24,480 research outputs found
A new coalgebraic semantics for positive modal logic
Positive Modal Logic is the restriction of the modal local consequence relation defined by the class of all Kripke models to the propositional negation-free modal language. The class of positive modal algebras is the one canonically associated with PML according to the theory of Abstract Algebraic Logic. Celani and Jansana established a Priestley-style duality the category of positive modal algebras and the category of K+-spaces. In this paper, we establish a categorical equivalence between the category K+ of K+-spaces and the category Coalg(V) of coalgebras of a suitable endofunctor V on the category of Priestley spaces
Positive Modal Logic Beyond Distributivity
We develop a duality for (modal) lattices that need not be distributive, and
use it to study positive (modal) logic beyond distributivity, which we call
weak positive (modal) logic. This duality builds on the Hofmann, Mislove and
Stralka duality for meet-semilattices. We introduce the notion of
-persistence and show that every weak positive modal logic is
-persistent. This approach leads to a new relational semantics for weak
positive modal logic, for which we prove an analogue of Sahlqvist
correspondence result
Named Models in Coalgebraic Hybrid Logic
Hybrid logic extends modal logic with support for reasoning about individual
states, designated by so-called nominals. We study hybrid logic in the broad
context of coalgebraic semantics, where Kripke frames are replaced with
coalgebras for a given functor, thus covering a wide range of reasoning
principles including, e.g., probabilistic, graded, default, or coalitional
operators. Specifically, we establish generic criteria for a given coalgebraic
hybrid logic to admit named canonical models, with ensuing completeness proofs
for pure extensions on the one hand, and for an extended hybrid language with
local binding on the other. We instantiate our framework with a number of
examples. Notably, we prove completeness of graded hybrid logic with local
binding
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
The Relevant Logic E and Some Close Neighbours: A Reinterpretation
This paper has two aims. First, it sets out an interpretation of the relevant logic E of relevant entailment based on the theory of situated inference. Second, it uses this interpretation, together with Anderson and Belnap’s natural deduc- tion system for E, to generalise E to a range of other systems of strict relevant implication. Routley–Meyer ternary relation semantics for these systems are produced and completeness theorems are proven
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