73 research outputs found
Changing a semantics: opportunism or courage?
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
Vietoris endofunctor for closed relations and its de Vries dual
We generalize the classic Vietoris endofunctor to the category of compact
Hausdorff spaces and closed relations. The lift of a closed relation is done by
generalizing the construction of the Egli-Milner order. We describe the dual
endofunctor on the category of de Vries algebras and subordinations. This is
done in several steps, by first generalizing the construction of Venema and
Vosmaer to the category of boolean algebras and subordinations, then lifting it
up to -subordination algebras, and finally using MacNeille
completions to further lift it to de Vries algebras. Among other things, this
yields a generalization of Johnstone's pointfree construction of the Vietoris
endofunctor to the category of compact regular frames and preframe
homomorphisms
A generalization of de Vries duality to closed relations between compact Hausdorff spaces
Stone duality generalizes to an equivalence between the categories StoneR of Stone spaces and closed relations and BAS of boolean algebras and subordination relations. Splitting equivalences in StoneR yields a category that is equivalent to the category KHausR of compact Hausdorff spaces and closed relations. Similarly, splitting equivalences in BAS yields a category that is equivalent to the category De VS of de Vries algebras and compatible subordination relations. Applying the machinery of allegories then yields that KHausR is equivalent to De VS, thus resolving a problem recently raised in the literature.The equivalence between KHausR and De VS further restricts to an equivalence between the category KHausR of compact Hausdorff spaces and continuous functions and the wide subcategory De VF of De VS whose morphisms satisfy additional conditions. This yields an alternative to de Vries duality. One advantage of this approach is that composition of morphisms is usual relation composition
From Contact Relations to Modal Operators, and Back
One of the standard axioms for Boolean contact algebras says that if a region
x is in contact with the join of y and z , then x is in contact with at least one of the two regions. Our intention is to examine a stronger version of this axiom according to which if x is in contact with the supremum of some family S of regions, then there is a y in S that is in contact with x. We study a modal possibility operator which is definable in complete algebras in the presence of the aforementioned axiom, and we prove that the class of complete algebras satisfying the axiom is closely related to the class of modal KTBalgebras. We also demonstrate that in the class of complete extensional contact algebras the axiom is equivalent to the statement: every region is isolated. Finally, we present an interpretation of the modal operator in the class of the so-called resolution contact algebras
On the structure of modal and tense operators on a boolean algebra
We study the poset NO(B) of necessity operators on a boolean algebra B. We
show that NO(B) is a meet-semilattice that need not be distributive. However,
when B is complete, NO(B) is necessarily a frame, which is spatial iff B is
atomic. In that case, NO(B) is a locally Stone frame. Dual results hold for the
poset PO(B) of possibility operators. We also obtain similar results for the
posets TNO(B) and TPO(B) of tense necessity and possibility operators on B. Our
main tool is Jonsson-Tarski duality, by which such operators correspond to
continuous and interior relations on the Stone space of B.Comment: 18 page
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