Dependency as Question Entailment

Abstract

In the past few years, a tight connection has emerged between logics of dependency and logics of questions. The aim of this paper is to show that this connection stems from a fundamental relation existing between dependency and questions. Once we expand our view on logic by bringing questions into the picture, dependency emerges as a facet of the fundamental logical notion of entailment, namely entailment among questions. Besides providing an insightful conceptual picture, this perspective yields a general and well-behaved way of dealing with dependency in logical systems.

Appendix

Proof of Proposition 22.

We first observe one basic property of our system. Replacement of equivalents:for all formulas $\varphi,\psi,\chi \in \mathcal{L}$,  ψ ⊣ ⊢ χ  implies $\;\varphi [\psi /p] \dashv \vdash \varphi [\chi /p]$.

This claim can be established straightforwardly by induction on φ. Equipped with this observation, we will proceed to prove Proposition 22by induction on φ. If φis an atom or ⊥ , then $\mathcal{R}(\varphi ) =\{\varphi \}$and the claim is trivially true. Let us now assume the claim holds for φand ψ, that is, let us assume

and

, and let us proceed to prove that it also holds for φ∧ψ, φ → ψ, φ∨ψ, and

.

• φ∧ψ. By induction hypothesis and replacement of equivalents, we have

  

  Proposition 21ensures that any entailment pattern which is valid in intuitionistic logic is also valid in QD _FO , when

  

  is interpreted as intuitionistic In particular, this ensures that the distributivity of ∧ over

  

  is

  

  is provable. We thus have

  

  Given that $\mathcal{R}(\varphi \wedge \psi ) =\{\alpha \wedge \beta \,\vert \,\alpha \in \mathcal{R}(\varphi )\text{ and }\beta \in \mathcal{R}(\psi )\}$, the latter formula is nothing but

  

  .

• φ → ψ. By induction hypothesis and replacement of equivalents, we have

  

  By reasoning intuitionistically with →, ∧ and

  

  we then have

  

  Now, for any $\alpha \in \mathcal{R}(\varphi )$, we have

  

  The right-to-left direction holds by simple intuitionistic reasoning, while the left-to-right direction simply amounts to the Split rule of our system (recall that resolutions are classical formulas, so this rule is indeed applicable). By replacement of equivalents, we then have the following:

  

  Finally, using again the distributivity of ∧ over

  

  we get that the right-hand side is provably equivalent with the following formula:

  

  Given that $\mathcal{R}(\varphi \longrightarrow \psi ) =\{\bigwedge _{\alpha \in \mathcal{R}(\varphi )}\alpha \longrightarrow f(\alpha )\,\vert \,f: \mathcal{R}(\varphi )\longrightarrow \mathcal{R}(\psi )\}$, the latter formula is nothing but

  

  .

• φ∨ψ. By induction hypothesis and replacement of equivalents, we have

  

  Notice that we have

  

  : the left-to-right direction of this equivalence is obtained by an application of the rule (∨d) of our system, while the right-to-left direction is easily proved using the rules

  

  ,

  

   (∨r).

  Also, notice that by using the rule (∨c), we can make sure that distributivity holds not only on the right side of a tensor, but also on the left side. Then, multiple applications of distributivity on both sides of the tensor yield

  

  Given that $\mathcal{R}(\varphi \vee \psi ) =\{\alpha \vee \beta \,\vert \,\alpha \in \mathcal{R}(\varphi )\text{ and }\beta \in \mathcal{R}(\psi )\}$, the formula on the right is nothing but

  

  .

• 

  . By induction hypothesis and replacement of equivalents we have

  

  Given that

  

  , the above formula is simply

  

  .