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Topological Completeness for HigherOrder Logic
 Journal of Symbolic Logic
, 1997
"... Using recent results in topos theory, two systems of higherorder logic are shown to be complete with respect to sheaf models over topological spacessocalled "topological semantics". The first is classical higherorder logic, with relational quantification of finitely high type; the second sy ..."
Abstract

Cited by 8 (3 self)
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Using recent results in topos theory, two systems of higherorder logic are shown to be complete with respect to sheaf models over topological spacessocalled "topological semantics". The first is classical higherorder logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.
Orthomodularvalued models for quantum set theory
, 908
"... Orthomodular logic represented by a complete orthomodular lattice has been studied as a pertinent generalization of the twovalued logic, Booleanvalued logic, and quantum logic. In this paper, we introduce orthomodular logic valued models for set theory generalizing quantum logic valued models intr ..."
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Orthomodular logic represented by a complete orthomodular lattice has been studied as a pertinent generalization of the twovalued logic, Booleanvalued logic, and quantum logic. In this paper, we introduce orthomodular logic valued models for set theory generalizing quantum logic valued models introduced by Takeuti as well as Booleanvalued models introduced by Scott and Solovay, and prove a general transfer principle that states that every theorem of ZFC set theory without free variable is, if modified by restricting every unbounded quantifier appropriately with the notion of commutators, valid in any orthomodular logic valued models for set theory. This extends the wellknown transfer principle for Booleanvalued models. In order to overcome an unsolved problem on the implication in quantum logic, we introduce the notion of generalized implications in orthomodular logic by simple requirements satisfied by the wellknown six polynomial implication candidates, and show that for every choice from generalized implications the above transfer principle holds. In view of the close connection between interpretations of quantum mechanics and quantum set theory, this opens an interesting problem as to how the choice of implication affects the interpretation of quantum mechanics. 1