# Donkey sentence

Donkey sentences are sentences that contain a certain type of anaphora, such as:

• Every farmer who owns a donkey beats it.
• Every police officer who arrested a murderer insulted him.

Donkey sentences became a major force in advancing semantic research in the 1980s, with the introduction of discourse representation theory (DRT). During that time, an effort was made to settle the inconsistencies which arose from the attempts to translate donkey sentences into first-order logic.

Donkey sentences present the following problem, when represented in first-order logic: The systematic translation of every existential expression in the sentence into existential quantifiers produces an incorrect representation of the sentence, since it leaves a free occurrence of the variable y in BEAT(x.y):

$\forall x\,({\text{FARMER}}(x)\land \exists y\,({\text{DONKEY}}(y)\land {\text{OWNS}}(x,y))\rightarrow {\text{BEAT}}(x,y))$

Trying to extend the scope of existential quantifier also does not solve the problem:

$\forall x\,\exists y\,({\text{FARMER}}(x)\land {\text{DONKEY}}(y)\land {\text{OWNS}}(x,y)\rightarrow {\text{BEAT}}(x,y))$

In this case, the logical translation fails to give correct truth conditions to donkey sentences: Imagine a situation where there is a farmer owning a donkey and a pig, and not beating any of them. The formula will be true in that situation, because for each farmer we need to find at least one object that either is not a donkey owned by this farmer, or is beaten by the farmer. Hence, if this object denotes the pig, the sentence will be true in that situation.

A correct translation into first-order logic for the donkey sentence seems to be:

$\forall x\,\forall y\,({\text{FARMER}}(x)\land {\text{DONKEY}}(y)\land {\text{OWNS}}(x,y)\rightarrow {\text{BEAT}}(x,y))$

Unfortunately, this translation leads to a serious problem of inconsistency. One possible interpretation, for example, might be that every farmer that owns any donkeys beats every donkey. Clearly this is rarely the intentional meaning. Indefinites must sometimes be interpreted as existential quantifiers, and other times as universal quantifiers, without any apparent regularity.

The solution that DRT provides for the donkey sentence problem can be roughly outlined as follows: The common semantic function of non-anaphoric noun phrases is the introduction of a new discourse referent, which is in turn available for the binding of anaphoric expressions. No quantifiers are introduced into the representation, thus overcoming the scope problem that the logical translations had.