First-order syntactic characterizations of minimal entailment, domain-minimal entailment, and Herbrand entailment

Abstract

This paper investigates a consistent versioncwa _s of Reiter's closed-world assumptioncwa. It provides a syntactic characterization of minimal entailment ⊢ _min : for every ∨-sentence ϕ and for every ∨-theory Σ,

$$\varphi \varepsilon cwa_S \left( \sum \right)iff\sum \vdash _{min} \varphi .$$

A version of this characterization remains valid if not all relations are subject to minimization. These two characterizations do not use the domain-closure axiom nor the unique-names assumptionuna, although they may be easily modified to ones that incorporateuna. A similar result for Herbrand entailment ⊢ _Her , by means of generalized domain-closure axiomdca _s , is provided: for every ∨-sentence ϕ and every ∨-theory Σ,

$$\varphi \varepsilon dca_S \left( \sum \right)iff\sum \vdash _{Her} \varphi .$$

Finally, a syntactic characterization of domain-minimal entailment ⊢ _dom in terms of a versionmda _s of minimal-domain assumption is shown: for every ∨-sentence ϕ and for every ∨-theory Σ,

$$\varphi \varepsilon mda_S \left( \sum \right)iff\sum \vdash _{dom} \varphi .$$

The proving power of these entailments is then evaluated. In particular, it is shown that (1) neither ⊢ _min nor its versions are strong enough to derive positive sentences from Σ unless they are first-order provable from Σ however, a double application of ⊢ _min has enough power to derive such positive sentences; (2) ⊢ _Her has the strength of infinitary rule of inference but cannot derive existential nor quantifier-free sentences from Σ unless they are first-order provable from Σ (3) ⊢ _Her and ⊢ _dom can derive from Σ certain positive facts about = which are otherwise unprovable from Σ and (4) ⊢ _dom cannot derive from Σ sentences without positive occurrences of = unless they are first-order provable from Σ. Moreover, the paper relatescwa _s to Reiter'scwa and to Minker's generalized closed-world assumptionGCWA and its extension.