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103
Utility Representation of an Incomplete Preference Relation
 Journal of Economic Theory
, 2002
"... We consider the problem of representing a (possibly) incomplete preference relation by means of a vectorvalued utility function. Continuous and semicontinuous representation results are reported in the case of preference relations that are, in a sense, not “too incomplete. ” These results generaliz ..."
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Cited by 65 (6 self)
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We consider the problem of representing a (possibly) incomplete preference relation by means of a vectorvalued utility function. Continuous and semicontinuous representation results are reported in the case of preference relations that are, in a sense, not “too incomplete. ” These results generalize some of the classical utility representation theorems of the theory of individual choice, and paves the way towards developing a consumer theory that realistically allows individuals to exhibit some “indecisiveness ” on occasion.
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Cited by 24 (0 self)
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
Quantum Mereotopology
 Annals of Mathematics and Artificial Intelligence
, 2000
"... While mereotopology  the theory of boundaries, contact and separation built up on a mereological foundation  has found fruitful applications in the realm of qualitative spatial reasoning, it faces problems when its methods are extended to deal with those kinds of spatial and nonspatial reas ..."
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Cited by 20 (8 self)
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While mereotopology  the theory of boundaries, contact and separation built up on a mereological foundation  has found fruitful applications in the realm of qualitative spatial reasoning, it faces problems when its methods are extended to deal with those kinds of spatial and nonspatial reasoning which involve a factor of granularity. This is because granularity cannot easily be represented within a mereologybased framework. We sketch how this problem can be solved by means of a theory of coarsegrained partitions, drawing on methods developed for the manipulation of partitions in the spatial realm and applying these to a range of partitions of nonspatial sorts. We then show how these same methods can be extended to apply to finite sequences of partitions evolving over time, or to what we shall call coarse and finegrained histories. Keywords: mereotopology, granularity, ontology, partitions, histories 1. Introduction As a result of a series of important contribut...
Computerizing Mathematical Text with MathLang
"... Mathematical texts can be computerized in many ways that capture differing amounts of the mathematical meaning. At one end, there is document imaging, which captures the arrangement of black marks on paper, while at the other end there are proof assistants (e.g., Mizar, Isabelle, Coq, etc.), which c ..."
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Cited by 18 (4 self)
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Mathematical texts can be computerized in many ways that capture differing amounts of the mathematical meaning. At one end, there is document imaging, which captures the arrangement of black marks on paper, while at the other end there are proof assistants (e.g., Mizar, Isabelle, Coq, etc.), which capture the full mathematical meaning and have proofs expressed in a formal foundation of mathematics. In between, there are computer typesetting systems (e.g., LATEX and Presentation MathML) and semantically oriented systems (e.g., Content MathML, OpenMath, OMDoc, etc.). The MathLang project was initiated in 2000 by Fairouz Kamareddine and Joe Wells with the aim of developing an approach for computerizing mathematical texts and knowledge which is flexible enough to connect the different approaches to computerization, which allows various degrees of formalization, and which is compatible with different logical frameworks (e.g., set theory, category theory, type theory, etc.) and proof systems. The approach is embodied in a computer representation, which we call MathLang, and associated software tools, which are being developed by ongoing work. Three Ph.D. students (Manuel Maarek (2002/2007), Krzysztof Retel (since 2004), and Robert Lamar (since 2006)) and over a dozen master’s degree and undergraduate students have worked on MathLang. The project’s progress and design choices are driven by the needs for computerizing real representative mathematical texts chosen from various
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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Cited by 11 (5 self)
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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
Algorithms for ordinal arithmetic
 In 19th International Conference on Automated Deduction (CADE
, 2003
"... Abstract. Proofs of termination are essential for establishing the correct behavior of computing systems. There are various ways of establishing termination, but the most general involves the use of ordinals. An example of a theorem proving system in which ordinals are used to prove termination is A ..."
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Cited by 11 (5 self)
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Abstract. Proofs of termination are essential for establishing the correct behavior of computing systems. There are various ways of establishing termination, but the most general involves the use of ordinals. An example of a theorem proving system in which ordinals are used to prove termination is ACL2. In ACL2, every function defined must be shown to terminate using the ordinals up to ɛ0. We use a compact notation for the ordinals up to ɛ0 (exponentially more succinct than the one used by ACL2) and define efficient algorithms for ordinal addition, subtraction, multiplication, and exponentiation. In this paper we describe our notation and algorithms, prove their correctness, and analyze their complexity. 1
Natural and Formal Infinities
 Educational Studies in Mathematics 48 (2001
"... Concepts of infinity usually arise by reflecting on finite experiences and imagining them extended to the infinite. This paper will refer to such personal conceptions as natural infinities. Research has shown that individuals’ natural conceptions of infinity are ‘labile and selfcontradictory ’ (Fis ..."
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Cited by 11 (0 self)
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Concepts of infinity usually arise by reflecting on finite experiences and imagining them extended to the infinite. This paper will refer to such personal conceptions as natural infinities. Research has shown that individuals’ natural conceptions of infinity are ‘labile and selfcontradictory ’ (Fischbein et al., 1979, p. 31). The formal approach to mathematics in the twentieth century attempted to rationalize these inconsistencies by selecting a finite list of specific properties (or axioms) from which the conception of a formal infinity is built by formal deduction. By beginning with different properties of finite numbers, such as counting, ordering or arithmetic, different formal systems may be developed. Counting and ordering lead to cardinal and ordinal number theory and the properties of arithmetic lead to ordered fields that may contain infinite and infinitesimal quantities. Cardinal and ordinal numbers can be added and multiplied but not divided or subtracted. The operations of cardinals are commutative, but the operations of ordinals are not. Meanwhile an ordered field has a full system of arithmetic in which the reciprocals of