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48
Belief Functions and Default Reasoning
, 2000
"... We present a new approach to deal with default information based on the theory of belief functions. Our semantic structures, inspired by Adams' epsilon semantics, are epsilonbelief assignments, where mass values are either close to 0 or close to 1. In the first part of this paper, we show t ..."
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Cited by 34 (4 self)
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We present a new approach to deal with default information based on the theory of belief functions. Our semantic structures, inspired by Adams' epsilon semantics, are epsilonbelief assignments, where mass values are either close to 0 or close to 1. In the first part of this paper, we show that these structures can be used to give a uniform semantics to several popular nonmonotonic systems, including Kraus, Lehmann and Magidor's system P, Pearl's system Z, Brewka's preferred subtheories, Geffner's conditional entailment, Pinkas' penalty logic, possibilistic logic and the lexicographic approach. In the second part, we use epsilonbelief assignments to build a new system, called LCD, and show that this system correctly addresses the wellknown problems of specificity, irrelevance, blocking of inheritance, ambiguity, and redundancy.
Looking at graphs through infinitesimal microscopes, windows and telescopes
 Math. Gaz
, 1980
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The notion of Infinite Measuring Number and its Relevance
 in the Intuition of Infinity", Educational Studies in Mathematics
, 1980
"... Abstract. In this paper a concept of infinity is described which extrapolates the measuring properties of number rather than counting aspects (which lead to cardinal number theory). Infinite measuring numbers are part of a coherent number system extending the real numbers, including both infinitely ..."
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Cited by 16 (7 self)
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Abstract. In this paper a concept of infinity is described which extrapolates the measuring properties of number rather than counting aspects (which lead to cardinal number theory). Infinite measuring numbers are part of a coherent number system extending the real numbers, including both infinitely large and infinitely small quantities, A suitable extension is the superreal number system described here; an alternative extension is the hyperreal number field used in nonstandard analysis which is also mentioned. Various theorems are proved in careful detail to illustrate that certain properties of infinity which might be considered 'false ' in a cardinal sense are 'true ' in a measuring sense, Thus cardinal infinity is now only one of a choice of possible extensions of the number concept to the infinite case, It is therefore inappropriate to judge the 'correctness ' of intuitions of infinity within a cardinal framework alone, especially those intuitions which relate to measurement rather than oneone correspondence. The same comments apply in general to the analysis of naive intuitions within an extrapolated formal framework. 1.
Leibniz’s infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond
, 2012
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Perceiving the infinite and the infinitesimal world: unveiling and optical diagrams and the construction of mathematical concepts
 Foundations of Science
, 2005
"... Many important concepts of the calculus are difficult to grasp, and they may appear epistemologically unjustified. For example, how does a real function appear in “small ” neighborhoods of its points? How does it appear at infinity? Diagrams allow us to overcome the difficulty in constructing rep ..."
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Cited by 12 (5 self)
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Many important concepts of the calculus are difficult to grasp, and they may appear epistemologically unjustified. For example, how does a real function appear in “small ” neighborhoods of its points? How does it appear at infinity? Diagrams allow us to overcome the difficulty in constructing representations of mathematical critical situations and objects. For example they actually reveal the behavior of a real function not “close to ” a point (as in the standard limit theory) but “in ” the point. We are interested in our research in the diagrams which play an optical role – microscopes and “microscopes within microscopes”, telescopes, windows, a mirror role (to externalize rough mental models), and an unveiling role (to help create new and interesting mathematical concepts, theories, and structures). In this paper we describe some examples of optical diagrams as a particular kind of epistemic mediator able to perform the explanatory abductive task of providing a better understanding of the calculus, through a nonstandard model of analysis. We also maintain they can be used in many other different epistemological and cognitive situations. The Explanatory and Abductive
A Combination of Nonstandard Analysis and Geometry Theorem Proving, with Application to Newton's Principia
 PROCEEDINGS OF THE 15TH INTERNATIONAL CONFERENCE ON AUTOMATED DEDUCTION (CADE15
, 1998
"... The theorem prover Isabelle is used to formalise and reproduce some of the styles of reasoning used by Newton in his Principia. The Principia's reasoning is resolutely geometric in nature but contains "infinitesimal" elements and the presence of motion that take it beyond the traditio ..."
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Cited by 10 (3 self)
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The theorem prover Isabelle is used to formalise and reproduce some of the styles of reasoning used by Newton in his Principia. The Principia's reasoning is resolutely geometric in nature but contains "infinitesimal" elements and the presence of motion that take it beyond the traditional boundaries of Euclidean Geometry. These present difficulties that prevent Newton's proofs from being mechanised using only the existing geometry theorem proving (GTP) techniques. Using
Open Maps (at) Work
 DEPARTMENT OF COMPUTER SCIENCE, UNIVERSITY OF AARHUS
, 1995
"... The notion of bisimilarity, as defined by Park and Milner, has turned out to be one of the most fundamental notions of operational equivalences in the field of process algebras. Not only does it induce a congruence (largest bisimulation) in CCS which have nice equational properties, it has also ..."
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Cited by 10 (3 self)
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The notion of bisimilarity, as defined by Park and Milner, has turned out to be one of the most fundamental notions of operational equivalences in the field of process algebras. Not only does it induce a congruence (largest bisimulation) in CCS which have nice equational properties, it has also proven itself applicable for numerous models of parallel computation and settings such as Petri Nets and semantics of functional languages. In an attempt to understand the relationships and differences between the extensive amount of research within the field, Joyal, Nielsen, and Winskel recently presented an abstract categorytheoretic definition of bisimulation. They identify spans of morphisms satisfying certain "path lifting" properties, socalled open maps, as a possible abstract definition of bisimilarity. In [JNW93] they show, that they can capture Park and Milner's bisimulation. The aim of this paper is to show that the abstract definition of bisimilarity is applicable "in practice" by showing how a representative selection of wellknown bisimulations and equivalences, such as e.g. Hennessy's testing equivalence, Milner and Sangiorgi's barbed bisimulation, and Larsen and Skou's probabilistic bisimulation, are captured in the setting of open maps and hence, that the proposed notion of open maps seems successful. Hence, we confirm that the treatment of strong bisimulation in [JNW93] is not a oneoff application of open maps.
Expected Qualitative Utility Maximization
 Games and Economic Behavior
, 2001
"... This paper proposes a deviation from the von NeumannMorgenstern (vNM) postulates paradigm in decision theory. It suggests that it may, in certain circumstances, be rational to judge alternatives according to the issues in their main focus. If these issues are deemed as equal, the alternatives will ..."
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Cited by 8 (0 self)
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This paper proposes a deviation from the von NeumannMorgenstern (vNM) postulates paradigm in decision theory. It suggests that it may, in certain circumstances, be rational to judge alternatives according to the issues in their main focus. If these issues are deemed as equal, the alternatives will continue to be judged as equal even if they are mixed with different sideeffects  sideeffects which would not be judged as equal were they to be in the main focus themselves. This contradicts the vNM Independence axiom. Moreover, outcomes of issues in the main focus may completely overshadow the outcomes of the sideissues, so alternatives of the former may be judged as infinitely more (or infinitely less) preferable than those of the latter. This contradicts the vNM Continuity axiom.
Forti  Hausdorff nonstandard extensions
"... We introduce a notion of topological extension of a given set X. The resulting class of topological spaces includes the StoneČech compactification βX of the discrete space X, as well as all nonstandard models of X in the sense of nonstandard analysis (when endowed with a “natural ” topology). In ..."
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We introduce a notion of topological extension of a given set X. The resulting class of topological spaces includes the StoneČech compactification βX of the discrete space X, as well as all nonstandard models of X in the sense of nonstandard analysis (when endowed with a “natural ” topology). In this context, we give a simple characterization of nonstandard extensions in purely topological terms, and we establish connections with special classes of ultrafilters whose existence is independent of ZFC.