EF vs AGM

The input contains an excerpt from Handbook of Philosophical Logic,edited by Gabbay and Guenthner, volume 4, p. 82.

The following is a conditional logic: whereas →,∧,⊤,⊥,¬ are classically defined connectives with their usual meaning, > is a new binary connective defined by the axioms. It is claimed that this new conditional “approximates AGM belief revision”. Explain how it does so.

A conditional logic that approximates AGM belief revision

Whereas Gärdenfors [1978] sets out to find epistemic models for Lewis’s system VC of conditional logic, Arló-Costa [1995] sets out to find a system of conditional logic whose primitive belief revision models are defined at least approximately by the AGM belief revision postulates for transitive relational partial meet contraction (see [Gärdenfors, 1988], Chapters 3–4). The result is the system EF, which is defined only on languages of type L₁ (languages of flat conditionals). EF has the following axioms and rules, where φ, ψ, and χ are conditional free:

Axiom schemata

Taut: All truth-functional tautologies
ID: φ > φ
MP: (φ > ψ) → (φ → ψ)
CC: [(φ > ψ) ∧ (φ > χ)] → [φ > (ψ ∧ χ)]
CA: [(φ > χ) ∧ (ψ > χ)] → [(φ ∨ ψ) > χ]
CV: [(φ > ψ) ∧ ¬(φ > ¬χ)] → [(φ ∧ χ) > ψ]
CN: φ > ⊤
CD: ¬(φ > ⊥) for all non-tautologous φ.

Rules of inference

Modus Ponens: From φ and φ → ψ to infer ψ.
RCM: From ψ → χ to infer (φ > ψ) → (φ > χ).
RCEA: From φ ↔ ψ to infer (φ > χ) ↔ (ψ > χ).

Here’s the usual bridge between the conditional “>” and AGM, and how each EF axiom lines up with a belief-revision postulate.

Read “>” as a Ramsey-test revision conditional

Fix a belief set $K$ and an AGM revision operator $*$. Read

$$ \varphi > \psi \quad\text{as}\quad \psi \in K*\varphi $$

i.e., “if you were to revise $K$ by $\varphi$, you would believe $\psi$.” With that reading, EF’s principles are exactly the things that must be true for every AGM-style (more specifically: transitive relational partial-meet) revision model. That’s the sense in which EF “approximates” AGM.

How each EF axiom/rule reflects an AGM postulate

Group the AGM postulates by what they say about $K*\varphi$, and compare to EF.

Closure & Consequence

• RCM (Right Consequence Monotonicity): from $\psi\to\chi$ infer $(\varphi>\psi)\to(\varphi>\chi)$. ↔ If $\psi\in K*\varphi$ and $\psi\vdash\chi$, then $\chi\in K*\varphi$ (closure under consequence).
• CC: $(\varphi>\psi)\land(\varphi>\chi)\to(\varphi>(\psi\land\chi))$. ↔ $K*\varphi$ is closed under conjunction.
• CN: $\varphi>\top$. ↔ Every revised set contains tautologies.

Success, Extensionality, and the “Ramsey bridge”

• ID: $\varphi>\varphi$. ↔ Success: $\varphi\in K*\varphi$.
• RCEA: from $\varphi\leftrightarrow\psi$ infer $(\varphi>\chi)\leftrightarrow(\psi>\chi)$. ↔ Extensionality: revision depends only on the proposition, not its form.
• MP: $(\varphi>\psi)\to(\varphi\to\psi)$. ↔ If revising by $\varphi$ yields $\psi$, then, conditional on $\varphi$, you’re committed to $\psi$. This is the standard Ramsey-test bridge from revision-beliefs back to an indicative conditional.

Consistency of revision

• CD: if $\varphi$ is not a tautology, then $\neg(\varphi>\bot)$. ↔ If $\varphi$ is classically consistent, then $K*\varphi$ is consistent (AGM consistency of revision).

Disjunction and cautious strengthening (the “relational/transitive” part)

• CA: $(\varphi>\chi)\land(\psi>\chi)\to((\varphi\lor\psi)>\chi)$. ↔ Disjunctive rationality: if either input would lead you to $\chi$, then so does their disjunction. This is validated by the relational (system-of-spheres / entrenchment) pictures used to represent partial-meet operators.
• CV: $(\varphi>\psi)\land\neg(\varphi>\neg\chi)\to((\varphi\land\chi)>\psi)$. ↔ Rational Monotony / cautious strengthening: if, after $\varphi$, $\psi$ holds and $\chi$ is not ruled out, then adding $\chi$ to $\varphi$ preserves $\psi$. This corresponds to transitivity properties of the underlying epistemic-entrenchment relation that generate AGM partial-meet contraction.

Why this is only an “approximation”

• Language is flat. EF only allows conditionals with non-conditional antecedents/consequents (no nesting), so it cannot talk about iterated revision (e.g., Darwiche–Pearl postulates) or capture identities like Harper/Levi inside the object language.
• Only universal consequences. EF captures what is true across all transitive relational partial-meet models; it doesn’t pin down any single revision operator. AGM postulates that compare entire belief sets (e.g., equalities/containments between $K*\varphi$ and $K$) don’t translate directly into a single conditional formula.
• Recovery and other contraction-specific nuances aren’t expressible with a single binary conditional either; EF abstracts away from those details.

Bottom line: interpret $\varphi>\psi$ as “$\psi$ would be believed after AGM-style revision by $\varphi$.” Under that reading, each EF axiom is the “shadow,” in the conditional language, of a core AGM principle (closure, success, consistency, extensionality) plus the extra cautious/rational behavior guaranteed by transitive relational partial-meet models. That is how EF approximates AGM belief revision.