Anthony rightly raised the problem of my ‘some men are not men’. He suggested (rightly I think) that this is a problem for ‘Brentano equivalence’, the thesis that ‘Some A is B’ is equivalent to ‘An A-B exists’, and that ‘some A is not B’ is equivalent to ‘An A-not-B exists’.
He is right. If Brentano is right, then ‘some men are not men’ is equivalent to ‘men that are not men exist’, which is clearly wrong. So what’s the problem? I suggest that Brentano is wrong. Clearly we can say that some of the men who landed on the moon have now died ( for example, Alan Shepard, the one who played golf on the moon). So, some men such as Shepard are no longer men. If Brentano is right, that implies that men who are non-men exist, which is false. Non valet consequentia, so Brentano is wrong.
The late thirteenth century philosophers of language, such as the early Scotus, were acutely aware of this problem. Many of them distinguished between so-called indefinite negation of the form ‘A is a non-B’, and pure negation ‘A is not B’. Indefinite negation is affirmative. It affirms the existence of an A that is non-B. In this sense ‘some man is a non-man’ is false. By contrast, pure negation denies everything, including the affirmation of existence. In that sense ‘some man is a not a man’ is true, pace Brentano.
The problem is to render this in predicate logic. The formal sentence ‘for some x, Ax and not Bx’ is affirmative in the traditional sense: it asserts that some x is both A and non-B. However, the pure negative for ‘not for some x, Ax and Bx’ is not equivalent to the medieval ‘some A is not B’. The predicate logic version simply denies the existence of anything that is both A and B, whereas the medievals understood it in the sense we understand ‘some men (such as Alan Shepard) are not men (i.e. are men no longer)’.
Brentano’s thesis was the first formulation of one of the key assumptions of the modern predicate calculus. It is wrong for the same reason the calculus is wrong. It does not translate the meaning of a standard English sentence in the way we want to translate it. So what is the meaning of the sentence, and into what formal language can we translate it?