  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.
