons 79] into a version of the one proposed in [Kamareddine, Klein 93] giving an interpretation of Parsons' system in a type free theory...

Call a quantifier unrestricted if it ranges over absolutely all things: not just over all physical