Analyzing Reliability Change in Legal Case

Abstract

Reliability among agents plays a significant role in both human and agent communications. An agent may change her reliability for the other agents, when she receives a new piece of information from one of them. In order to analyze such reliability change, this paper proposes a logical formalization with two dynamic operators, i.e., downgrade and upgrade operators. The downgrade operator allows an agent to downgrade some specified agents to be less reliable in terms of the degree of reliability, while the upgrade operator allows the agent to upgrade them to be more reliable. Furthermore, we demonstrate our formalization by a legal case from Thailand.

A Complete Axiomatization of Dynamic Logic

1.1 A.1 Proof of Theorem 1

Proof

Let us write our axiomatization by $\mathbf {BS}_{\leqslant }$. We show that any unprovable formula $\varphi$ in $\mathbf {BS}_{\leqslant }$ is falsified in some si-model and we basically follow the standard techniques, e.g. found in [13]. Let $\varphi$ be an unprovable formula in $\mathbf {BS}_{\leqslant }$. We define the canonical model $\mathfrak {M}$ where $\varphi$ is falsified at some point of $\mathfrak {M}$. We say that a set $\varGamma$ of formulas is $\mathbf {BS}_{\leqslant }$ -consistent (for short, consistent) if $\bigwedge \varGamma '$ is unprovable in $\mathbf {BS}_{\leqslant }$, for all finite subsets $\varGamma '$ of $\varGamma$, and that $\varGamma$ is maximally consistent if $\varGamma$ is consistent and $\varphi \in \varGamma$ or $\lnot \varphi \in \varGamma$ for all formulas $\varphi$. Note that $\psi$ is unprovable in $\mathbf {BS}_{\leqslant }$ iff $\lnot \psi$ is $\mathbf {BS}_{\leqslant }$-consistent, for any formula $\psi$. We define the canonical model for $\mathbf {BS}_{\leqslant }$ by:

• W is the set of all maximal consistent sets;

• $\varGamma R_{a} \varDelta$ iff ($\mathsf {Bel}(a,\psi ) \in \varGamma$ implies $\psi \in \varDelta$) for all $\psi$;

• $\varGamma S_{a} \varDelta$ iff ($\mathsf {Sign}(a,\psi ) \in \varGamma$ implies $\psi \in \varDelta$) for all $\psi$;

• $b \preccurlyeq _{a}^{\varGamma } c$ iff $b \leqslant _a c \in \varGamma$;

• $\varGamma \in V(p)$ iff $p \in \varGamma$.

Then, we can show the following equivalence (Truth Lemma [13, Lemma 4.21]): $\mathfrak {M},\varGamma \; \models \; \psi$ iff $\psi \in \varGamma$ for all formulas $\psi$ and $\varGamma \in W$. Given any unprovable formula $\varphi$ in $\mathbf {BS}_{\leqslant }$, we can find a maximal consistent set $\varDelta$ such that $\lnot \varphi \in \varGamma$. Then, by the equivalence above, $\varphi$ is falsified at $\varDelta$ of the canonical model $\mathfrak {M}$ for $\mathbf {BS}_{\leqslant }$, where we can assure that $\mathfrak {M}$ is our intended si-model by axioms of Proposition 1. $\quad \square$

1.2 A.2 Proof of Theorem 2

Proof

By $\vdash \psi$ (or $\vdash ^{+} \psi$), we mean that $\psi$ is a theorem of the axiomatization $\mathbf {BS}_{\leqslant }$ in the previous proof (or, the axiomatization $\mathbf {BS}_{\leqslant }^{+}$ given in the statement of Theorem 2, respectively.) As for the completeness part, we can reduce the completeness of our dynamic extension to the static counterpart (i.e., Theorem 1) as follows. With the help of the axioms of Propositions 3, 4, and 6, we can define a mapping t sending a formula $\psi$ of the expanded syntax (we denote this by $\mathcal {L}^{+}$ below) possibly with three kinds of dynamic operators (i.e., $[ \mathop {H\Downarrow ^a_{\varphi }} ]$, $[ \mathop {H\Uparrow ^a_{\varphi }} ]$, and $[\varphi \leadsto a]$) to a formula $t(\psi )$ of the original syntax $\mathcal {L}$. For this aim, we employ inside-out strategy, i.e., we start rewriting the innermost occurrences of three kinds of dynamic operators. (So, we do not need to consider an axiom for iterated dynamic operators such as $[\varphi \leadsto a][\psi \leadsto a]$ or $[\varphi \leadsto a][ \mathop {H\Uparrow ^a_{\varphi }} ]$.) For example, if one of the innermost dynamic operators is $[\varphi \leadsto a]$, then we cannot find any occurrences of three kinds of dynamic operators. For inside-out strategy, we need to have the following inference rules for dynamic operators:

$$\begin{aligned} \frac{\psi \leftrightarrow \psi ' }{ [ \mathop {H\Downarrow ^a_{\varphi }}] \psi \leftrightarrow [ \mathop {H\Downarrow ^a_{\varphi }}] \psi ' }\quad \frac{\psi \leftrightarrow \psi ' }{ [ \mathop {H\Uparrow ^a_{\varphi }}] \psi \leftrightarrow [ \mathop {H\Uparrow ^a_{\varphi }}] \psi ' } \quad \frac{\psi \leftrightarrow \psi ' }{ [ \varphi \leadsto a] \psi \leftrightarrow [ \varphi \leadsto a] \psi ' }, \end{aligned}$$

to assure the replacement of equivalent formulas inside of a formula. But, these rules are derivable from the corresponding necessitation laws and the reduction axioms for the negation and the conjunction in Propositions 3, 4, and 6. Then, for this mapping t, we can show that $\psi \leftrightarrow t(\psi )$ is valid on all si-models and $\vdash ^{+} \psi \leftrightarrow t(\psi )$. Then, we can proceed as follows. Fix any formula $\psi$ of $\mathcal {L}^{+}$ such that $\psi$ is valid on all si-models. By the validity of $\psi \leftrightarrow t(\psi )$ on all si-models, we obtain that $t(\psi )$ is valid on all si-models. By Theorem 1, $\vdash t(\psi )$, which implies $\vdash ^{+} t(\psi )$. Finally, it follows from $\vdash ^{+} \psi \leftrightarrow t(\psi )$ that $\vdash ^{+} \psi$, as desired. $\quad \square$