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.

Buridan on logical form

I posted four extensive works by Jean Buridan at the weekend, and just noticed this interesting argument in Book I question 6 of his Questions on the Prior Analytics, which is connected with my earlier discussion on context and indexicals.  He asks whether an expository syllogism - a syllogism in which the middle term is an expression that demonstratively identifies a subject - is valid in virtue of its form.

Latin	English
Sexta quaestio est utrum syllogismus expositorius sit bonus gratia formae.	The sixth question is whether the expository syllogism is good [i.e. valid] in virtue of its form
Et arguitur quod non: quia iste syllogismus videtur esse expositorius 'hic homo est albus, hic homo est niger; ergo nigrum est album', et tamen consequentia non est bona, quia conclusio est manifeste falsa et tamen possibile est quod ambae praemissae sint simul verae, scilicet si in maiori demonstratur Socrates et in minori Plato.	And it is argued that it is not, for the syllogism “this man is white, this man is black, therefore a black thing is a white thing, and yet the consequence is not good, because the conclusion is manifestly false and yet it is possible that both premisses are true at the same time, namely if Socrates is pointed to in the major, and Plato is pointed to in the minor

He replies that if one suppositum is signified in the major and another in the minor, the middle term is varied, and if the middle term is varied, it is not a good syllogism, nor is it a good expository syllogism. I don’t quite understand how this shows that the syllogism is valid in virtue of its form, however.