Doxastic Kripke Models

• Doxastic Kripke Models are mathematical frameworks that represent agents' beliefs and counterfactuals using modal logic and selection functions.
• They integrate belief update and revision by employing doxastic priorities that align with both AGM and Katsuno–Mendelzon paradigms.
• Their model-theoretic equivalence offers a unified view of conditional belief change, supporting advanced applications in dynamic epistemic logic.

A doxastic Kripke model is a mathematical structure for modeling agents’ beliefs and belief changes within a modal logic framework. The Kripke–Lewis extension incorporates a selection function analogous to counterfactual semantics, providing a unified model-theoretic basis for both belief update (Katsuno–Mendelzon) and belief revision (AGM). This approach subsumes the standard KD45 modal semantics for belief and enables a rigorous representation of complex change operations, such as conditional belief and rational prioritization of evidence.

1. Formal Structure of Doxastic Kripke–Lewis Models

A doxastic Kripke–Lewis frame is a triple

$F = (W, R, f)$

where:

• $W$ is a nonempty set of states (“worlds”).
• $R \subseteq W \times W$ is a serial accessibility relation (for every $w \in W$, there exists $v$ such that $w R v$), representing the agent’s doxastic alternatives.
• $f: W \times 2^W \to 2^W$ is a Lewis-style selection function, satisfying:
  • Non-emptiness (Consistency): $f(w, E) \neq \emptyset$ for all $w$ and $E \neq \emptyset$.
  • Inclusion (Success): $f(w, E) \subseteq E$.
  • Weak Centering: if $w \in E$, then $w \in f(w, E)$.

A model adds a valuation

$M = (W, R, f, V), \quad V: \text{At} \to 2^W$

where $V$ extends inductively to Boolean formulas in the classic way. The belief operator is interpreted as

$$M, w \models B\varphi \iff \forall v \in R(w),\; M, v \models \varphi.$$

The initial belief set at $w$ is the deductively closed set

$K_w = \{\varphi \mid M, w \models B\varphi \}.$

Given any formulas $A, \psi$, the conditional $(A > \psi)$ holds at $w$ iff $f(w, [A]_M) \subseteq [\psi]_M$, where $[A]_M = \{ v : M, v \models A \}$ (Bonanno, 2023).

2. Semantics of Belief, Conditionals, Update, and Revision

The Kripke–Lewis model semantics supports both classical and conditional (counterfactual) belief reasoning. For belief update/revision: $$\begin{aligned} K_w \circ_U A &= \{ \psi \mid M, w \models (A > \psi) \land B(A > \psi) \} \ K_w \!*_R A &= \{ \psi \mid M, w \models (A > \psi) \land B(A > \psi) \} \end{aligned}$$ Formally, $K_w \circ_U A$ is the updated belief set, $K_w \!*_R A$ is the revised belief set. Both collect formulas $\psi$ such that the agent already believes the conditional $A > \psi$ (Bonanno, 2023).

The distinction between update and revision is not in the syntactic clause but arises from different constraints on the frame $(W, R, f)$, as detailed below.

3. Doxastic-Priority Properties: Distinguishing Update from Revision

Bonanno’s framework identifies two semantic constraints responsible for the difference between update and revision:

• Doxastic Priority 1 (DP1):

$$\text{If } R(w) \cap E \neq \emptyset, \quad \forall v \in R(w) \;\; f(v, E) \subseteq R(w) \cap E$$

DP1 formalizes the AGM revision axiom $(K*_4)$: if evidence $A$ is consistent with prior beliefs, old beliefs are not discarded in favor of new information.

• Doxastic Priority 2 (DP2):

$$\text{If } \exists v \in R(w)\!:\;f(v, E) \cap F \neq \emptyset,\; \forall v \in R(w)\;\; f(v, E \cap F) \subseteq \bigcup_{u \in R(w)} (f(u, E) \cap F)$$

DP2 corresponds to AGM $(K*_8)$, enforcing that when choosing among $A \wedge B$-worlds, priority is given to those consistent with prior beliefs.

Update-frames (KM) satisfy only weaker, often local or pointed versions of these conditions (Bonanno, 2023).

4. Representation Theorems and Model-Theoretic Equivalence

Two central representation results hold:

• Model-to-Function: Each model gives rise to a partial belief update (KM) or revision (AGM) function, which extends uniquely to a full KM or AGM function on theories.
• Function-to-Model: For any KM update or AGM revision function, there exists a Kripke–Lewis model whose partial belief change operator coincides with the given function for every consistent input.

This model-theoretic equivalence rigorously demonstrates that the Kripke–Lewis semantics precisely characterizes both major approaches to belief change, unifying update and revision under a single semantic architecture (Bonanno, 2023).

5. Illustrative Example: Semantic Dynamics of Update and Revision

Consider $W = \{w_1, w_2, w_3\}$, with initial belief $K_{w_1} = \{ p \}$, and $R(w_1) = \{w_1, w_2\}$ where $w_1, w_2 \models p$. Upon learning $q$:

• Update: The selection function $f(w_1, [q]) = \{w_3\}$ (where only $w_3$ satisfies $q$ but not necessarily $p$), so

$K_{w_1} \circ_U q = \{ \psi : w_3 \models \psi \}$

• Revision: DP1 requires prioritizing prior $p$-worlds that are also $q$-worlds. If $w_2$ is a $p \wedge q$-world, then

$f(w_1, [q]) \subseteq R(w_1) \cap [q] = \{ w_2 \}$

so

$K_{w_1} \!*_R q = \{ \psi : w_2 \models \psi \}$

This example demonstrates how doxastic priorities realize the classical AGM revision and KM update dynamics (Bonanno, 2023).

6. Significance, Expressivity, and Extensions

Doxastic Kripke–Lewis models provide a foundational semantic framework unifying belief, conditionals, update, and revision. The explicit inclusion of a selection function enables modeling of counterfactuals and supports the full expressive range required for conditional belief logics. The encoded doxastic-priority conditions are necessary and sufficient for capturing the essential differences between AGM and KM paradigms, thus serving as a touchstone for generalizing belief change in modal and dynamic epistemic logic.

Extensions, such as fuzzy semantics and many-valued belief logics, apply the Kripke–style approach with infinitely valued accessibility and belief-strength (e.g., [0,1]-valued Kripke models), generalizing the binary logic to better model uncertainty and degrees of belief (Dastgheib et al., 2021). However, such generalizations preserve the underlying frame and accessibility structure conceptually derived from classical doxastic Kripke–Lewis models.

7. Comparison with Alternative Semantics

While classical doxastic Kripke models ground belief in serial, transitive, and Euclidean accessibility relations, the Kripke–Lewis semantics extends this tradition by adding conditional selection functions, allowing direct encoding of both the AGM and KM approaches, and supporting more sophisticated belief change logics. In contrast to, for example, belief-base or probability-based approaches, the Kripke–Lewis model yields a principled, compositional, and representation-theoretic account of both update and revision within a unified modal logic framework (Bonanno, 2023).