Dynamic Doxastic Logic

Origin. Segerberg (1995) proposed putting AGM revision inside the object language as a dynamic modality [*φ] rather than as a conditional, which is what allows the theory to escape Gärdenfors' triviality result for the Ramsey test. Van Benthem (2007) and Baltag and Smets (2008) rebuilt the idea on plausibility models, adding conditional belief, safe belief and a family of upgrade operators, and joining belief revision to the reduction-axiom method of dynamic epistemic logic.

Models. A single relation orders worlds by plausibility, and belief is truth throughout the most plausible worlds. Conditioning that order on a hypothesis gives conditional belief, so the static language already encodes what the agent would believe on each supposition; a dynamic operator then changes the order itself. Hard information deletes worlds and cannot be undone; soft information only reorders them, so a belief given up can be recovered by a later upgrade.

Formalism.

Plausibility models: M = (W, ≤, V). W: worlds. ≤: plausibility ordering (more plausible ≤ less). Total preorder: reflexive, transitive, connected. Well-founded, so minima exist. V: valuation.

Belief operator: Bφ: agent believes φ. True at w iff φ true at all most plausible worlds. Min_≤(W) ⊆ [[φ]]. Best worlds satisfy φ.

Conditional belief: B^ψ φ: believes φ given ψ. Most plausible ψ-worlds satisfy φ. Min_≤([[ψ]]) ⊆ [[φ]]. Hypothetical belief. Bφ ≡ B^⊤φ.

Safe belief: □φ: φ holds at all worlds at least as plausible. Belief no true information can defeat. Bφ is the weakest, safe belief stronger, knowledge strongest.

Revision action: [*φ]: revise with φ. Reorders plausibility. φ-worlds become most plausible. Dynamic modality.

Semantics of revision: After [*φ], new ordering ≤' where: φ-worlds more plausible than ¬φ-worlds. Within φ-worlds: old order. Within ¬φ-worlds: old order. This is radical (lexicographic) upgrade ⇑φ.

Conservative upgrade: ↑φ: only the best φ-worlds move to the front. Rest of the order unchanged. Weaker policy, same resulting plain belief. ⇑ and ↑ differ on conditional belief and on iteration.

Reduction axioms: [*φ]p ↔ p (atoms unchanged). [*φ]¬ψ ↔ ¬[*φ]ψ. [*φ]Bψ ↔ B^φ[*φ]ψ. Recursive reduction.

Public announcement: [!φ]: truthful announcement. Different from revision. Eliminates ¬φ-worlds. Hard information.

Symbols.

Symbol	Unicode	Name	Meaning
B	—	Belief	True at all most plausible worlds
B^φ	—	Conditional belief	Belief on the supposition that φ
[*φ]	—	Revision modality	After revising with φ
≤	U+2264	Plausibility ordering	w ≤ v: w at least as plausible as v
⇑	U+21D1	Radical upgrade	All φ-worlds moved above all ¬φ-worlds
↑	U+2191	Conservative upgrade	Only the best φ-worlds promoted
[!φ]	—	Public announcement	Hard information: delete ¬φ-worlds
□	U+25A1	Safe belief	True at all worlds at least as plausible
Min_≤	—	Minimal worlds	The most plausible worlds of a set
[[φ]]	—	Truth set	Worlds where φ holds
⊆	U+2286	Inclusion	Min_≤(W) ⊆ [[φ]] defines belief
↔	U+2194	Biconditional	Form of the reduction axioms

Metatheory. Belief and conditional belief are read off the total preorder, so the static base is a conditional doxastic logic, and van Benthem (2007) and Baltag and Smets (2008) axiomatize the dynamic operators by reduction axioms — [⇑φ]p ↔ p, commutation with the Boolean connectives, and a clause turning [⇑φ]B into a conditional belief in the pre-update model. Every formula with upgrades is therefore provably equivalent to one without, and completeness and decidability transfer from the static logic. On a well-founded total preorder the induced revision satisfies all eight AGM postulates, and iterated radical upgrade satisfies the Darwiche–Pearl postulates that AGM alone leaves open, which is the point of representing the whole ordering rather than a belief set. Because [*φ] is a dynamic modality and not a conditional connective, Gärdenfors' triviality theorem does not apply. Announcement and upgrade are distinct operations: the first shrinks the domain irreversibly, the second only permutes it.

Applies to. Belief dynamics. Multi-agent epistemics. AI. Game theory.

Limitations. The plausibility order is given with the model and the logic says nothing about where it comes from or how a total preorder on all worlds could be held by any agent; nor does it choose among revision policies, since radical and conservative upgrade agree on plain belief and diverge on conditional belief, and the framework licenses both. Success fails for doxastic input: after upgrading with a Moore sentence p ∧ ¬Bp the agent believes p and so believes the announced sentence false. Well-foundedness is required for minima to exist, so infinite descending chains of plausibility make belief undefined. Belief sets are deductively closed and the agent is logically omniscient. Reduction axioms are available only for the operators explicitly listed; adding common belief breaks them, as it does in dynamic epistemic logic generally.