Sahlqvist via Translation

Abstract

In recent years, unified correspondence has been developed as a generalized Sahlqvist theory which applies uniformly to all signatures of normal and regular (distributive) lattice expansions. A fundamental tool for attaining this level of generality and uniformity is a principled way, based on order theory, to define the Sahlqvist and inductive formulas and inequalities in every such signature. This definition covers in particular all (bi-)intuitionistic modal logics. The theory of these logics has been intensively studied over the past seventy years in connection with classical polyadic modal logics, using versions of Gödel-McKinsey-Tarski translations, suitably defined in each signature, as main tools. In view of this state-of-the-art, it is natural to ask (1) whether a general perspective on Gödel-McKinsey-Tarski translations can be attained, also based on order-theoretic principles like those underlying the general definition of Sahlqvist and inductive formulas and inequalities, which accounts for the known Gödel-McKinsey-Tarski translations and applies uniformly to all signatures of normal (distributive) lattice expansions; (2) whether this general perspective can be used to transfer correspondence and canonicity theorems for Sahlqvist and inductive formulas and inequalities in all signatures described above under Gödel-McKinsey-Tarski translations.In the present paper, we set out to answer these questions. We answer (1) in the affirmative; as to (2), we prove the transfer of the correspondence theorem for inductive inequalities of arbitrary signatures of normal distributive lattice expansions. We also prove the transfer of canonicity for inductive inequalities, but only restricted to arbitrary normal modal expansions of bi-intuitionistic logic. We also analyze the difficulties involved in obtaining the transfer of canonicity outside this setting, and indicate a route to extend the transfer of canonicity to all signatures of normal distributive lattice expansions.

Slanted Canonicity of Analytic Inductive Inequalities

“…Example 2. 16. In light of the previous definitions and the discussion in Example 2.12, the modal bi-intuitionistic inequality p ∨ q ≤ q ∧ ( r − p) is ε-Sahlqvist (and hence inductive) for ε(p, q, r ) = (1, 1, ∂).…”

“…In [16], Gödel-McKinsey-Tarski type translations (GMT-type translations) are used to obtain Sahlqvist correspondence and canonicity as transfer results in a number of settings. Specifically, GMT-type translations τ ε are defined parametrically in each order-type on a set PROP of propositional variables so as to preserve the syntactic shape of (Ω, ε)-inductive inequalities in passing from arbitrary DLE-languages to corresponding target Boolean algebra expansion languages (BAElanguages) enriched with additional S4-modalities ≥ and ≤ .…”

“…While correspondence via translation is obtained in full generality for inductive inequalities in arbitrary DLE-languages (cf. [16, Theorem 6.1]), the canonicity via translation of inductive inequalities is obtained in [16] only in the restricted setting of normal modal expansions of bi-Heyting algebras (bHAEs) (cf. [16, Theorem 7.1]).…”

“…As to the problem of obtaining Sahlqvist-type results for certain non-classical logics by reduction to classical Sahlqvist theory by means of GMT-type translations, in [24], the correspondencevia-translation problem has been completely solved for Sahlqvist inequalities in the signature of Distributive Modal Logic, but the corresponding canonicity-via-translation problem, reported to be much harder, was not addressed there, and the canonicity result was obtained following the methodology introduced by Jónsson [31]. In [16], results on both correspondence-via-translation and canonicity-via-translation for inductive inequalities in arbitrary signatures of normal distributive lattice expansions (aka normal DLE-logics) are presented, but the canonicity via translation is restricted to arbitrary normal expansions of bi-Heyting algebras. The source of the additional difficulties was identified in the fact that the algebraic interpretations of the S4-modal operators used to define the GMT-type translations are not defined on each original algebra but might map elements of it to closed or open elements of its canonical extension.…”

“…Namely, a strengthening of the canonicity result for subordination algebras of [17] is obtained as a direct application, simply by recognizing that the s-Sahlqvist formulas exactly coincide with the analytic 1-Sahlqvist formulas in the classical normal modal/tense logic signature. Moreover, the canonicity-via-translation result of [16] is extended to normal DLE-logics in arbitrary signatures for a subclass of analytic inductive inequalities referred to as transferable (cf. Definition 5.1); the syntactic shape of the formulas in this subclass guarantees that the suitable parametric translation of each formula in this class is analytic inductive, so that the generalized canonicity result obtained in this article applies to them.…”