Characterizing Strongly First Order Dependencies: The Non-Jumping Relativizable Case

Abstract

Team Semantics generalizes Tarski's Semantics for First Order Logic by allowing formulas to be satisfied or not satisfied by sets of assignments rather than by single assignments. Because of this, in Team Semantics it is possible to extend the language of First Order Logic via new types of atomic formulas that express dependencies between different assignments.Some of these extensions are much more expressive than First Order Logic proper; but the problem of which atoms can instead be added to First Order Logic without increasing its expressive power is still unsolved.In this work, I provide an answer to this question under the additional assumptions (true of most atoms studied so far) that the dependency atoms are relativizable and non-jumping. Furthermore, I show that the global (or Boolean) disjunction connective can be added to any strongly first order family of dependencies without increasing the expressive power, but that the same is not true in general for non strongly first order dependencies.

Strongly First Order, Domain Independent Dependencies: the Union-Closed Case

“…By Theorem 21 of [3], D ↑ ∪ =(•) is strongly first order. By Proposition 14 of [5], if D is a strongly first order family of dependencies, every sentence of FO(D, ⊔) is equivalent to some first order sentence. The result follows at once.…”

“…In [5], a characterization was found for the strongly first order dependencies that are domain independent 13 and have the (somewhat technical) property of being non-jumping, which in particular is true of strongly first order downwards closed dependencies (even of those without the empty team property): Theorem 3 ([5], Corollary 31). Let D be a non-jumping (or, in particular, downwards closed), domain independent 14 , strongly first order dependency.…”