1,413 research outputs found

    On the Tensor Product of Archimedean Riesz Spaces

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    We characterize when, for any infinite cardinal α, Fremlin’s Archimedean Riesz space tensor product of two Archimedean Riesz spaces is Dedekind α-complete. We provide an example of an ideal I in an Archimedean Riesz space E such that the Fremlin tensor product of I with itself is not an ideal in the Fremlin tensor product of E with itself. On the other hand, we present conditions for which the Fremlin tensor product of ideals is an ideal and that the Fremlin tensor product of projection bands is a projection band. Lastly, we prove that the CarathĂ©odory space of place functions on the free product of two Boolean algebras is Riesz isomorphic with the Fremlin tensor product of their respective CarathĂ©odory spaces of place functions. This result is used to provide a solution to Fremlin’s problem 315Y(f) in [16] concerning completeness in the free product of Boolean algebras by applying our results on the Fremlin tensor product to CarathĂ©odory spaces of place functions

    Changing a semantics: opportunism or courage?

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    The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

    A mind of a non-countable set of ideas

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    The paper is dedicated to the 80th birthday of the outstanding Russian logician A.V. Kuznetsov. It is addressing a history of the ideas and research conducted by him in non-classical and intermediate logics

    Do Hard SAT-Related Reasoning Tasks Become Easier in the Krom Fragment?

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    Many reasoning problems are based on the problem of satisfiability (SAT). While SAT itself becomes easy when restricting the structure of the formulas in a certain way, the situation is more opaque for more involved decision problems. We consider here the CardMinSat problem which asks, given a propositional formula ϕ\phi and an atom xx, whether xx is true in some cardinality-minimal model of ϕ\phi. This problem is easy for the Horn fragment, but, as we will show in this paper, remains Θ2\Theta_2-complete (and thus NP\mathrm{NP}-hard) for the Krom fragment (which is given by formulas in CNF where clauses have at most two literals). We will make use of this fact to study the complexity of reasoning tasks in belief revision and logic-based abduction and show that, while in some cases the restriction to Krom formulas leads to a decrease of complexity, in others it does not. We thus also consider the CardMinSat problem with respect to additional restrictions to Krom formulas towards a better understanding of the tractability frontier of such problems
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