Donkey Anaphora

The puzzle

Two innocent-looking sentences from Geach (1962):

(1) Every farmer who owns a donkey beats it.
(2) If a farmer owns a donkey, he beats it.

Both are perfectly natural English. Both are interpretable in exactly the way you'd expect: every farmer-donkey pair where the farmer owns the donkey, the farmer beats that donkey. But standard predicate logic, applied compositionally, cannot get this reading.

The problem is that a donkey is a Russellian indefinite — it should translate as ∃y. The smallest scope for that ∃y is the relative clause. So a naive translation of (1) is:

∀x. (FARMER(x) ∧ ∃y. (DONKEY(y) ∧ OWNS(x, y))) → BEATS(x, it)

The pronoun it is supposed to bind to y. But y is trapped inside the existential, which is itself trapped inside the antecedent of a conditional inside a universal. There is no scope path from it back to y. The pronoun is logically unbound.

Yet speakers process (1) and (2) effortlessly with the universal reading: ∀x. ∀y. (FARMER(x) ∧ DONKEY(y) ∧ OWNS(x, y)) → BEATS(x, y). The indefinite has acquired universal force, and the pronoun reaches inside the relative clause to bind the trapped variable.

That gap — between what the standard apparatus predicts and what speakers compute — is the donkey puzzle. It animated semantics from the 1970s onward and produced two of the most influential frameworks in modern formal semantics.

Discourse Representation Theory (Kamp 1981)

Hans Kamp's solution: stop trying to translate sentences into self-contained logical formulas. Translate them incrementally into Discourse Representation Structures (DRSs), boxed records with two parts — a list of discourse referents (entities currently "in play") and a list of conditions on them.

A simple example. "A farmer arrived. He whistled."

┌─────────────────┐
│  x              │
├─────────────────┤
│  farmer(x)      │
│  arrived(x)     │
│  whistled(x)    │
└─────────────────┘

The indefinite introduces discourse referent x; the pronoun retrieves it. No special binding apparatus needed.

For donkey sentences, DRT introduces duplex conditions. The relative-clause donkey sentence (1) becomes:

┌─────────────────────┐    ┌─────────────────┐
│  x, y               │    │                 │
├─────────────────────┤  ⇒ ├─────────────────┤
│  farmer(x)          │    │  beats(x, y)    │
│  donkey(y)          │    │                 │
│  owns(x, y)         │    │                 │
└─────────────────────┘    └─────────────────┘

The duplex condition is read "every assignment to x and y that satisfies the left-hand DRS satisfies the right-hand DRS." This delivers ∀x∀y. The discourse referents introduced by the indefinites (a donkey, a farmer) live inside the universal, and the pronoun in the right-hand DRS finds y by accessibility from the surrounding restrictor.

File Change Semantics (Heim 1982)

Irene Heim, working independently, gave a structurally parallel solution using a different metaphor. A discourse is a file. Each individual gets a file card. Indefinites open new cards; definites and pronouns retrieve old ones.

"A farmer owns a donkey."
   → file: [card 1: farmer; card 2: donkey, owned by 1]

"He beats it."
   → "he" finds card 1 (farmer); "it" finds card 2 (donkey)
   → update card 1: beats card 2

Quantifiers create sub-files; donkey anaphora works because the sub-file's indefinites are accessible to pronouns in the corresponding nuclear-scope sub-file. Like DRT, FCS abandons the requirement that all binding be sentence-internal, replacing it with discourse-level accessibility.

DRT and FCS converge on essentially the same predictions for donkey anaphora — and both are typically grouped under dynamic semantics. Later refinements (Groenendijk and Stokhof's Dynamic Predicate Logic 1991, Beaver's Update Semantics) formalize the dynamic moves directly in terms of context updates.

Donkey-anaphora theories at a glance

	DRT (Kamp)	FCS (Heim)	E-type (Evans)	DPL (Groenendijk & Stokhof)
Type of theory	Representational dynamic	Representational dynamic	Static, descriptive proxy	Direct dynamic logic
Pronoun is	Discourse referent	File-card lookup	Hidden definite description	Bound variable, dynamically
Indefinite is	Variable + condition	New file card	Existential w/ accommodated description	Dynamic existential
Strong reading	Default	Default	Requires extra machinery	Default
Weak reading	Available with relative-clause adjustments	Available with adjustments	Default	Available
Compositionality	Two-level (DRT then truth)	Two-level (file then truth)	One-level, classical	One-level, dynamic
Empirical coverage	Wide; classic donkey, modal subordination	Wide; same as DRT	Narrow; struggles with proportion	Wide; most parallel to DRT
Foundational text	Kamp 1981, Kamp & Reyle 1993	Heim 1982 dissertation	Evans 1977, 1980; Heim 1990	Groenendijk & Stokhof 1991

Strong vs weak readings

Donkey sentences are systematically ambiguous between strong and weak readings:

• Strong (universal): "Every farmer who owns a donkey beats it" = every farmer beats every donkey he owns.
• Weak (existential): "Every farmer who owns a donkey beats it" = every farmer beats at least one donkey he owns.

Some sentences favor strong, others weak. Kanazawa (1994) and Geurts (2002) survey the data. Compare:

(3) Every farmer who has a credit card uses it.    [favors weak]
(4) Every man who has a quarter puts it in the meter. [favors weak]
(5) Every farmer who owns a donkey beats it.        [favors strong]

The contrast tracks the predicate's plausibility profile: a person with multiple credit cards typically uses some of them, not all; a farmer with multiple donkeys plausibly beats all of them. The strong/weak ambiguity is genuine and remains theory-resistant.

Cross-linguistic variation

• English / German / Dutch / French. Standard donkey readings as described.
• Mandarin. Cheng and Huang (1996) — Mandarin uses bare wh-words ("wh-conditionals") where English uses indefinites: 谁先来谁先吃 shéi xiān lái shéi xiān chī ("whoever comes first eats first"). The bare wh has the same dual life as English indefinites in donkey contexts.
• Japanese. Bare nouns and the wh-mo construction (誰でも dare-mo "anyone") handle donkey-like contexts. Nishigauchi (1990) gives a detailed treatment.
• Hindi. Conditional-correlative constructions (jo... vo) explicitly mark the donkey relation: jo kisaan ke paas gadhaa hai vo use peeTtaa hai "the farmer who has a donkey, he beats it."
• Tlingit and other languages with overt switch-reference. Some show explicit grammatical marking of the antecedent-anaphor link, providing morphological evidence for dynamic-style accessibility.

Beyond donkeys: modal subordination

Roberts (1989) showed that the same dynamic mechanism handles modal subordination:

A wolf might walk in. It would eat me first.

The pronoun it in the second sentence cannot pick up the wolf if "a wolf might walk in" is read with narrow-scope existential — there is no actual wolf to refer to. Yet the pronoun works. Dynamic semantics handles this by letting the modal operator (might) license a temporary discourse referent that subsequent modal sentences (would) can access. DRT extends naturally to model this and other long-distance binding phenomena.

Related phenomena

• Bishop sentences. "If a bishop meets a bishop, he blesses him" — two indefinites, two pronouns. Tests the symmetry of dynamic binding (Heim 1990, Elbourne 2005).
• Sage-plant sentences. "Everyone who bought a sage plant from Mary's nursery thinks it will outlast the year" (Heim 1982) — singular indefinite, plural reference; tests the genericity-individuation interface.
• Telescoping. "Every chess set comes with a spare pawn. They are all kept in a separate box." The plural they picks up the kind, not the individuals — argues for kind-referring discourse referents.
• Paycheck sentences. "The man who gave his paycheck to his wife is wiser than the man who gave it to his mistress" (Karttunen 1969) — "it" is a paycheck-of-some-other-man. Treated as E-type lookalike or as a special form of dynamic binding.

Common pitfalls

• Treating the donkey indefinite as a true universal. It's an indefinite that acquires universal force from the surrounding quantifier — its lexical meaning is still existential.
• Ignoring the strong/weak ambiguity. Sentences with credit cards, quarters, light switches favor weak; sentences with donkeys, mistakes, books favor strong. The choice is a real empirical fact.
• Conflating DRT with FCS. They make near-identical predictions but use different formal apparatus; cross-citing their predictions without flagging the framework hides where they actually differ.
• Assuming E-type theories are dead. They handle some cases (paycheck sentences) more naturally than dynamic theories. Modern E-type theories (Elbourne 2005) are still active.
• Reading dynamic semantics as a representational claim. Some versions are; others (like DPL) treat dynamics as a property of the logic, not a mental representation.
• Forgetting the conditional version. "If a farmer owns a donkey, he beats it" is a donkey sentence too — without the relative clause. Both relative-clause and conditional donkey sentences need to be covered by any complete theory.