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### Page 1534 of Mathematical Reviews Vol. , Issue 2002C [page]

2002 Mathematical Reviews
equivalence of canonical structures.  ...  (G4: « an infinite cardinal) is the sequence of canonical structures for A of size 2*. '¥4 is the set of quasi- modal sentences true in G*.  ...

### Correspondence, Canonicity, and Model Theory for Monotonic Modal Logics [article]

Kentarô Yamamoto
2020 arXiv   pre-print
The elementary equivalence here can be relativized to the classes of monotonic, quasi-filter, augmented quasi-filter, filter, or augmented filter neighborhood frames, respectively.  ...  We prove analogues of the Goldblatt-Thomason Theorem and Fine's Canonicity Theorem for classes of monotonic neighborhood frames closed under elementary equivalence in coalgebraic predicate logic.  ...  Fine's Canonicity Theorem By the dual equivalence between monotonic modal logics and varieties of BAMs [13, Chapter 7], we will state our version of Fine's Canonicity Theorem in an algebraic manner.  ...

### Fine's Theorem on First-Order Complete Modal Logics [article]

Robert Goldblatt
2017 arXiv   pre-print
The ultimate point is that the construction of the canonical frame of a modal algebra does not commute with the ultrapower construction.  ...  Fine's influential Canonicity Theorem states that if a modal logic is determined by a first-order definable class of Kripke frames, then it is valid in its canonical frames.  ...  F Lκ for infinite κ all satisfy the same quasi-modal sentences. • If C qm (L) is the class of all frames that satisfy the same quasi-modal sentences as the canonical frames F Lκ , then the modal logic  ...

### Strong Completeness of Coalgebraic Modal Logics

Lutz Schröder, Dirk Pattinson, Marc Herbstritt
2009 Symposium on Theoretical Aspects of Computer Science
But there is more to modal logic than Kripke semantics, and indeed the natural semantic structures used to interpret a large class of modal logics go beyond pure relations.  ...  Canonical models are of central importance in modal logic, in particular as they witness strong completeness and hence compactness.  ...  By Thm. 2.5, there exists a quasi-canonical Kripke model for all normal modal logics.  ...

### Strong Completeness of Coalgebraic Modal Logics [article]

Lutz Schröder, Dirk Pattinson
2009 arXiv   pre-print
Canonical models are of central importance in modal logic, in particular as they witness strong completeness and hence compactness.  ...  Here, we present a generic canonical model construction in the semantic framework of coalgebraic modal logic, which pinpoints coherence conditions between syntax and semantics of modal logics that guarantee  ...  By Thm. 2.5, there exists a quasi-canonical Kripke model for all normal modal logics.  ...

### Algebraic polymodal logic: a survey

R Goldblatt
2000 Logic Journal of the IGPL
Issues discussed include validity in canonical structures, completeness and incompleteness under the relational semantics, and characterisations of logics by elementary classes of structures and by finite  ...  powerset algebras of an ultraproduct-closed class of structures generate a variety of BAO's closed under canonical extensions. • The question of whether a logic is complete with respect to some class of  ...  Since Λ e is canonical (5.5.2) , Λ e ⊆ Λ c . • The canonical structures of Λ c also have Ψ Λ as their quasi-modal theory.  ...

### On Sahlqvist theory for hybrid logics

2015 Journal of Logic and Computation
of Kripke frames and is canonical.  ...  Since modal logic on the frame level is essentially second-order, computing the first-order correspondence of a modal formula is a kind of second-order quantifier elimination.  ...  In [15, Theorem 7.20] , it is shown that all inductive formulas are filter-canonical and hence every normal modal logic axiomatized by inductive formulas is sound and complete with respect to its canonical  ...

### Completeness of Pledger's modal logics of one-sorted projective and elliptic planes

Rob Goldblatt
2021 The Australasian Journal of Logic
Ken Pledger devised a one-sorted approach to the incidence relation of plane geometries, using structures that also support models of propositional modal logic.  ...  He introduced a modal system 12g that is valid in one-sorted projective planes, proved that it has finitely many non-equivalent modalities, and identified all possible modality patterns of its extensions  ...  The ones with a single 2 -equivalance class are called elliptic. Now a structure of the form ℱ = ( , ) is known in modal logic as a Kripke frame, although Kripke called it a model structure [9] .  ...

### Subordination algebras in modal logic [article]

Laurent De Rudder and Georges Hansoul and Valentine Stetenfeld
2020 arXiv   pre-print
This motivates for an algebraic (in the sense of universal algebra) study of those relational structures that are subordinate algebras.  ...  The aim of this paper is to show that even if the natural algebraic semantic for modal (normal) logic is modal algebra, the more general class of subordination algebras (roughly speaking, the non symmetric  ...  Such a structure is called descriptive quasi-modal space by Celani in [7, Definition 10]. Remark 1.5.  ...

### A Coalgebraic Approach to Process Equivalence and a Coinduction Principle for Traces

Bartek Klin
2004 Electronical Notes in Theoretical Computer Science
An abstract coalgebraic approach to well-structured relations on processes is presented, based on notions of tests and test suites.  ...  It turns out that most equivalences from the so-called van Glabbeek spectrum can be described by well-structured test suites.  ...  The most popular coalgebraic approach to bisimulation, based on coalgebra spans [1, 15] , focuses on a single, canonical notion of equivalence for every notion of behaviour.  ...

### Canonical Extensions, Esakia Spaces, and Universal Models [chapter]

Mai Gehrke
2014 Leo Esakia on Duality in Modal and Intuitionistic Logics
In preparation for this we show that the categories of Heyting and modal algebras are both equivalent to certain categories of maps between distributive lattices and Boolean algebras.  ...  Finally we relate the N -universal model of intuitionistic logic to the Esakia space of the corresponding Heyting algebra via bicompletion of quasi-uniform spaces.  ...  U, or equivalently, given by the partial order reflection of the quasi-order corresponding to U.  ...

### Algorithmic Correspondence and Canonicity for Possibility Semantics [article]

Zhiguang Zhao
2016 arXiv   pre-print
Specifically, we define the possibility semantics version of the algorithm ALBA, and an adapted interpretation of the expanded modal language used in the algorithm.  ...  We prove the soundness of the algorithm with respect to both (the dual algebras of) full possibility frames and (the dual algebras of) filter-descriptive possibility frames.  ...  in B RO ; • An algorithm which transforms a given modal formula ϕ( p) into equivalent pure quasi-inequalities Pure(ϕ( p)); • A soundness proof of the algorithm with respect to B RO ; • A syntactically  ...

### Algorithmic correspondence for relevance logics, bunched implication logics, and relation algebras: the algorithm PEARL and its implementation (Technical Report) [article]

Willem Conradie, Valntin Goranko, Peter Jipsen
2021 arXiv   pre-print
We also show that all formulae on which PEARL succeeds are canonical, i.e., preserved under canonical extensions of relevant algebras.  ...  of formulas of the language of relevance logics RL in terms of the standard Routley-Meyer relational semantics.  ...  It gives a syntactic characterization of a class of modal formulas which define first-order conditions on Kripke frames and which are canonical, hence, when added to the basic normal modal logic K, they  ...

### Explicit non-normal modal logic [article]

Atefeh Rohani, Thomas Studer
2022 arXiv   pre-print
Faroldi argues that deontic modals are hyperintensional and thus traditional modal logic cannot provide an appropriate formalization of deontic situations.  ...  We establish soundness and completeness with respect to various models and we study the problem of realization.  ...  As usual the proof is by induction on the structure of F . We only show the case when F is [t]G.  ...

### Admissible Bases Via Stable Canonical Rules

Nick Bezhanishvili, David Gabelaia, Silvio Ghilardi, Mamuka Jibladze
2016 Studia Logica: An International Journal for Symbolic Logic