Log In Sign Up

Gödel-McKinsey-Tarski and Blok-Esakia for Heyting-Lewis Implication

by   Jim de Groot, et al.

Heyting-Lewis Logic is the extension of intuitionistic propositional logic with a strict implication connective that satisfies the constructive counterparts of axioms for strict implication provable in classical modal logics. Variants of this logic are surprisingly widespread: they appear as Curry-Howard correspondents of (simple type theory extended with) Haskell-style arrows, in preservativity logic of Heyting arithmetic, in the proof theory of guarded (co)recursion, and in the generalization of intuitionistic epistemic logic. Heyting-Lewis Logic can be interpreted in intuitionistic Kripke frames extended with a binary relation to account for strict implication. We use this semantics to define descriptive frames (generalisations of Esakia spaces), and establish a categorical duality between the algebraic interpretation and the frame semantics. We then adapt a transformation by Wolter and Zakharyaschev to translate Heyting-Lewis Logic to classical modal logic with two unary operators. This allows us to prove a Blok-Esakia theorem that we then use to obtain both known and new canonicity and correspondence theorems, and the finite model property and decidability for a large family of Heyting-Lewis logics.


page 1

page 2

page 3

page 4


Wijesekera-style constructive modal logics

We define a family of propositional constructive modal logics correspond...

Relevant Reasoners in a Classical World

We develop a framework for epistemic logic that combines relevant modal ...

Positive (Modal) Logic Beyond Distributivity

We present a duality for non-necessarily-distributive (modal) lattices a...

Probabilistic logics based on Riesz spaces

We introduce a novel real-valued endogenous logic for expressing propert...

Stone-Type Dualities for Separation Logics

Stone-type duality theorems, which relate algebraic and relational/topol...

From Propositional Logic to Plausible Reasoning: A Uniqueness Theorem

We consider the question of extending propositional logic to a logic of ...

Computability logic: Giving Caesar what belongs to Caesar

The present article is a brief informal survey of computability logic --...