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Descripción
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| What are variables, and what is universal quanti?cation over a variable? Nominal sets are a notion of ?sets with names?, and using equational axioms in nominal algebra these names can be given substitution and quanti?cation actions. So we can axiomatise ?rst-order logic as a nominal logical theory. We can then seek a nominal sets representation theorem in which predicates are interpreted as sets; logical conjunction is interpreted as sets intersection; negation as complement. Now what about substitution; what is it for substitution to act on a predicate-interpreted-as-a-set, in which case universal quanti?cation becomes an in?nite sets intersection? Given answers to these questions, we can seek notions of topology. What is the general notion of topological space of which our sets representation of predicates makes predicates into ?open sets?; and what speci?c class of topological spaces corresponds to the image of nominal algebras for ?rst-order logic? The classic Stone duality answers these questions for Boolean algebras, representing them as Stone spaces. Nominal algebra lets us extend Boolean algebras to ?FOL-algebras?, and nominal sets let us correspondingly extend Stone spaces to ?8-Stone spaces?. These extensions reveal a wealth of structure, and we obtain an attractive and self-contained account of logic and topology in which variables directly populate the denotation, and open predicates are interpreted as sets rather than functions from valuations to sets. Keywords: Stone duality, nominal sets, ?rst-order logic, topology, variables | |
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Internacional
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Si |
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DOI
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Edición del Libro
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Editorial del Libro
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ISBN
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978-3-642-22943-5 |
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Serie
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Título del Libro
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Festschrifft in honour of Edward Barringer |
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Desde página
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20 |
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Hasta página
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49 |