6,562 research outputs found
MacNeille completion and profinite completion can coincide on finitely generated modal algebras
Following Bezhanishvili & Vosmaer, we confirm a conjecture of Yde Venema by
piecing together results from various authors. Specifically, we show that if
is a residually finite, finitely generated modal algebra such that
has equationally definable principal
congruences, then the profinite completion of is isomorphic to its
MacNeille completion, and is smooth. Specific examples of such modal
algebras are the free -algebra and the free
-algebra.Comment: 5 page
Categories for Dynamic Epistemic Logic
The primary goal of this paper is to recast the semantics of modal logic, and
dynamic epistemic logic (DEL) in particular, in category-theoretic terms. We
first review the category of relations and categories of Kripke frames, with
particular emphasis on the duality between relations and adjoint homomorphisms.
Using these categories, we then reformulate the semantics of DEL in a more
categorical and algebraic form. Several virtues of the new formulation will be
demonstrated: The DEL idea of updating a model into another is captured
naturally by the categorical perspective -- which emphasizes a family of
objects and structural relationships among them, as opposed to a single object
and structure on it. Also, the categorical semantics of DEL can be merged
straightforwardly with a standard categorical semantics for first-order logic,
providing a semantics for first-order DEL.Comment: In Proceedings TARK 2017, arXiv:1707.0825
Canonical extension and canonicity via DCPO presentations
The canonical extension of a lattice is in an essential way a two-sided
completion. Domain theory, on the contrary, is primarily concerned with
one-sided completeness. In this paper, we show two things. Firstly, that the
canonical extension of a lattice can be given an asymmetric description in two
stages: a free co-directed meet completion, followed by a completion by
\emph{selected} directed joins. Secondly, we show that the general techniques
for dcpo presentations of dcpo algebras used in the second stage of the
construction immediately give us the well-known canonicity result for bounded
lattices with operators.Comment: 17 pages. Definition 5 was revised slightly, without changing any of
the result
Modal logic of planar polygons
We study the modal logic of the closure algebra , generated by the set
of all polygons in the Euclidean plane . We show that this logic
is finitely axiomatizable, is complete with respect to the class of frames we
call "crown" frames, is not first order definable, does not have the Craig
interpolation property, and its validity problem is PSPACE-complete
Topological Models of Columnar Vagueness
This paper intends to further the understanding of the formal properties of (higher-order) vagueness by connecting theories of (higher-order) vagueness with more recent work in topology. First, we provide a “translation” of Bobzien's account of columnar higher-order vagueness into the logic of topological spaces. Since columnar vagueness is an essential ingredient of her solution to the Sorites paradox, a central problem of any theory of vagueness comes into contact with the modern mathematical theory of topology. Second, Rumfitt’s recent topological reconstruction of Sainsbury’s theory of prototypically defined concepts is shown to lead to the same class of spaces that characterize Bobzien’s account of columnar vagueness, namely, weakly scattered spaces. Rumfitt calls these spaces polar spaces. They turn out to be closely related to Gärdenfors’ conceptual spaces, which have come to play an ever more important role in cognitive science and related disciplines. Finally, Williamson’s “logic of clarity” is explicated in terms of a generalized topology (“locology”) that can be considered an alternative to standard topology. Arguably, locology has some conceptual advantages over topology with respect to the conceptualization of a boundary and a borderline. Moreover, in Williamson’s logic of clarity, vague concepts with respect to a notion of a locologically inspired notion of a “slim boundary” are (stably) columnar. Thus, Williamson’s logic of clarity also exhibits a certain affinity for columnar vagueness. In sum, a topological perspective is useful for a conceptual elucidation and unification of central aspects of a variety of contemporary accounts of vagueness
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
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