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Why isn't $\forall(f,d)[\mathrm{farmer?}(f) \land \mathrm{donkey?}(d) \land \mathrm{owns?}(f, d) \implies \mathrm{beats?}(f,d)]$ a valid formalization of, "Every farmer who owns a donkey beats it?"

The Wikipeida article on donkey sentences suggests that "this translation leads to a serious problem of inconsistency," but even with the provided example, it is not at all clear to me why this is the case.

What am I missing here?

Thanks!

anarchocurious
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    who beats $d$? ${}$ – user251257 Aug 17 '15 at 00:29
  • @user251257 Good catch. I edited my post to include $f$. – anarchocurious Aug 17 '15 at 00:48
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    My interpretation of the comment on that Wikipedia page is essentially that the given form means "A farmer beats every donkey (s)he owns", while the original sentence is not as clear: to what does "it" refer? Every donkey owned by the farmer? Some donkey owned by the farmer? – rogerl Aug 17 '15 at 01:09
  • I've asked a few different people, and they have all said that they believe "it" refers to the donkey owned by the farmer which seems to be exactly what my formulation asserts. I don't understand the distinction between every donkey owned by the farmer or some donkey owned by the farmer. – anarchocurious Aug 17 '15 at 02:56
  • Wow, that Wikipedia section is seriously confused. I've added a "needs attention by an expert" note; let's hope someone clears it up a bit. – joriki Aug 17 '15 at 06:18

1 Answers1

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The Wikipedia article does not do a good job of explaining what it is trying to say. That has been known to happen. Your translation looks fine; a natural reading of "every farmer who owns a donkey beats it" is "every farmer who owns a donkey beats that donkey".

My guess is that the article was written by someone who had read something, but they didn't summarize their source very well, and didn't mention what the source was, leaving the article in a confused state.

The deeper issue here is in linguistics, rather than mathematics. Very similar English sentences can have very different meanings. Consider:

  1. If you have a quarter, pay your parking meter with it.
  2. If you borrowed a book from me, return it.

Sentence (1) does not suggest to put all your quarters in the parking meter, but sentence (2) suggests returning all the books you borrowed.

One benefit of formal languages and formal quantifiers is that they do not have the same ambiguity that natural languages can have.

Finally, here is an interesting dichotomy that shows things are more complicated than they might appear. Consider:

  1. Some woman who owns a bicycle rides it.
  2. Every woman who owns a bicycle rides it.

To me, (3) suggests the woman rides at least one bicycle, but (4) suggests she rides every bicycle she owns. How is that, when the quantifier that was changed seems to be about the woman, not the bicycle? This sort of issue is what makes linguists interested in these sentences.

Carl Mummert
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