Contribution
Two things. First, a compact statement of update semantics as a framework, with a criterion for when it matters: an update system is equivalent to a static, truth-conditional one exactly when it is additive, and additivity fails precisely for sentences, like epistemic might, whose effect depends on the state they meet. Second, a semantics for default rules (“P’s normally are Q”, “if φ then normally ψ”) in which conflicts between rules are resolved by what the rules mean, not by a priority ordering added on top, and which yields a decidable consequence relation with predictions that differ, case by case, from every rival the author compares.
Key insight
Some sentences do not tell you about the world; they ask your information state a question. “Might φ” passes if φ is still open and fails otherwise; “presumably φ” passes if φ holds in every most-normal world you cannot rule out. Because they test rather than inform, they can be accepted now and rejected after more is learnt, and that non-persistence, not any special mode of reasoning, is where non-monotonic inference comes from. Default rules themselves are ordinary persistent information, about normality rather than fact.
Assumptions
- Finite propositional language: a finite set A of atoms; worlds are subsets of A (W = ℘(A)), so every logic here is decidable. Lemma 2.5 and Proposition 2.6 show the choice of A does not matter.
- Information states as elimination: in §2 a state is a set of worlds; in §3 a pair ⟨ε, s⟩ of an expectation pattern (reflexive, transitive relation on W) and a set of worlds; in §4 a pair ⟨π, s⟩ of a frame (a pattern πd for every d ⊆ W) and a set of worlds.
- No revision: an update that would be inconsistent yields the absurd state 1; the system says when revision is needed but not how to do it.
- Knowledge, not belief: states are what the agent takes to be knowledge (fn. 3).
- One situation: s is knowledge of the actual situation at one time, so “normally it rains, but not today; tomorrow presumably it will” is outside the system (fn. 4).
- Restricted syntax: might, presumably and normally occur only outermost, over sentences of the base language.
Key results
- Proposition 1.2. ⟨L, Σ, [ ]⟩ is additive iff Σ is an information lattice on which [ ] is total and Idempotence (σ[φ] ⊩ φ), Persistence (σ ⊩ φ and σ ≤ τ ⇒ τ ⊩ φ), Monotony (σ ≤ τ ⇒ σ[φ] ≤ τ[φ]) and Strengthening (σ ≤ σ[φ]) hold.
- Proposition 1.3. In an additive system valid₁ (from the minimal state), valid₂ (from every state) and valid₃ (acceptance-preserving) coincide. In general they do not; valid₁ satisfies Sequential Monotony, Sequential Cut and Reflexivity, which characterise it under Idempotence (citing van Benthem 1991).
- Definition 2.3 and Examples 2.7. σ[might φ] = σ if σ[φ] ≠ 1, else 1. “might ¬p, p” is consistent; “p, might ¬p” is not. Both right and left monotonicity fail. The base language without might is additive and its logic is classical (Lemma 2.8).
- Definition 3.9. σ[normally ψ] refines ε by ‖ψ‖ (removing pairs ⟨v, w⟩ with w ∈ ‖ψ‖, v ∉ ‖ψ‖), absurd if no normal world satisfies ψ; σ[presumably ψ] = σ if all optimal worlds of s satisfy ψ, else 1. Normally p ⊩ presumably p; normally p, ¬p ⊮ presumably p but ⊩ normally p (Examples 3.10). Rules are idempotent, persistent and monotone (Lemma 3.12); presumably is neither persistent nor monotone. “normally” validates conjunction of rules and necessitation but not normally φ ⊩ φ or normally φ ⊩ normally (φ ∨ ψ).
- §4. “If q, normally ¬p” is not definable from unary normally (normally(q ⊃ ¬p) creates an ambiguous state). Restricted rules refine the pattern of their own domain (Definition 4.6); a frame is coherent iff every nonempty domain has a normal world (4.3), with Proposition 4.7 giving when a refinement stays coherent. A set of defaults applies within s iff every domain above s has a normal world complying with all of them (4.9); Proposition 4.12 puts the existential quantifier over refinements outside the universal over defaults. Optimal worlds are those complying with a maximal applicable set (4.13); Proposition 4.14 restricts the search to the explicitly given defaults.
- Worked consequences (4.11, §5). Specificity: normally p, q ~> ¬p, q ⊩ presumably ¬p, and an exception to the exception restores p. Nixon diamond: neither presumably r nor presumably ¬r. Students/adults/employed: presumably adult and not employed (Reiter gives two extensions; this theory one). Defeasible modus tollens (p ~> q, ¬q ⊩ presumably ¬p), with modus ponens taking precedence over it in a cycle. Independence holds, which selection-function theories (Delgrande; Asher and Morreau) lose.
- Failures for ~>. Hypothetical syllogism, contraposition and strengthening the antecedent fail; of the variable-strict principles only Conditional Identity and Conjunction of Consequents hold, while Weakening the Consequent, ASC and Disjunction of Antecedents hold only defeasibly, defeatable by facts alone.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | A dynamic semantics is reducible to a static one exactly when it is additive, characterised by totality and four structural principles | strong (stated proposition) | Proposition 1.2; proof not given |
| C2 | Epistemic might and presumably are tests on information states and are non-persistent, so their content is not a proposition | strong within the model | Definitions 2.3, 3.9; Examples 2.7, 3.10 |
| C3 | Default rules carry context-independent (persistent) information; only the conclusions drawn from them are defeasible | strong within the model | Lemma 3.12 |
| C4 | Priority between conflicting defaults, including specificity, is a consequence of the meaning of the rules (coherence and applicability), not an extra stipulation | moderate | §4 definitions and worked examples; argued by cases, not by a general theorem about priority |
| C5 | The theory’s predictions differ from Reiter, Delgrande, Asher–Morreau and Horty–Thomason–Touretzky, and match intuition better | weak to moderate | a handful of benchmark examples (§5); “better” is the author’s judgement |
| C6 | Selection functions cannot model knowledge of rules, because they lose Independence | moderate | §5 examples |
Method
Definitions of states, updates and orderings over a finite space of worlds, with propositions stated (mostly without proof) and checked against small worked examples drawn as diagrams of normality orderings. The comparison with other default logics is by benchmark examples.
Concepts
- update system: ⟨L, Σ, [ ]⟩, a language, a set of information states, and for each sentence an operation on states (postfix, σ[φ]).
- acceptance (σ ⊩ φ): σ[φ] = σ. acceptable: σ[φ] ≠ 1 (the absurd state).
- additive: σ[φ] = σ + 0[φ] on an information lattice; the case where dynamic meaning reduces to static content.
- test: an update that returns the input state or the absurd state, as might and presumably do.
- expectation pattern: a preorder on worlds, “w ≤ε v” meaning w conforms to every rule v conforms to.
- frame: a pattern for every domain d ⊆ W, so rules can be restricted.
- coherent: every nonempty domain has a normal world.
- applies within s: no domain extending s has all its normal worlds violating the default(s).
- optimal world: a world of s complying with a maximal applicable set of defaults. An ambiguous state has more than one.
- valid₁: updating the minimal state with the premises in order yields acceptance of the conclusion.
Connections
The dynamic notion of meaning is credited (fn. 1) to Stalnaker’s work on presupposition and assertion, Kamp’s and Heim’s work on anaphora, and Gärdenfors’s dynamics of belief; the direct inspiration is Groenendijk and Stokhof’s dynamic predicate logic. Presupposition as definedness of an update is noted and referred to Beaver and Zeevat. The default theory sits beside Reiter’s default logic, Delgrande’s conditional logic, Asher and Morreau’s commonsense entailment (all built on Lewis’s conditionals) and the skeptical inheritance of Horty, Thomason and Touretzky. Generics are left to Carlson and Krifka (1987).
Bearing on the record
- The record held no account of dynamic semantics before the readings of 2026-10-09. This paper, with Krifka’s (LIT-835) and Heim’s dissertation (LIT-838), gives it one. Krifka’s informative update is Veltman’s propositional (additive) update; Krifka’s performative update is a non-eliminative update of a kind Veltman’s systems do not contain (every update here only shrinks s or refines the pattern).
- CLAIM-061 (proposition neither necessary nor sufficient). A weak, indirect bearing: might φ and presumably φ make a conversational contribution that has no propositional content at all, only a test, so what a sentence does in context is not fixed by a proposition it expresses. It does not speak to footing or force.
- A source, with the Heim and Krifka readings of the same day, of THEORY-162: what an utterance does to a context is part of its meaning and is not fixed by its truth conditions. This paper supplies the tests and the additivity criterion.
- No instruction for ML practice; nothing for the anthology.
Limitations
- Most propositions are stated without proof.
- Finite propositional language; the predicate reading of §5 treats “possible objects” as types, not as a first-order semantics.
- No revision: the theory says when an update is absurd, not how to recover.
- One time only (fn. 4), and modal operators only outermost.
- The author says the formalisation is not the only possible one and hopes for a more elegant one, and that it does not explain when a generic sentence gets a default reading (§6).
- The superiority over rival default theories rests on a few examples and on intuitions about them.
Open questions
- Why ASC and Disjunction of Antecedents fail; the author says he has no intuitive explanation (§5).
- Whether a system can have normally p ⊩ normally (p ∨ q) together with the two intuitive inferences of fn. 7 without giving up Sequential Cut.
- How the default semantics relates to the semantics of bare plurals, definite and indefinite generics (§6).