In logic, a modal companion of a superintuitionistic (intermediate) logic L is a normal modal logic which interprets L by a certain canonical translation, described below. Modal companions share various properties of the original intermediate logic, which enables to study intermediate logics using tools developed for modal logic.
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Gödel–McKinsey–Tarski translation
Let A be a propositional intuitionistic formula. A modal formula T(A) is defined by induction on the complexity of A:
As negation is in intuitionistic logic defined by
T is called the Gödel translation or Gödel–McKinsey–Tarski translation. The translation is sometimes presented in slightly different ways: for example, one may insert
Modal companions
For any normal modal logic M which extends S4, we define its si-fragment ρM as
The si-fragment of any normal extension of S4 is a superintuitionistic logic. A modal logic M is a modal companion of a superintuitionistic logic L if
Every superintuitionistic logic has modal companions. The smallest modal companion of L is
where
For example, S4 itself is the smallest modal companion of the intuitionistic logic (IPC). The largest modal companion of IPC is the Grzegorczyk logic Grz, axiomatized by the axiom
over K. The smallest modal companion of the classical logic (CPC) is Lewis' S5, whereas its largest modal companion is the logic
More examples:
Blok–Esakia isomorphism
The set of extensions of a superintuitionistic logic L ordered by inclusion forms a complete lattice, denoted ExtL. Similarly, the set of normal extensions of a modal logic M is a complete lattice NExtM. The companion operators ρM, τL, and σL can be considered as mappings between the lattices ExtIPC and NExtS4:
It is easy to see that all three are monotone, and
Accordingly, σ and the restriction of ρ to NExtGrz are called the Blok–Esakia isomorphism. An important corollary to the Blok–Esakia theorem is a simple syntactic description of largest modal companions: for every superintuitionistic logic L,
Semantic description
The Gödel translation has a frame-theoretic counterpart. Let
on F, which identifies points belonging to the same cluster. Let
Then
Therefore, the si-fragment of a modal logic M can be defined semantically: if M is complete with respect to a class C of transitive reflexive general frames, then ρM is complete with respect to the class
The largest modal companions also have a semantic description. For any intuitionistic general frame
The skeleton of a Kripke frame is itself a Kripke frame. On the other hand, σF is never a Kripke frame if F is a Kripke frame of infinite depth.
Preservation theorems
The value of modal companions and the Blok–Esakia theorem as a tool for investigation of intermediate logics comes from the fact that many interesting properties of logics are preserved by some or all of the mappings ρ, σ, and τ. For example,
Other properties
Every intermediate logic L has an infinite number of modal companions, and moreover, the set
The Gödel translation can be applied to rules as well as formulas: the translation of a rule
is the rule
A rule R is admissible in a logic L if the set of theorems of L is closed under R. It is easy to see that R is admissible in a superintuitionistic logic L whenever T(R) is admissible in a modal companion of L. The converse is not true in general, but it holds for the largest modal companion of L.