Interpolation and Beth Definability in Default Logics

Abstract

We investigate interpolation and Beth definability in default logics. To this end, we start by defining a general framework which is sufficiently abstract to encompass most of the usual definitions of a default logic. In this framework a default logic $\mathscr {D}\mathfrak {L}$ is built on a base, monotonic, logic $\mathfrak {L}$. We then investigate the question of when interpolation and Beth definability results transfer from $\mathfrak {L}$ to $\mathscr {D}\mathfrak {L}$. This investigation needs suitable notions of interpolation and Beth definability for default logics. We show both positive and negative general results: depending on how $\mathscr {D}\mathfrak {L}$ is defined and of the kind of interpolation/Beth definability involved, the property might or might not transfer from $\mathfrak {L}$ to $\mathscr {D}\mathfrak {L}$.

A Selected Proofs

Remark 3

Let $\mathfrak {L}$ be any logic, and $\varPhi$ and $\varPsi$ be sets of sentences; we say that $\varPsi$ is a conservative extension of $\varPhi$, notation $\varPsi \ge \varPhi$, iff and .

Lemma 4

Let $\mathfrak {L}$ be any logic, $\varPhi$ be a set of sentences defined on an alphabet A, and $q \notin A$; if consequence in $\mathfrak {L}$ has $\mathsf {SIP}$, and .

Proof

Trivially, . In turn, let ; then, , alt., . From $\mathsf {SIP}$, there is $\varepsilon$ defined on $A \setminus \{p\}$ s.t. and . Since , results in . Then, and ; and so, , alt., . Therefore, .

Lemma 1

Any strongly saturated default logic $\mathscr {D}\mathfrak {L}$ is stable.

Proof

Let $\Theta$ be a default theory defined on an alphabet A, and and $q \notin A$. In addition, let $E \in \mathscr {E}(\Theta )$ be s.t. for some . Since $\mathscr {D}\mathfrak {L}$ is strongly saturated, $\varDelta$ is saturated in $\Theta$. The result follows immediately if is saturated in ; as is our extension. The proof proceeds by contradiction. Let be not saturated in ; w.l.o.g. there is a default s.t. $\delta$ is detached by . Clearly, or for some . If , from Lemma 4, $\delta$ is detached by $\varDelta$; and so $\varDelta$ is not saturated. This yields a contradiction. If for some , from Lemma 4, is detached by ; and so is not saturated. But by substitution, $\delta '$ is detached by $\varDelta$, and so $\varDelta$ is not saturated. This also yields a contradiction. Thus, is saturated in .

Lemma 2

Let $\mathscr {D}\mathfrak {L}$ be a default logic; for all default theories $\Theta$ and all , if $\varDelta$ is self coherent in $\Theta$, then, is self coherent in .

Proof

Similar to that of Lemma 1.

Lemma 3

Let be a set of sets of formulas s.t. for all $i \in I$, and ; if consequence in $\mathfrak {L}$ has $\mathsf {SIP}$, $q \notin A$, and , then .

Proof

(by contradiction). Let us assume that ; by definition, it follows that all ($*$) . At the same time, let ; then, for all $\xi$ defined on ${A \setminus \{p\}}$, either ($\dagger$) or ($\ddagger$) . From ($\dagger$), there is ; and from Lemma 4, ($\S$) . But ($\S$) leads to a contradiction; since from ($*$) , by $\mathsf {SIP}$, there is in fact $\xi$ defined on $A \setminus \{p\}$ s.t. ! Similarly, we obtain a contradiction from ($\ddagger$). Thus, .