CSI 5386 Donkey Sentence Discussion

Here is the well-known donkey sentence:

 (1) Every farmer who owns a donkey beats it

I have claimed in class that the donkey sentence poses problems when we try to translate it into standard predicate logic. Before we can see why, let's do some warmup exercises. Here are some English sentences and their predicate logic representations:

 (2a) Ken plays hockey
 (2b) plays(ken, hockey)

 (3a) Ken drinks beer
 (3b) drinks(ken, beer)

 (4a) Everybody who plays hockey drinks beer
 (4b) " X ( plays(X, hockey) Þ drinks(X, beer) )

 (5a) Every man who owns a train is happy
 (5b) " X ( man(X) Ù \$ Y ( train(Y) Ù owns(X, Y) ) Þ happy(X) )

(5b) can be interpreted as follows: for every X, if X is a man and there exists some Y (where Y is a train that X owns), then X is happy.

Now we're ready to look at the donkey sentence again:

 (1) Every farmer who owns a donkey beats it

We could try to translate it the same way that we translated the train sentence above:

 (6) " X ( farmer(X) Ù \$ Y ( donkey(Y) Ù owns(X, Y) ) Þ beats(X, Y) )

The problem with (6) is that the beats(X, Y) is not within the scope of the variable Y. That is, the variable Y only has meaning within the red parentheses.

There are three ways that we might attempt to extend the scope of Y to include beats(X, Y). First, we could just try to extend Y's scope to the right:

 (7) " X ( farmer(X) Ù \$ Y ( donkey(Y) Ù owns(X, Y) Þ beats(X, Y) ) )

But that removes farmer(X) from the scope of the implication, giving us the interpretation: for every X in the universe, X is a farmer. Also for every X, there exists some Y such that if Y is a donkey and X owns Y, X beats Y.

Instead, we could keep farmer(X) within the scope of the implication and extend the scope of Y by moving it outside:

 (8) " X \$ Y ( farmer(X) Ù donkey(Y) Ù owns(X, Y) Þ beats(X, Y) )

But this translation has the following interpretation: for every X in the universe, there absolutely is a Y; if the X is a farmer and the Y is a donkey and X owns Y, then X beats Y.

Finally, we could go all the way and move Y outside the scope of X even:

 (9) \$ Y " X ( farmer(X) Ù donkey(Y) Ù owns(X, Y) Þ beats(X, Y) )

But (9) is worst of all. It says: there is some single thing Y in the universe such that for every X in the universe if X is a farmer and Y is a donkey and X owns Y, then X beats Y.

So the problem with the donkey sentence is that the scope of the variable corresponding to the donkey must be contained within the antecedent of the implication to prevent requiring the unconditional existence of the donkey. But the scope of the donkey variable must contain the consequent of the implication to allow the anaphoric reference!