An Application to Default Logic

Abstract

Given a default theory $(\Delta ,D),$ where $\Delta$ is a theory in propositional logic and D is a set of defaults, we will define three kinds of the minimal change in default logic: a theory $\Theta$ is a $\subseteq$-/$\preceq$-/$\vdash _\preceq$-minimal change of D by $\Delta ,$ where $\Theta$ is a $\subseteq$-minimal change of D by $\Delta$ if and only if $\mathrm{Th}(\Theta \cup \Delta )$ is an extension of $(\Delta ,D).$ $\preceq$-minimal change has the actual importance. For example, let $(\Delta ,D)$ be a default theory, where $\Delta$ says that someone is a human (denoted by p) and this man has no arms (denoted by $\lnot q$); and D contains one default $\dfrac{p:q\wedge r}{q\wedge r},$ which says that a man defaultly has arms and legs, where r denotes that this man has legs. By default logic, $(\Delta ,D)$ has an extension $\{p\},$ that is, we know that this man is a human.