Preferential Accessibility and Preferred Worlds

Abstract

Modal accounts of normality in non-monotonic reasoning traditionally have an underlying semantics based on a notion of preference amongst worlds. In this paper, we motivate and investigate an alternative semantics, based on ordered accessibility relations in Kripke frames. The underlying intuition is that some world tuples may be seen as more normal, while others may be seen as more exceptional. We show that this delivers an elegant and intuitive semantic construction, which gives a new perspective on defeasible necessity. Technically, the revisited logic does not change the expressive power of our previously defined preferential modalities. This conclusion follows from an analysis of both semantic constructions via a generalisation of bisimulations to the preferential case. Reasoners based on the previous semantics therefore also suffice for reasoning over the new semantics. We complete the picture by investigating different notions of defeasible conditionals in modal logic that can also be captured within our framework.

Proof of Theorem 1

In the following, we denote the dual of in the usual sense, i.e., .

Definition 14

(Subsentence) Let . The set of subsentences of $\alpha$, denoted by ${\text { sub}(\alpha )}$, is defined inductively by:

• If $\alpha =\top$ or $\alpha =\bot$, then ${\text { sub}(\alpha )}\mathrel {\mathop :}=\{\alpha \}$;

• If $\alpha =p\in \mathcal {P}$, then ${\text { sub}(\alpha )}\mathrel {\mathop :}=\{p\}$;

• If $\alpha =\beta \wedge \gamma$ or $\alpha =\beta \vee \gamma$ or $\alpha =\beta \rightarrow \gamma$, then ${\text { sub}(\alpha )}\mathrel {\mathop :}=\{\alpha \}\cup {\text { sub}(\beta )}\cup {\text { sub}(\gamma )}$;

• If $\alpha =\lnot \beta$ or $\alpha =\Diamond _{i}\beta$ or $\alpha =\Box _{i}\beta$ or , then ${\text { sub}(\alpha )}\mathrel {\mathop :}=\{\alpha \}\cup {\text { sub}(\beta )}$.

Definition 15

(Closure under subsentences) Let . We say $\mathcal {C}$ is closed under its subsentences if and only if $\bigcup \{{\text { sub}(\alpha )} \mid \alpha \in \mathcal {C}\}\subseteq \mathcal {C}$.

Definition 16

($\mathcal {C}$-type) Let and let $\mathscr {R}=\langle { W},{ R},{ V},\ll \rangle$ be an ${ R}$-ordered model. For every $w\in { W}$,

$$\begin{aligned} t_{\mathcal {C}}(w)\mathrel {\mathop :}=\{\alpha \in \mathcal {C}\mid \mathscr {R},w\Vdash \alpha \} \end{aligned}$$

is the $\mathcal {C}$-type of w in $\mathscr {R}$.

Definition 17

($\mathcal {C}$-equivalence) Let be finite and let $\mathscr {R}=\langle { W},{ R},{ V},\ll \rangle$ be an ${ R}$-ordered model. For every $w,w'\in { W}$, $w\simeq _{\mathcal {C}}w'$ if and only if $t_{\mathcal {C}}(w)=t_{\mathcal {C}}(w')$. (It is easy to see that $\simeq _{\mathcal {C}}$ is an equivalence relation on ${ W}$.) With $[w]_{\mathcal {C}}\mathrel {\mathop :}=\{w' \mid w\simeq _{\mathcal {C}}w'\}$, we denote the $\mathcal {C}$-equivalence class of $w\in { W}$.

Definition 18

($\mathcal {C}$-filtration) Let be finite and let $\mathscr {R}=\langle { W},{ R},{ V},\ll \rangle$ be an ${ R}$-ordered model. Moreover, let:

• ${ W}'\mathrel {\mathop :}={ W}/\simeq _{\mathcal {C}}$ (i.e., ${ W}'=\{[w]_{\mathcal {C}} \mid w\in { W}\}$);

• ${ R}'\mathrel {\mathop :}=\langle { R}'_{1},\ldots ,{ R}'_{n} \rangle$, where ${ R}'_{i}\mathrel {\mathop :}=\{([w]_{\mathcal {C}},[w']_{\mathcal {C}}) \mid (w,w')\in { R}_{i}\}$, $i=1,\ldots ,n$;

• ${ V}':{ W}'\longrightarrow \{0,1\}^{\mathcal {P}}$, with ${ V}'([w]_{\mathcal {C}})={ V}(w)$;

• $\ll '=\langle \ll '_{1},\ldots ,\ll '_{n} \rangle$, where $\ll '_{i}\mathrel {\mathop :}=\{([w]_{\mathcal {C}}[v]_{\mathcal {C}},[w']_{\mathcal {C}}[v']_{\mathcal {C}}) \mid (wv,w'v')\in \ll _{i}$ and either $w\notin [w']_{\mathcal {C}}$ or $v\notin [v']_{\mathcal {C}}\}$, $i=1,\ldots ,n$. (It is not hard to check that each $\ll '_{i}$ is a strict well-founded partial order).

With $\mathscr {R}'\mathrel {\mathop :}=\langle { W}',{ R}',{ V}',\ll ' \rangle$ we denote the $\mathcal {C}$-filtration of $\mathscr {R}$.

In Definition 18, the purpose of the clause “either $w\notin [w']_{\mathcal {C}}$ or $v\notin [v']_{\mathcal {C}}$” is to prevent two indistinguishable (w.r.t. $\mathcal {C}$) pairs of worlds from inducing a reflexive $\ll '_{i}$-edge, which would violate the stated condition that $\ll '_{i}$ must be a strict partial order.

Lemma 5

(Minimality preservation) Let be finite and closed under subsentences, let $\mathscr {R}=\langle { W},{ R},{ V},\ll \rangle$ be an ${ R}$-ordered model, and let $\mathscr {R}'=\langle { W}',{ R}',{ V}',\ll ' \rangle$ be the $\mathcal {C}$-filtration of $\mathscr {R}$. Then, for $i=1,\ldots ,n$, (i) if $(w,v)\in \min _{\ll _{i}}{ R}_{i}$, then $([w]_{\mathcal {C}},[v]_{\mathcal {C}})\in \min _{\ll '_{i}}{ R}'_{i}$, and (ii) if $([w]_{\mathcal {C}},[v]_{\mathcal {C}})\in \min _{\ll '_{i}}{ R}'_{i}$, then there are $w'\in [w]_{\mathcal {C}}$ and $v'\in [v]_{\mathcal {C}}$ such that $(w',v')\in \min _{\ll _{i}}{ R}_{i}$.

Proof

Showing (i): Assume $([w]_{\mathcal {C}},[v]_{\mathcal {C}})\notin \min _{\ll '_{i}}{ R}'_{i}$. If $([w]_{\mathcal {C}},[v]_{\mathcal {C}})\notin { R}'_{i}$, then $(w,v)\notin { R}_{i}$ and we are done. If there is $([w']_{\mathcal {C}},[v']_{\mathcal {C}})\in { R}'_{i}$ such that $([w']_{\mathcal {C}},[v']_{\mathcal {C}})\ll '_{i}([w]_{\mathcal {C}},[v]_{\mathcal {C}})$, then, by the construction of $\ll '_{i}$, we have $(x,y)\ll _{i}(w,v)$ for some $x\in [w]_{\mathcal {C}}$ and $y\in [v]_{\mathcal {C}}$. Hence $(w,v)\notin \min _{\ll _{i}}{ R}_{i}$.

Showing (ii): Let $X\mathrel {\mathop :}=\{(w',v') \mid w'\in [w]_{\mathcal {C}}$ and $v'\in [v]_{\mathcal {C}}\}$. Assume $X\cap \min _{\ll _{i}}{ R}_{i}=\emptyset$. Then for every $(w',v')\in X$ there must be $(x,y)\in { R}_{i}$ such that $(x,y)\ll _{i}(w',v')$. Moreover, $x\notin [w]_{\mathcal {C}}$ or $y\notin [v]_{\mathcal {C}}$, otherwise $(x,y)\in X$. Hence $([x]_{\mathcal {C}},[y]_{\mathcal {C}})\ll '_{i}([w']_{\mathcal {C}},[v']_{\mathcal {C}})$, by the construction of $\mathscr {R}'$. Since $[w']_{\mathcal {C}}=[w]_{\mathcal {C}}$ and $[v']_{\mathcal {C}}=[v]_{\mathcal {C}}$, it follows that $([x]_{\mathcal {C}},[y]_{\mathcal {C}})\ll '_{i}([w]_{\mathcal {C}},[v]_{\mathcal {C}})$, and therefore $([w]_{\mathcal {C}},[v]_{\mathcal {C}})\notin \min _{\ll '_{i}}{ R}'_{i}$. $\square$

Lemma 6

(Satisfaction preservation) Let be finite and closed under subsentences, let $\mathscr {R}=\langle { W},{ R},{ V},\ll \rangle$ be an ${ R}$-ordered model, and let $\mathscr {R}'=\langle { W}',{ R}',{ V}',\ll ' \rangle$ be the $\mathcal {C}$-filtration of $\mathscr {R}$. For every $w\in { W}$ and every $\alpha \in \mathcal {C}$, $\mathscr {R},w\Vdash \alpha$ if and only if $\mathscr {R}',[w]_{\mathcal {C}}\Vdash \alpha$.

Proof

The proof is by induction on the structure of the sentence $\alpha$. Below we only show the case of via its dual . (The classical cases are as usual.)

Assume . Since $\mathcal {C}$ is closed, we have $\beta \in \mathcal {C}$, and thus the induction hypothesis applies to $\beta$.

For the only-if part, let . Then there is $w'\in { W}$ such that $(w,w')\in { R}_{i}$, $(w,w')\in \min _{\ll _{i}}{ R}_{i}^{w}$ and $\mathscr {R},w'\Vdash \beta$. We have $([w]_{\mathcal {C}},[w']_{\mathcal {C}})\in { R}'_{i}$, since $w\in [w]_{\mathcal {C}}$ and $w'\in [w']_{\mathcal {C}}$. By induction, we get $\mathscr {R}',[w']_{\mathcal {C}}\Vdash \beta$. By Lemma 5, $([w]_{\mathcal {C}},[w']_{\mathcal {C}})\in \min _{\ll '_{i}}{ R}_{i}^{[w]_{\mathcal {C}}}$. Hence .

For the if part, let . Then there is $[v]_{\mathcal {C}}\in { W}'$ such that $([w]_{\mathcal {C}},[v]_{\mathcal {C}})\in { R}'_{i}$, $([w]_{\mathcal {C}},[v]_{\mathcal {C}})\in \min _{\ll '_{i}}{{ R}'_{i}}^{[w]_{\mathcal {C}}}$ and $\mathscr {R}',[v]_{\mathcal {C}}\Vdash \beta$. By induction, we get $\mathscr {R},v\Vdash \beta$. Moreover, by Lemma 5, there are $w'\in [w]_{\mathcal {C}}$ and $v'\in [v]_{\mathcal {C}}$ such that $(w',v')\in \min _{\ll _{i}}{ R}_{i}^{w'}$, from which follows $(w',v')\in { R}_{i}$. Since $v\simeq _{\mathcal {C}}v'$ and $\beta \in \mathcal {C}$, $\mathscr {R},v\Vdash \beta$ implies $\mathscr {R},v'\Vdash \beta$. Hence we have . Since , we get . $\square$

Lemma 7

If is satisfiable, then it has a finite model.

Proof

Let $\mathscr {R}$ be such that $\mathscr {R}\Vdash \alpha$, and let $\mathcal {C}={\text { sub}(\alpha )}$. From finiteness of $\alpha$, it follows that $\mathcal {C}$ is finite, too. Hence Definition 18 guarantees the existence of a $\mathcal {C}$-filtration $\mathscr {R}'$ of $\mathscr {R}$ having a finite set of worlds. It remains to show that $\mathscr {R}'$ is a model of $\alpha$. Let $w\in { W}$ be such that $\mathscr {R},w\Vdash \alpha$. Since $\alpha \in \mathcal {C}$, by Lemma 6 it follows that $\mathscr {R}',[w]_{\mathcal {C}}\Vdash \alpha$. $\square$

The proof of Theorem 1 follows immediately from Lemma 7.