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Types in logic and mathematics before 1940
 Bulletin of Symbolic Logic
, 2002
"... 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 λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, thou ..."
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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 λcalculus 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 λcalculus (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
Set Theory and Nominalisation, Part I
 Journal of Logic and Computation
, 1996
"... 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, ..."
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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 wellformed 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...
Set Theory and Nominalisation, Part II
 Journal of Logic and Computation
, 1992
"... 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 ..."
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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 nary 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...
Are Types needed for Natural Language?
, 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 ..."
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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 selfreferential 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...