11.1.2 Donkey Sentences

Motivation of discourse phenomena: Donkey Sentences.

A similar problem is also manifested in so-called donkey sentence s. (Incidentally, the example sentences that led to increased study of anaphoric pronouns in discourse in formal semantics, due to Peter T. Geach, staged as main characters donkeys and farmers.) One of the most famous donkey sentences is: `` Every farmer that owns a donkey beats it. ''

If we use our systematic method of translating into first-order logic and use free-variables to translate pronouns, we would get something similar to:

However, this is not a correct translation since there is an occurrence of a free variable (namely, the occurrence of y in (x,y)). In other words, it is not a logical sentence. So let's try to repeat the trick that we applied in our previous example and extend the scope of the existential quantifier.

Unfortunately, our ``scope-extending-trick'' doesn't work in this case. This formula does not assign the right truth conditions for the donkey sentence (Why is this so? Try to imagine a situation where there is a farmer owning a donkey and a pig, and not beating any of them. The above 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:

Recall from Chapter 3 that we translate determiners such as ``every'' with universal quantifiers, and that the indefinite articles (such as ``a'' and ``some'') are translated into existential quantifiers. However, this seems not the case for donkey sentences. As the translation above shows, the indefinite noun phrase ``a donkey'' is translated as a universal quantifier.

In other words, our approach about being systematic when it comes to translating natural language to semantic representations seems not to rhyme with a correct treatment of indefinite noun phrases: depending on the context, sometimes they are translated as existential quantifiers, and sometimes as universal quantifiers. But exactly when?