First-Order Default Logic

We propose a model theory for full first-order default logic that allows both closed and non-closed default theories. Beginning with first-order languages without logical equality, we note how Henkin’s proof of the completeness theorem for first-order logic yields complete algebras; that is, algebras over which models of consistent theories may always be found. The uniformity is what is interesting here. The algebra is constructed independently of the theory for which a model is sought and depends only on the underlying first-order language.. With these observations in place, the model theory for first-order defaults can be treated. Reiter [Rei80] has already told us what the extensions of closed first-order default theories are. With these extensions as a guide we introduce models, and extensions, of first-order default theories (D,W ) where these theories may be closed or non-closed. Beginning with closed default theories, the principal issue is how to check for consistency of the justifications in the defaults. A justification is consistent with a set of structures iff it is satisfied by some structure in the set. Let Γ be a set of structures. A Γ-model of (D,W ) is a set of structures over an algebra A which individually satisfy W and collectively satisfy D with respect to using Γ to check the consistency of the justifications. We describe these notions in detail in Section 5. The family of Γ-models of (D,W ) is closed under arbitrary union. Hence, over a given algebra A there is a unique largest Γ-model. Γ is a model of (D,W ) if Γ is the unique largest Γ-model of (D,W ). A maximal model of (D,W ) is a stable model of (D,W ) and the theory of a stable model, over a complete algebra, of (D,W ) is an extension of (D,W ). All complete algebras determine the same set of extensions. What about non-closed default theories? If one is going to assign values to freely occurring variables in default rules, we assume that one has a domain in mind where these values are to be found, i.e. an algebra A. One may then close (D,W ) with respect to A, fundamentally by adjoining the theory of A to W , and instantiating the freely occurring variables in the defaults in D, and