AGM and black swans

Suppose my knowledge base is:

1. All swans from Europe are white.
2. Sweden is part of Europe.
3. The bird caught in the trap is a swan.
4. The bird caught in the trap is from Sweden.

But then despite what it may seem, I look at the trap and learn: 5. The bird caught in the trap is black.

Explain AGM belief revision.

Here’s the AGM (Alchourrón–Gärdenfors–Makinson) picture, using your swan story.

The idea

A belief set \(K\) is the (logically closed) collection of sentences you currently accept. AGM studies three ways to change \(K\):

• Expansion \(K + \varphi\): just add \(\varphi\) and close under logic (no regard for consistency).
• Contraction \(K - \varphi\): give up enough of \(K\) so that \(\varphi\) is no longer entailed.
• Revision \(K * \varphi\): incorporate \(\varphi\) while keeping \(K\) as consistent and unchanged as possible.

Two identities connect them:

• Levi identity: \(K * \varphi = (K - \neg \varphi) + \varphi\) (to accept \(\varphi\), first remove commitment to its negation, then add \(\varphi\)).
• Harper identity: \(K - \varphi = K \cap (K * \neg \varphi)\).

AGM gives rationality postulates for revision (closure, success \(\varphi \in K * \varphi\), minimal change via inclusion/vacuity, consistency when \(\varphi\) is consistent, and extensionality).

Your case, step by step

Let (b) be “the trapped bird.” Your initial (K) contains:

1. \(\forall x,[(\text{Swan}(x) \wedge \text{FromEurope}(x)) \rightarrow \text{White}(x)]\)
2. \(\text{FromEurope}(\text{Sweden})\)
3. \(\text{Swan}(b)\)
4. \(\text{From}(b,\text{Sweden})\) (hence \(\text{FromEurope}(b)\))

From 1–4 you deduce \(\text{White}(b)\).

Then you observe:

5. \(\text{Black}(b)\).

Assume “black” and “white” are mutually exclusive. So \(K \cup {5}\) is inconsistent, and you must revise by \(\varphi=\text{Black}(b)\).

By the Levi identity: \[ K * \text{Black}(b) = (K - \neg \text{Black}(b)) + \text{Black}(b). \] Contracting by \(\neg\text{Black}(b)\) means: remove just enough from \(K\) so that you no longer (even indirectly) commit to “\(b\) isn’t black.” Right now \(K\) entails \(\neg\text{Black}(b)\) via \(\text{White}(b)\). To block that entailment you could drop any one of the premises that make \(b\) white:

• drop (1) the universal rule,
• or drop (2) Sweden⊆Europe,
• or drop (3) that (b) is a swan,
• or drop (4) that (b) is from Sweden.

Each of these yields a remainder set (a maximal subset of (K) that no longer implies \(\neg\text{Black}(b))\). AGM lets a selection function choose among such remainders according to your epistemic entrenchment (how hard a belief is to give up). Then you expand by adding \(\text{Black}(b)\).

A natural entrenchment here is:

• very hard to give up: geographical fact (2) and the direct particulars (3) & (4),
• easier to give up: the overbold universal (1).

With that entrenchment, the chosen remainder drops (1). So:

• Contraction: \(K - \neg\text{Black}(b) = K \setminus \{(1)\}\).
• Revision (final): \(K * \text{Black}(b) = \text{Cn}\big(\{(2),(3),(4),\text{Black}(b)\}\big)\).

Consequences now include \(\neg(1)\) (i.e., not all European swans are white), since from (2–4) and \(\text{Black}(b)\) you get a counterexample to (1). You’ve satisfied success (you believe the bird is black) and changed your beliefs minimally (you gave up just the generalization that the new observation falsified).

What AGM adds conceptually

• Minimal change via selection: Among all ways to restore consistency, pick those that sacrifice the least entrenched beliefs.
• Representation theorems: AGM shows revision is equivalent to choosing models minimal in a faithful total preorder (or, dually, to using an epistemic entrenchment relation). This formalizes “keep as much as you reasonably can.”
• Underdetermination: Different entrenchments → different (but still rational) revisions. If, for some reason, you trusted the universal law more than the reports about \(b\), AGM would instead suggest dropping (3) or (4) and then adding \(\text{Black}(b)\).

Quick glossary

• Belief set \(K\): logically closed set of sentences.
• Expansion \(+\): add only.
• Contraction \(-\): give up just enough to avoid \(\varphi\).
• Revision \(*\): adopt \(\varphi\) with minimal disturbance; \(K * \varphi = (K - \neg\varphi) + \varphi\).
• Remainder sets: maximal subsets of \(K\) that don’t entail \(\varphi\).
• Selection function / entrenchment: tells which remainders to keep; captures what you find most trustworthy.

In short: AGM tells you how to rationally absorb “the bird is black” by first retracting whatever commits you to “it isn’t black,” guided by which beliefs you treat as most defensible—here, naturally, the sweeping Rule (1) gives way to the observed counterexample.

But how to revise the universal?

Great question. AGM tells you what to drop to accept the new fact, but it doesn’t itself say how to re-formulate a universal generalization once it’s been falsified. Here are the standard, rational ways to “revise the universal,” from most conservative to more conceptual changes:

1) Name the exception (syntactic weakening in the same language)

If your language can name the bird (b), replace \[ \forall x,[(\text{Swan}(x)\wedge \text{Europe}(x))\to \text{White}(x)] \] with \[ \forall x,[(\text{Swan}(x)\wedge \text{Europe}(x)\wedge x\neq b)\to \text{White}(x)]. \] This is the smallest weakening that preserves the rule for every other case. In AGM terms: after revising by \(\text{Black}(b)\), you’ve dropped the old universal; adding the “all-but-\(b\)” version is an admissible, minimal repair.

If you can’t name \(b\), you can’t write “\(x\neq b\)”. Then the safest move is simply to not keep a universal at all (see #2).

2) Retreat from “all” to “not all”

Keep only what your new beliefs logically support: \[ \exists x,(\text{Swan}(x)\wedge \text{Europe}(x)\wedge \neg \text{White}(x)). \] That is, don’t replace the universal; just accept that it was too strong. This is the default AGM outcome if you don’t add any new generalization.

3) Guard the rule with an “abnormality” (defeasible/default move)

Introduce a guard: \[ \forall x,[(\text{Swan}(x)\wedge \text{Europe}(x)\wedge \neg \text{Ab}(x))\to \text{White}(x)],\qquad \text{Ab}(b). \] Now the rule reads “normally, European swans are white,” and \(b\) is marked as an exception. This is the classic fix in nonmonotonic/default logics (e.g., rational closure/System Z) and aligns well with the intuition that universal laws in the wild are default rules.

4) Strengthen the antecedent (narrow the scope)

Replace “from Europe” with a narrower condition supported by evidence, e.g., \[ \forall x,[(\text{Swan}(x)\wedge \text{Europe}(x)\wedge \neg \text{Sweden}(x))\to \text{White}(x)]. \] This is justified only if you have reasons to localize the exception (more than a single counterexample).

5) Switch to a statistical/inductive universal

Replace \(\forall\) with a quantifier of proportion or a probability claim, e.g., \[ \Pr(\text{White}\mid \text{Swan}\wedge \text{Europe}) \ge 1-\varepsilon, \] and update that probability when you see a black swan. This preserves the spirit of the generalization while acknowledging uncertainty.

---

Where AGM fits technically

• Revision step: \(K * \text{Black}(b) = (K - \neg\text{Black}(b)) + \text{Black}(b)\). Dropping the old universal (1) is the natural contraction.

• What to put back: AGM is agnostic; your entrenchment/selection policy decides.

  • If you favor exception-by-name → option #1.
  • If you avoid new symbols or dislike ad-hoc exceptions → option #2.
  • If you treat “laws” as defaults → option #3.
  • If you’re doing induction/statistics → option #5.

A crisp way to remember it:

• Logical repair: “all” → “all-but-(b)” (if you can name (b)), else drop “all”.
• Knowledge-engineering repair: add an abnormality guard.
• Scientific/inductive repair: replace universals with high-confidence generalizations and keep updating.