## Appendix J: Conditional Doxastic Logic

The language \eqref{CDL} of conditional doxastic logic is defined by the following grammar:

$\begin{gather*}\taglabel{CDL} F \ccoloneqq p \mid F\land F \mid \lnot F\mid B_a^FF \\ \small p\in\sP,\; a\in\sA \end{gather*}$

We think of the formula $$B_a^FG$$ as saying that agent a believes F after conditionalization (or static belief revision) by G. The satisfaction relation $$\models$$ between pointed plausibility models and formulas of \eqref{CDL} and the set $$\sem{F}_M$$ of worlds in the plausibility model M at which formula F is true are defined by the following recursion.

• $$\sem{F}_M\coloneqq\{w\in W\mid M,w\models F\}$$.
• $$M,w\models p$$ holds if and only if $$w\in V(p)$$.
• $$M,w\models F\land G$$ holds if and only if $$M,w\models F$$ and $$M,w\models G$$.
• $$M,w\models\lnot F$$ holds if and only if $$M,w\not\models F$$.
• $$M,w\models B_a^GF$$ holds if and only if $$\min_a(\sem{G}_M\cap\cc_a(w))\subseteq\sem{F}_M$$.
$$B_a^GF$$ means that “F is true at the most plausible G-worlds consistent with a’s information”.

In the language \eqref{CDL}, the operator $$K_a$$ for possession of information may be defined as follows:

$K_a F \text{ denotes } B_a^{\lnot F}\bot,$

where $$\bot$$ is the propositional constant for falsehood. We then have the following.

Theorem (Baltag and Smets 2008b). For each pointed plausibility model $$(M,w)$$, we have: $M,w\models B_a^{\lnot F}\bot \text{ iff } M,v\models F \text{ for each }v \text{ with } v\simeq_a w.$ So the \eqref{CDL}-abbreviation of $$K_aF$$ as $$B_a^{\lnot F}\bot$$ gives the same meaning in \eqref{CDL} for “possession of information” as we had in $$(K\Box)$$.

The validities of \eqref{CDL} are axiomatized as follows.

The axiomatic theory $$\CDL$$ (Baltag, Renne, and Smets 2015).

• Axiom schemes and rules for classical propositional logic
• $$B_a^F(G_1\to G_2)\to (B_a^FG_1\to B_a^FG_2)$$
• $$B_a^FF$$
• $$B_a^F\bot\to B_a^{F\land G}\bot$$
• $$\lnot B_a^F\lnot G\to( B_a^FH\to B_a^{F\land G}H)$$
• $$B_a^{F\land G}H\to B_a^F(G\to H)$$
• $$B_a^{F\land G}H\to B_a^{G\land F}H$$
• $$B_a^FH\to B_a^G B_a^FH$$
• $$\lnot B_a^FH\to B_a^G\lnot B_a^FH$$
• $$B_a^F\bot\to\lnot F$$

$$\CDL$$ Soundness and Completeness (Board 2004; Baltag and Smets 2008b; Baltag, Renne, and Smets 2015). $$\CDL$$ is sound and complete with respect to the collection $$\sC_*$$ of pointed plausibility models. That is, for each \eqref{CDL}-formula F, we have $$\CDL\vdash F$$ if and only if $$\sC_*\models F$$.

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