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35
An introduction to substructural logics
, 2000
"... Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1 ..."
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Cited by 139 (16 self)
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Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1
Implications of largecardinal principles in homotopical localization
 Adv. Math
"... The existence of arbitrary cohomological localizations on the homotopy category of spaces has remained unproved since Bousfield settled the same problem for homology theories in the decade of 1970. This is related with another open question, namely whether or not every homotopy idempotent functor on ..."
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Cited by 19 (3 self)
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The existence of arbitrary cohomological localizations on the homotopy category of spaces has remained unproved since Bousfield settled the same problem for homology theories in the decade of 1970. This is related with another open question, namely whether or not every homotopy idempotent functor on spaces is an flocalization for some map f. We prove that both questions have an affirmative answer assuming the validity of a suitable largecardinal axiom from set theory (Vopěnka’s principle). We also show that it is impossible to prove that all homotopy idempotent functors are flocalizations using the ordinary ZFC axioms of set theory (Zermelo–Fraenkel axioms with the axiom of choice), since a counterexample can be displayed under the assumption that all cardinals are nonmeasurable, which is consistent with ZFC.
Transfers between Logics and their Applications
 STUDIA LOGICA
, 2002
"... In this paper, logics are conceived as twosorted firstorder structures, and we argue that this broad definition encompasses a wide class of logics with theoretical interest as well as interest from the point of view of applications. The language, concepts and methods of model theory can thus be ..."
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Cited by 11 (4 self)
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In this paper, logics are conceived as twosorted firstorder structures, and we argue that this broad definition encompasses a wide class of logics with theoretical interest as well as interest from the point of view of applications. The language, concepts and methods of model theory can thus be used to describe the relationship between logics through morphisms of structures called transfers. This leads to a formal framework for studying several properties of abstract logics and their attributes such as consequence operator, syntactical structure, and internal transformations. In particular,
SpaceTime Foam Dense Singularities and de Rham Cohomology
, 2004
"... In an earlier paper of the authors it was shown that the sheaf theoretically based recently developed abstract differential geometry of the first author can in an easy and natural manner incorporate singularities on arbitrary closed nowhere dense sets in Euclidean spaces, singularities which therefo ..."
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Cited by 10 (4 self)
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In an earlier paper of the authors it was shown that the sheaf theoretically based recently developed abstract differential geometry of the first author can in an easy and natural manner incorporate singularities on arbitrary closed nowhere dense sets in Euclidean spaces, singularities which therefore can have arbitrary large positive Lebesgue measure. As also shown, one can construct in such a singular context a de Rham cohomology, as well as a short exponential sequence, both of which are fundamental in differential geometry. In this paper, these results are significantly strengthened, motivated by the so called spacetime foam structures in general relativity, where singularities can be dense. In fact, this time one can deal with singularities on arbitrary sets, provided that their complementaries are dense, as well. In particular, the cardinal of the set of singularities can be larger than that of the nonsingular points. Note: This paper is an augmented version of the paper with the same title, published in Acta Applicandae Mathematicae 67(1):5989,2001, and it is posted here with the kind permission of Kluwer Academic Publishers. ’We do not possess any method at all to derive systematically solutions that are free of singularities... ’ 1
SpaceTime Foam Differential Algebras of Generalized Functions and a Global CauchyKovalevskaia Theorem
, 2008
"... The new global version of the CauchyKovalevskaia theorem presented here is a strengthening and extension of the regularity of similar global solutions obtained earlier by the author. Recently the spacetime foam differential algebras of generalized functions with dense singularities were introduced. ..."
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Cited by 6 (4 self)
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The new global version of the CauchyKovalevskaia theorem presented here is a strengthening and extension of the regularity of similar global solutions obtained earlier by the author. Recently the spacetime foam differential algebras of generalized functions with dense singularities were introduced. A main motivation for these algebras comes from the so called spacetime foam structures in General Relativity, where the set of singularities can be dense. A variety of applications of these algebras have been presented elsewhere, including in de Rham cohomology, Abstract Differential Geometry, Quantum Gravity, etc. Here a global CauchyKovalevskaia theorem is presented for arbitrary analytic nonlinear systems of PDEs. The respective global generalized solutions are analytic on the whole of the domain of the equations considered, except for singularity sets which are closed and nowhere dense, and which upon convenience can be chosen to have zero Lebesgue measure. In view of the severe limitations due to the polynomial type growth conditions in the definition of Colombeau algebras, the class of singularities such algebras can deal with is considerably limited. Consequently, in such algebras one cannot even formulate, let alone obtain, the global version of the CauchyKovalevskaia theorem presented in this paper. 1 “We do not possess any method at all to derive systematically solutions that are free of singularities...”
A representation theorem for voting with logical consequences
 Economics and Philosophy
"... Much of this article was written while the author was a fellow at the Swedish Collegium for Advanced Study in the Social Sciences (SCASSS) in Uppsala. I want to thank the Collegium for providing me with excellent working conditions. Wlodek Rabinowicz and other fellows gave me valuable comments at a ..."
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Cited by 6 (0 self)
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Much of this article was written while the author was a fellow at the Swedish Collegium for Advanced Study in the Social Sciences (SCASSS) in Uppsala. I want to thank the Collegium for providing me with excellent working conditions. Wlodek Rabinowicz and other fellows gave me valuable comments at a seminar at SCASSS when an early version of the paper was presented. I also want to thank Luc Bovens, Christian List and two anonymous referees for their excellent comments on a later version. The final version was prepared during a stay at Oxford University for which I am grateful to the British Academy. 1 A representation theorem for voting with logical consequences 1.
Algebraic Terminological Representation
, 1991
"... This thesis investigates terminological representation languages, as used in klonetype knowledge representation systems, from an algebraic point of view. Terminological representation languages are based on two primitive syntactic types, called concepts and roles, which are usually interpreted mo ..."
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Cited by 5 (1 self)
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This thesis investigates terminological representation languages, as used in klonetype knowledge representation systems, from an algebraic point of view. Terminological representation languages are based on two primitive syntactic types, called concepts and roles, which are usually interpreted modeltheoretically as sets and relations, respectively. I propose an algebraic rather than a modeltheoretic approach. I show that terminological representations can be naturally accommodated in equational algebras of sets interacting with relations, and I use equational logic as a vehicle for reasoning about concepts interacting with roles.
E [16] : Differential algebras with dense singularities
, 1999
"... Abstract. Recently the spacetime foam differential algebras of generalized functions with dense singularities were introduced, motivated by the so called spacetime foam structures in General Relativity with dense singularities, and by Quantum Gravity. A variety of applications of these algebras ha ..."
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Cited by 5 (2 self)
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Abstract. Recently the spacetime foam differential algebras of generalized functions with dense singularities were introduced, motivated by the so called spacetime foam structures in General Relativity with dense singularities, and by Quantum Gravity. A variety of applications of these algebras has been presented, among them, a global CauchyKovalevskaia theorem, de Rham cohomology in abstract differential geometry, and so on. So far the spacetime foam algebras have only been constructed on Euclidean spaces. In this paper, owing to their relevance in General Relativity among others, the construction of these algebras is extended to arbitrary finite dimensional smooth manifolds. Since these algebras contain the Schwartz distributions, the extension of their construction to manifolds also solves the long outstanding problem of defining distributions on manifolds, and doing so in ways compatible with nonlinear operations. Earlier, similar attempts were made in the literature with respect to the extension of the Colombeau algebras to manifolds, algebras which also contain the distributions. These attempts have encountered significant technical difficulties, owing to the growth condition type limitations the elements of Colombeau algebras have to satisfy near singularities. Since in this paper no any type of such or other growth conditions are required in the construction of spacetime foam algebras, their extension to manifolds proceeds in a surprisingly easy and natural way. It is also shown that the spacetime foam algebras form a fine and flabby sheaf, properties which are important in securing a considerably large class of singularities which generalized functions can handle. ”We do not possess any method at all to derive systematically solutions that are free of singularities...”
Universal homogeneous causal sets
 Computational Structures for Modelling Space, Time and Causality, Dagstuhl Seminar Proceedings, Internationales Begegnungs und Forschungszentrum (IBFI), Schloss Dagstuhl
, 2006
"... Causal sets are particular partially ordered sets which have been proposed as a basic model for discrete spacetime in quantum gravity. We show that the class C of all countable pastfinite causal sets contains a unique causal set (U, ≤) which is universal (i.e., any member of C can be embedded into ..."
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Cited by 4 (1 self)
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Causal sets are particular partially ordered sets which have been proposed as a basic model for discrete spacetime in quantum gravity. We show that the class C of all countable pastfinite causal sets contains a unique causal set (U, ≤) which is universal (i.e., any member of C can be embedded into (U, ≤)) and homogeneous (i.e., (U, ≤) has maximal degree of symmetry). Moreover, (U, ≤) can be constructed both probabilistically and explicitly. 1