Slanted Canonicity of Analytic Inductive Inequalities

Abstract

We prove an algebraic canonicity theorem for normal LE-logics of arbitrary signature, in a generalized setting in which the non-lattice connectives are interpreted as operations mapping tuples of elements of the given lattice to closed or open elements of its canonical extension. Interestingly, the syntactic shape of LE-inequalities which guarantees their canonicity in this generalized setting turns out to coincide with the syntactic shape of analytic inductive inequalities , which guarantees LE-inequalities to be equivalently captured by analytic structural rules of a proper display calculus. We show that this canonicity result connects and strengthens a number of recent canonicity results in two different areas: subordination algebras, and transfer results via Gödel-McKinsey-Tarski translations.

Unified inverse correspondence for DLE-Logics

“…In what follows, we adapt the ALBA run of (ϕ ≤ ψ)[α/!x, β/!y, γ/!z, δ/!w] described in [15,Section 1.6]. In the DLE-setting, we can drop the assumption that at least one vector among γ and δ be nonempty, since we can assume w.l.o.g.…”

“…Approximation rules. Here below we provide the approximation rules 15 relative to each f ∈ F and g ∈ G of arity at least 1: for each 1…”

“…those logics the algebraic semantics of which is given by varieties of normal/regular lattice expansions (LEs), and their expansions with fixed points [5, 3] [5]. Not only this very high level of generality allows to extend the benefits of correspondence and canonicity results to many well known logical systems such as bi-intuitionistic (modal) logic, the Lambek-Grishin calculus [24], and the multiplicative-additive fragment of linear logic [16], but this new point of view paves the way to several developments and connections among the meta-theories of several logical frameworks, examples of which are a general perspective on Gödel-McKinsey-Tarski translations and correspondence/canonicity transfer results [13,15], systematic connections among different relational semantics of a given logic [4], and systematic connections between correspondence-theoretic results and the proof-theoretic behaviour of logical frameworks [21,23,2,22,20].…”

Correspondence and Canonicity Theory of Quasi-Inequalities and $Π_2$-Statements in Modal Subordination Algebras

“…

In the present paper, we study the correspondence and canonicity theory of modal subordination algebras and their dual Stone space with two relations, generalizing correspondence results for subordination algebras in [13,14,15,25]. Due to the fact that the language of modal subordination algebras involves a binary subordination relation, we will find it convenient to use the so-called quasi-inequalities and Π 2 -statements.

…”

“…Indeed, in the literature, there are works of correspondence theory for subordination algebras and subordination spaces. In [13,14,15], de Rudder et al studied correspondence theory of subordination algebras in the perspective of quasi-modal operators. In [25], Santoli studied the topological correspondence theory between conditions on algebras and first-order conditions on the dual subordination spaces, in the language of a binary connective definable from the squigarrow associated with the subordination relation, using the so-called ∀∃-statements [3,5,6] (which we call Π 2 -statements in the present paper).…”