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Generalized Quantifiers in Logic Programs
 In Proceedings of the ESSLLI Workshop on Generalized Quantifiers, AixenProvence
, 1997
"... Generalized quantifiers are an important concept in modeltheoretic logic which has applications in different fields such as linguistics, philosophical logic and computer science. In this paper, we consider a novel application in the field of logic programming, which has been presented recently. ..."
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Generalized quantifiers are an important concept in modeltheoretic logic which has applications in different fields such as linguistics, philosophical logic and computer science. In this paper, we consider a novel application in the field of logic programming, which has been presented recently. The enhancement of logic programs by generalized quantifiers is a convenient tool for interfacing extralogical functions and provides a natural framework for the definition of modular logic programs.
An Intensional Type Theory: Motivation and CutElimination
, 2001
"... By the theory TT is meant the higher order predicate logic with the following recursively defined types: (1) 1 is the type ofindividuals and [] is the type ofthe truth values; (2) [# 1 ,..., n ] is the type ofthe predicates with arguments ofthe types #1 ,...,# n . The theory ITT described in this pa ..."
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Cited by 5 (0 self)
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By the theory TT is meant the higher order predicate logic with the following recursively defined types: (1) 1 is the type ofindividuals and [] is the type ofthe truth values; (2) [# 1 ,..., n ] is the type ofthe predicates with arguments ofthe types #1 ,...,# n . The theory ITT described in this paper is an intensional version ofTT. The types ofITT are the same as the types ofTT, but the membership ofthe type 1 ofindividuals in ITT is an extension ofthe membership in TT. The extension consists ofallowing any higher order term, in which only variables oftype 1 have a free occurrence, to be a term oftype 1. This feature ofITT is motivated by a nominalist interpretation ofhigher order predication. In ITT both wellfounded and nonwellfounded recursive predicates can be defined as abstraction terms from which all the properties of the predicates can be derived without the use ofnonlogical axioms. The elementary syntax, semantics, and prooftheory for ITT are defined. A semantic consistency prooffor ITT is provided and the completeness proofofTakahashi and Prawitz for a version of TT without cut is adapted for ITT; a consequence is the redundancy of cut. 1.
An Editor Recalls Some Hopeless Papers
, 1998
"... set theory' [12] as his source, and another refers to Barrow `Theories of everything' [2]. One contents himself with references to two earlier unpublished papers of his own. Others give no source. For definiteness let me write down a proof, not in Cantor's words, which contains all the points we sh ..."
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Cited by 3 (2 self)
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set theory' [12] as his source, and another refers to Barrow `Theories of everything' [2]. One contents himself with references to two earlier unpublished papers of his own. Others give no source. For definiteness let me write down a proof, not in Cantor's words, which contains all the points we shall need to comment on. (1) We claim first that for every map f from the set {1, 2, . . . } of positive integers to the open unit interval (0, 1) of the real numbers, there is some real number which is in (0, 1) but not in the image of f. (2) Assume that f is a map from the set of positive integers to (0, 1). (3) Write 0 . a n1 a n2 a n3 . . . for the decimal expansion of f(n), where each a ni is a numeral between 0 and 9. (Where it applies, we choose the expansion which is eventually 0, not that which is eventually 9.) (4) For each positive integer n, let b n be 5 if a nn #= 5, and 4 otherwise. (5) Let b be the real number whose decimal expansion is 0 . b 1 b 2 b 3 . . . . (6...
Vagueness and Truth
"... In philosophy of logic and elsewhere, it is generally thought that similar problems should be solved by similar means. This advice is sometimes elevated to the status of a principle: the principle of uniform solution. In this paper I will explore the question of what counts as a similar problem and ..."
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In philosophy of logic and elsewhere, it is generally thought that similar problems should be solved by similar means. This advice is sometimes elevated to the status of a principle: the principle of uniform solution. In this paper I will explore the question of what counts as a similar problem and consider reasons for subscribing to the principle of uniform solution. 1 Introducing the Principle of Uniform Solution It would be very odd to give different responses to two paradoxes depending on minor, seeminglyirrelevant details of their presentation. For example, it would be unacceptable to deal with the paradox of the heap by invoking a multivalued logic, ̷L∞, say, and yet, when faced with the paradox of the bald man, invoke a supervaluational logic. Clearly these two paradoxes are of a kind—they are both instances of the sorites paradox. And whether the sorites paradox is couched in terms of heaps and grains of sand, or in terms of baldness and the number of hairs on the head, it is essentially the same problem and therefore must be solved by the same means. More generally, we might suggest that similar paradoxes should be resolved by similar means. This advice is sometimes elevated to the status of a principle, which usually goes by the name of the principle of uniform solution. This principle and its motivation will occupy us for much of the discussion in this paper. In particular, I will defend a rather general form of this principle. I will argue that two paradoxes can be thought to be of the same kind because (at a suitable level of abstraction) they share a similar internal structure, or because of external considerations such as the relationships of the paradoxes in question to other paradoxes in the vicinity, or the way they respond to proposed solutions. I will then use this reading of the principle of uniform solution to make a case for the sorites and the liar paradox being of a kind.
Alternative Set Theories
, 2006
"... By “alternative set theories ” we mean systems of set theory differing significantly from the dominant ZF (ZermeloFrankel set theory) and its close relatives (though we will review these systems in the article). Among the systems we will review are typed theories of sets, Zermelo set theory and its ..."
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By “alternative set theories ” we mean systems of set theory differing significantly from the dominant ZF (ZermeloFrankel set theory) and its close relatives (though we will review these systems in the article). Among the systems we will review are typed theories of sets, Zermelo set theory and its variations, New Foundations and related systems, positive set theories, and constructive set theories. An interest in the range of alternative set theories does not presuppose an interest in replacing the dominant set theory with one of the alternatives; acquainting ourselves with foundations of mathematics formulated in terms of an alternative system can be instructive as showing us what any set theory (including the usual one) is supposed to do for us. The study of alternative set theories can dispel a facile identification of “set theory ” with “ZermeloFraenkel set theory”; they are not the same thing. Contents 1 Why set theory? 2 1.1 The Dedekind construction of the reals............... 3 1.2 The FregeRussell definition of the natural numbers....... 4
Informal and Formal Representations in Mathematics
, 2007
"... In this paper we discuss the importance of good representations in mathematics and relate them to general design issues. Good design makes life easy, bad design difficult. For this reason experienced mathematicians spend a significant amount of their time on the design of their concepts. While many ..."
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In this paper we discuss the importance of good representations in mathematics and relate them to general design issues. Good design makes life easy, bad design difficult. For this reason experienced mathematicians spend a significant amount of their time on the design of their concepts. While many formal systems try to support this by providing a highlevel language, we argue that more should be learned from the mathematical practice in order to improve the applicability of formal systems.