Structural completeness in propositional logics of dependence

Abstract

In this paper we prove that three of the main propositional logics of dependence (including propositional dependence logic and inquisitive logic), none of which is structural, are structurally complete with respect to a class of substitutions under which the logics are closed. We obtain an analogues result with respect to stable substitutions, for the negative variants of some well-known intermediate logics, which are intermediate theories that are closely related to inquisitive logic.

Propositional team logics

“…A routine inductive proof shows that ¬α ≡ α ∼ for all classical formulas α (or see [20] for the proof), i.e., our negation as defined in Definition 4.2 coincides with the above syntactic negation when applied to classical formulas. It is also worth noting that our negation corresponds to the defined connective ∼↓ in Hodges [17,18].…”

“…The logics PU + and PU are not closed under uniform substitution either (for at least the trivial reason that strings of the form ¬(φ ψ) are not well-formed formulas), but nontrivial counter-examples of the above kind for the two logics are yet to be found. It was shown in [3] and [20] that propositional logics of dependence are, nevertheless, closed under flat substitutions, i.e., substitutions σ such that σ(p) has the flatness property for any propositional variable p. Using the method in [20], we will prove in this section that propositional team logics are closed under classical substitutions, i.e., substitutions σ such that σ(p) is a classical formula (i.e., a formula in the language of CPL) for any propositional variable p. Let us start by examining in detail the notion of substitution in our logics. The well-formed formulas of the propositional team logics we consider in this paper are assumed to be in negation normal form and we do not allow arbitrary formulas to occur in a dependence or independence atom.…”

Propositional logics of dependence

“…For this reason in the deduction systems of these logics to be given the axioms or rules should not be read as schemata unless otherwise specified. On the other hand, it is shown in [5] and [20] that the consequence relations of these logics are closed under flat substitutions, i.e., substitutions σ such that σ(p) is flat for all propositional variables p.…”

Modal dependence logics: axiomatizations and model-theoretic properties

“…, φ n , ψ) with arbitrary arguments are not necessarily well-formed formulas of the logics. For more details on substitution in dependence logics, we refer the reader to [5,15].…”

“…In the context of propositional logics of dependence, flat formulas admit a certain characterization theorem; see [5,15] for the proof. We now generalize this characterization result to the modal case by using the disjunctive normal form.…”