Structural completeness in propositional logics of dependence

In this paper we prove that three of the main propositional logics of dependence are structurally complete with respect to a class of substitutions under which the logics are closed. As these logics are not structural, the notions of admissibility and structural completeness have to be considered relative to classes of substitutions with respect to which they are, as we do in this paper. Dependence logic is a new logical formalism that characterizes the notion of “dependence” in social and natural sciences. First-order dependence logic was introduced by Väänänen [9] as a development of Henkin quantifier [2] and independence-friendly logic [3]. Recently, propositional dependence logic was studied and axiomatized in [10][8]. With a different motivation, Ciardelli and Roelofsen [1] introduced and axiomatized inquisitive logic, which turned out to be essentially equivalent to propositional intuitionistic dependence logic, a natural variant of propositional dependence logic. Dependency relations are characterized in these propositional logics of dependence by a new type of atoms =(~ p,q), called dependence atoms. Intuitively, the atom specifies that the proposition q depends completely on the propositions ~ p. The semantics of these logics is called team semantics, introduced by Hodges [4][5]. The basic idea of this new semantics is that properties of dependence cannot be manifested in single valuations, therefore unlike the case of classical propositional logic, formulas in propositional logics of dependence are evaluated on sets of valuations (called teams) instead. Propositional (intuitionistic) dependence logic as well as inquisitive logic characterize all downwards closed nonempty collections of teams. Therefore the three logics have the same expressive power. As a result of the feature of team semantics, the sets of theorems of these logics are closed under flat substitutions, but not closed under uniform substitution. In this paper, we prove that all admissible rules with respect to flat substitutions in these logics are derivable, that is, the three logics are structurally complete with respect to flat substitutions. There is a close connection between inquisitive logic and certain intermediate logics. The set of theorems of the former equals the negative variant of Kreisel-Putnam logic (KP), which is equal to the negative variant of Medvedev logic (ML). The logic KP is not structurally complete, whereas ML is known to be structurally complete but not hereditarily structurally complete. An interesting corollary we obtain in this paper is that the negative variants of both ML and KP are hereditarily structurally complete with respect to negative substitutions. Related research has been carried out in [6][7]. Our methods are of a syntactic nature, but we do think that the problems could also be approached from an algebraic point of view, an issue that we hope will be addressed in the future.