Upwards closed dependencies in team semantics

Abstract

We prove that adding upwards closed first-order dependency atoms to first-order logic with team semantics does not increase its expressive power (with respect to sentences), and that the same remains true if we also add constancy atoms. As a consequence, the negations of functional dependence, conditional independence, inclusion and exclusion atoms can all be added to first-order logic without increasing its expressive power.Furthermore, we define a class of bounded upwards closed dependencies and we prove that unbounded dependencies cannot be defined in terms of bounded ones.

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

“…This, however, has not been answered yet. In [9], a very general family of dependencies was found that does not increase the expressive power of First Order Logic if added to it; but it is an open question whether any dependency that has this property is definable in terms of dependencies in that family (and, in fact, in this work we will show that this is false).…”

“…It certainly is true that every Inclusion Logic sentence is equivalent to some Independence Logic sentence, that every Dependence Logic sentence is equivalent to some Independence Logic sentence, and that every Independence Logic sentence is equivalent to some Dependence Logic sentence; but on the other hand, it is not true that every Inclusion Logic formula, or every Independence Logic one, is equivalent to some Dependence Logic formula. This follows at once from the following classification: Definition 9 (Empty Team Property, Closure Properties - [29,8,9]) Let D be a generalized dependency. Then…”

“…10 Additionally, it is clear that any dependency E that is definable in FO(D ↑ , =(·)), in the sense that there exists some formula φ (v) ∈ FO(D ↑ , =(·)) over the empty signature such that M |= X Ev ⇔ M |= X φ (v), is itself strongly first order. This can be used, as discussed in [9], to show that for instance the negated inclusion atoms…”

“…The choice of the variable v is of course irrelevant here, and we could have defined NE as a 0-ary dependency instead; but treating it as a 1-ary dependency is formally simpler 9. That is, M |= X =(v) iff for all s, s ′ ∈ X, s(v) = s ′ (v) 10.…”

Interventionist Counterfactuals on Causal Teams

“…Notice that it might be impossible to define consistently the union of two causal teams 6. To the best of our knowledge, this connective has been used, with different notation, in[8], as a special case of the maximal implication introduced in[15].…”