Thursday, March 15, 2012

Every man's donkey is running

Suppose that every man has two donkeys, one running and one not running. Then every man's donkey is running, for the donkey of this man runs, the donkey of that man runs, and so on for each individual man. But on the other hand, a donkey of this man does not run, (namely the other one of his which is not running), a donkey of that man does not run, and so on for each individual man. Therefore every man's donkey is not running. Therefore it is not the case that every man's donkey is running.

This is one of those puzzles which caused medieval logicians all sorts of mental strain, but which is completely resolved by translation to modern predicate logic. It can easily be shown that the scenario of each man having two donkeys, one running and one not running, implies the following two propositions of predicate logic

(1) (x) Ey x owns y and y runs
(2) (x) Ey x owns y and not (y runs)

where x ranges over men and y over donkeys. Obviously the two propositions are not contraries: they can both be true at the same time. Yet the English sentences which they translate ('every man's donkey runs'and 'every man's donkey does not run') do appear contraries. This is clearly a problem for English, not for logic.

The program of modern analytic philosophy was to resolve all philosophical puzzles by means of the same kind of translation into modern predicate logic. I think this has failed, but that does not imply there can't be some way of formalising paradoxical or aporetic sets of English sentences in a way that dissolves the aporia.


Anthony said...

How is it a problem for English? The sentence you are looking for is "every man has a donkey which runs", not "every man's donkey runs". The latter would be "there exists a donkey which is owned by every man and which runs".

Furthermore, "the donkey of this man runs" does not mean the same thing as "a donkey of this man runs". The former implies there is only one donkey.

This may have been a problem for medieval language (the lack of a distinction between "the donkey" and "a donkey"?), but it isn't a problem for English.

Anthony said...

>> The latter would be "there exists a donkey which is owned by every man and which runs".

"there exists exactly one donkey which is owned by every man, and that donkey runs", even.

Edward Ockham said...

Of course we can disambiguate any English sentence which is ambiguous. The virtue of formal languages is that they are not ambiguous at all.

David Brightly said...

I confess my Latin is jolly rusty. I'd translate 'Cuiuslibet hominis asinus currit' as '[a?/the?] donkey of whichever man you choose runs'. The Latin does not make the intended article explicit. If we take the indefinite option 'a' and see it as 'some' then no problem arises. If we take the definite option 'the' then the language misrepresents the context, falsely suggesting that each man has a single donkey (I think this is how 'the' works in modern English). Alternatively we might say the 'donkey' picks out a function from men to donkeys and we can render the sentences in predicate logic as '(x) runs(donkey(x)' and '(x) ~runs(donkey(x)'. But for consistency the sentences must denote a distinct donkey() functions.

I'd be interested to know if the medievals understood this ambiguity/lack of expressive power in their language. The fact that this puzzle caused them all sorts of mental strain suggests not, but I find this hard to believe. Surely one can resolve these issues in any language by expanded paraphrasing? Or does one need new concepts, like the modern idea of function? A big topic perhaps.

Edward Ockham said...

I'm currently transcribing one of the 'donkey' questions to understand why it was such a puzzle. I think the problem was representing the ambiguity within the 'two term' framework. Note Wittgenstein's remark that mathematical logic has 'deformed' our thinking, and the old Aristotelian logic was no better.