📄 Identity Merging and Identity Revision in Talmudic Logic: An Outline Paper

Summary

The paper tackles a narrow kind of belief revision that the authors call "identity merging." A consistent monadic theory describes two individuals that were assumed to be distinct. New input then says they are the same, for example x = y, and the merged theory may become contradictory. General AGM revision would just drop one side of each clashing pair, such as P(z) versus ¬P(z), and that choice is arbitrary. The authors instead propose two richer mechanisms. The first is an intermediate non-monotonic approach (ANM). Here the surface theory is treated as the output of a deeper defeasible theory ∆ in an expanded language. After the merge, ∆ is consulted again over the combined vocabulary, and specificity determines which literal survives. The second is the Talmudic approach, a calculus of cancellations. Policies label transitions between states, and policies can cancel each other's support. Merging two cases then becomes a union of these graphs. Three examples drive the discussion: a rector acting as department head, the Boston bombers who were also US citizens, and a plumber call. The authors argue the cancellation calculus captures commonsense outcomes that specificity-based defeasible logic misses. They offer it as an alternative AI methodology to AGM.

Key points

• Setup. The theory T is built in classical logic from unary predicates P1, P2, …. The input x = y can also be simulated without equality: substitute a fresh variable z for both x and y, giving T(x = y = z).
• Three options compared. These are AGM contraction, a specialised method exploiting the form of the problem (here the ANM model), and the Talmudic cancellation calculus.
• ANM definitions.
  • Each surface predicate-constant pair (P, c) is mapped to a family F(P, c) of deep-language symbols.
  • ∆ "supports" T when T proves P(a) exactly if ∆ restricted to F(P, a) yields P(a).
  • The non-monotonic consequence relation must satisfy reflexivity, cut and restricted monotonicity.
  • Entailment is defined through finite minimal subsets that decide a formula.
• Example 3. Separately, the rules give P(a) and ¬P(b). After the merge, a more specific rule using both A(b) and B(b) wins, so ¬P(b) is contracted. If ∆ cannot settle the conflict, the authors propose a recursive hierarchy in which T(n+1) acts as the ∆ for T(n).
• Plumber example. The plumber is called for the toilet, and the $50 call fee is paid regardless. Defeasible logic with specificity would still have the residents fix the sink themselves. The union of the cancellation figures (Figs. 2–4) instead has the plumber fix both. The authors frame this as a predicate argumentation network with joint and higher-order attacks.

Interesting ideas

• I1. When two referents merge and their attributes clash, resolve the clash by re-deriving each attribute from its sources over the joint evidence, rather than discarding one side at random.
• I2. Treat a surface knowledge base as the output of a deeper, richer theory, and stack such levels so that each one can arbitrate conflicts in the level below.
• I3. Pull priority principles (save money, use experts) out of individual rules and handle them as meta-level policies that support or cancel actions, including cancelling other cancellations.
• I4. Merging two cases as a union of policy graphs lets a shared fixed cost, such as the plumber already coming, propagate to the other case, so existentials pick the same individual.
• I5. The paper tests its formalism against concrete commonsense cases where a competing logic gives the intuitively wrong answer.

For Mova

How these ideas could serve Mova, our small open language model — its files, stages and tests:

• I1 (frames): when coreference links two mentions with conflicting properties, the resolver could re-run the rules that produced each property on the merged mention and keep the better-supported one, logging the reason.
• I2 (world): entity states in the text world could store which compiled-knowledge rules (verb classes, ontology) produced them, so merges can be audited and settled one layer down.
• I3 (answer): word-problem and fable readers could justify a chosen action by showing which motives supported it and which were overridden, giving an explainable trace.
• I4 (world): when two event descriptions turn out to share an agent or resource, unifying their graphs could let downstream inferences reuse that entity instead of creating duplicates.
• I5 (learn): build small evaluation suites of identity-merge scenarios with known commonsense outcomes, used as negative controls for coreference and world-merging rules.