A Sahlqvist theorem for distributive modal logic

  title={A Sahlqvist theorem for distributive modal logic},
  author={Mai Gehrke and Hideo Nagahashi and Yde Venema},
  journal={Ann. Pure Appl. Log.},

Modal and temporal extensions of non-distributive propositional logics

It is established that the absence of a distributivity assumption of conjunctions over disjunctions and conversely has no effect on the interpretation of boxes and diamonds, which are interpreted exactly as in classical normal modal logics.

The Distributive Full Lambek Calculus with Modal Operators

In this paper, we study logics of bounded distributive residuated lattices with modal operators considering $\Box$ and $\Diamond$ in a noncommutative setting. We introduce relational semantics for

A Duality for Distributive Unimodal Logic

A Hennessy-Milner theorem is obtained which is the expansion of distributive unimodal logic by bi-intuitionistic connectives and a completeness theorem is proved which embeds each such algebra in the complex algebra of its canonical modal frame.

An application of Sahlqvist Theory to Bisorted Modal Logic

We axiomatize two bisorted modal logics: basic hybrid logic and sub- set modal logic. The systems are presented as normal multimodal logics allowing us to freely apply Sahlqvist Theory. The proposed

Sahlqvist theory for impossible worlds

Unified correspondence theory is extended to Kripke frames with impossible worlds and their associated regular modal logics and it is shown that additivity and multiplicativity turn out to be key to extend Jonsson’s original proof of canonicity to the full Sahlqvist class of certain regular distributives naturally generalizing distributive modal logic.

Algorithmic correspondence and canonicity for distributive modal logic


The motivation of this thesis is the idea of extending the results in [13] to the case of logics with fixpoints and with a non-classical base. We focus in particular on the intuitionistic modal

Algorithmic correspondence and canonicity for non-distributive logics

Extending Łukasiewicz Logics with a Modality: Algebraic Approach to Relational Semantics

Algebraic tools are applied and developed to generate complete modal n + 1-valued logics and the many-valued counterparts of Shalqvist canonicity result are obtained.

Unified correspondence and proof theory for strict implication

Gentzen-style cut-free sequent calculi for BDFNL and its extensions with analytic rules which are transformed from strict implication sequents, are developed.



Priestley Duality, a Sahlqvist Theorem and a Goldblatt-Thomason Theorem for Positive Modal Logic

A GoldblattThomason theorem that characterizes the elementary classes of frames of that semantics that are definable by sets of sequents is proved and can be seen also as arising from the Kripke semantics for a suitable intuitionistic modal logic.

Intuitionistic Modal Logics as Fragments of Classical Bimodal Logics

Godel's translation of intuitionistic formulas into modal ones provides the well-known embedding of intermediate logics into extensions of Lewis' system S4, which re ects and sometimes preserves

Duality and Canonical Extensions of Bounded Distributive Lattices with Operators, and Applications to the Semantics of Non-Classical Logics II

This paper considers logics that are sound and complete with respect to varieties of distributive lattices with certain classes of well-behaved operators for which a Priestley-style duality holds, and presents a way of constructing topological and non-topological Kripke-style models for these types of logics.

Duality and Canonical Extensions of Bounded Distributive Lattices with Operators, and Applications to the Semantics of Non-Classical Logics I

The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of propositional logics and show that finitely-generated varieties of distributive lattices with operators are closed under canonical embedding algebras.

A New Proof of Sahlqvist's Theorem on Modal Definability and Completeness

There are not many global results on modal logics. One of these is the following theorem by Sahlqvist on completeness and correspondence for a wide class of modal formulae (including many well known

A New Semantics for Positive Modal Logic

The paper provides a new semantics for positive modal logicusing Kripke frames having a quasi ordering ≤ on the set of possible worlds and an accessibility relation R connected to the quasi ordering

The McKinsey axiom is not canonical

It is shown here that this axiom is not valid in the canonical frame for KM, answering a question first posed in the Lemmon-Scott manuscript [Lemmon, 1966], and has been of historical importance in the development of the authors' understanding of intensional model theory.

Simulating polyadic modal logics by monadic ones

It is proved that this simulation operator transfers several useful properties of modal logics, such as finite/recursive axiomatizability, frame completeness and the finite model property, canonicity and first-order definability.

Categories of frames for modal logic

The category theory a reader needs to know is in the first twenty pages of [7], and the proofs of duality involve some rather detailed calculations, which have been omitted.