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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. More ..."
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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...
The Felix Klein protocols
 Notices AMS
, 2007
"... Two plain shelves in Göttingen, in the entrance room of the Mathematisches Institut library, hold an astounding collection of mathematical manuscripts and rare books. In this locked Giftschrank, or poison cabinet, stand several hundred volumes, largely handwritten and mostly unique, that form an ext ..."
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Two plain shelves in Göttingen, in the entrance room of the Mathematisches Institut library, hold an astounding collection of mathematical manuscripts and rare books. In this locked Giftschrank, or poison cabinet, stand several hundred volumes, largely handwritten and mostly unique, that form an extensive archive documenting the development of one of the world’s most important mathematical centers – the home of Gauss, Riemann, Dirichlet, Klein, Hilbert, Minkowski, Courant, Weyl, and other leading mathematicians and physicists of the 19th and early 20th centuries. A recent Report on the Göttingen Mathematical Institute Archive [2] cites “a range of material unrivalled in quantity and quality”: “No single archive is even remotely comparable”, not only because Göttingen was “the leading place for mathematics in the world”, but also because “no other community has left such a detailed record of its activity... usually we are lucky to have lecture lists, with no indication of the contents.”
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...
Gödel on Intuition and on Hilbert’s finitism
"... There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the con ..."
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There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, Gödel’s writings represent a smooth evolution, with just one rather small doublereversal, of his view of finitism. He used the term “finit ” (in German) or “finitary ” or “finitistic ” primarily to refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [Gödel, 1938a] and the lecture notes for a lecture at Yale [Gödel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of firstorder number theory, P A; but starting in the Dialectica paper
Are Types needed for Natural Language? In Applied Logic: How, What and Why, Polos and Masuch
, 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
oating 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 where logic is present. In other words, we take the standpoint that type freeness is needed yet types are also indispensable. On this ground, by constructing types in the type free theory, we obtain a framework which can be seen as a formalisation of Parsons ' claim that Natural Language needs type freeness in order to accommodate self referentiality yet many sentences should be understood as implicitly typed.