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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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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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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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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,
E.E.: Spacetime foam dense singularities and de rham cohomology
 Acta. Appl. Math
, 2001
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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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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.
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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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...”
EFFECTIVE MATCHMAKING (RECURSION THEORETIC ASPECTS OF A THEOREM OF Philip Hall)
, 1971
"... Given a set B of boys and a set 0 of girls, we call a subset S of BxO a society and we say that b knows g when (b,gy e S. The marriage problem for the society S is said to be solvable if it is possible to marry, in the traditional onetoone manner, each boy to a girl whom he knows. We are concerned ..."
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Given a set B of boys and a set 0 of girls, we call a subset S of BxO a society and we say that b knows g when (b,gy e S. The marriage problem for the society S is said to be solvable if it is possible to marry, in the traditional onetoone manner, each boy to a girl whom he knows. We are concerned here with the computable analogues of these notions. Thus a society is recursive if there exists an algorithm which, when presented with a boy 6 and a girl g, effectively determines whether 6 knows g. Similarly, the marriage problem for the society S is said to be recursively solvable if there exists a onetoone algorithm which, when presented with a boy 6, effectively marries him to a girl whom he knows. We first show that, even if (the marriage problem for) a recursive society is solvable, it need not be recursively solvable. We then consider several conditions on solvable recursive societies; for each we determine whether such a society must be recursively solvable and, if not, how computationally complex its solutions need be. We also discuss some sociological variations of the marriage problem and indicate how our results can be applied to them. We have drawn upon ideas from two branches of mathematics— combinatorics and recursive function theory. The combinatorial motivation has its source in a famous theorem of Philip Hall ([4]) which implies that if there are only a finite number of boys, then the society S is solvable if and only if, for each natural number k, any k distinct boys know among them at least k different girls. Using a compactness argument one can show that this same condition is necessary and sufficient even if there are an infinite number of boys, so long as no boy knows infinitely many girls. (See either [5] for a combinatorial argument or [1], p. 47, for a proof based on the propositional calculus. This generalization was first proved by M. Hall ([3]). L. Mirsky's new book ([10]) contains an exhaustive