If the utterer says more than one thing then there is no paradox. If one of the other things said is false then the utter does say something false, and so the sophism is true. If one of the other things said is true then the sophism is false because the utterer has not said anything false. (At least, it is consistent to say that the sophism is false in this case, though it is not clear that we are compelled to say so.) So the sophism's being problematic depends on a contingent fact, namely, that the utterer say nothing else. This relates to a similar recent discovery concerning contingent liar paradoxes. It was widely believed that paradoxical sentences pose a problem that can be weeded out a priori--by looking just at the syntactic structure of a sentence, we can figure out whether it is paradoxical and e.g. ban it from being grammatical or meaningful. Sophisms like this one, however, show this simply cannot be the case. Likewise if everything Jane utters is "Everything John says is true" and John utters "Everything Jane says is false", we land in paradox, but only due to the contingent fact that each has said what they've said. (If one failed to make such an utterance pointing back to the other in this way, there would be no paradox.) For an influential paper on these matters, see Saul Kripke's "Outline of a theory of truth", The Journal of Philosophy, Vol. 72, No. 19, pp. 690-716.
He says that the 'that'-clause, i.e. that man is a donkey, supposits for nothing, since it is impossible for a man to be a donkey. Indeed, the proposition 'A man is a donkey' does not even signify that a man is a donkey. However, according to the fifth conclusion, 'A man is a donkey' still signifies something, namely that a man exists and that a donkey exists, since each is respectively signified by the terms 'man' and 'donkey'.
In the seventh conclusion Buridan says "Since the verb 'to signify' or 'to be signified' ampliates supposition to past and future [things], it follows that if Aristotle is signified by the term 'man', and Aristotle was that Aristotle exists, then that Aristotle exists is signified by the term 'man'" [853-4]. It is clear that there is no object-/meta-language distinction for Buridan and that the one language he uses contains expressions like 'signify' to speak about the semantic properties of that very language. We can only conclude that Buridan is likely more Meinongian than Quinean!
This is an interesting one. It claims that a true affirmative proposition about actuality and about the present is not true on the grounds that every proposition it signifies is true. As we will see in a later reading, this is in contrast to Thomas Bradwardine's view according to which this is so. Bradwardine even uses this theory of truth (that p is true just in case whatever it signifies is so) to solve paradoxes like the liar. The reason Buridan rejects this principle is due to his views concerning ampliation and signification. In particular, 'man' signifies Aristotle which signifies that Aristotle exists, and so 'A man is an animal' signifies that Aristotle exists which is not so. So true propositions can signify what is not the case. However and relatedly, he goes on to say that a proposition cannot be true if it "asserts something that implies that implies [...] that it is false" , which sounds closer to what Bradwardine has in mind. Indeed, what he says toward the end of the eleventh conclusion is very much indeed in the spirit of Bradwardine.
It has been said in places that Buridan is committed to possiblia and things that don't presently exist, but passages like this suggest a different view. It seems that in order to get the right truth conditions for sentences, Buridan believes in ampliation, but only in a Meinongian sort of way, since he is not committed to the existence of the supposita of sentences, just like Meinong is not committed to the existence of referents of expressions. A more careful analysis of this would be most interesting.
To the fourth sophism, Buridan claims that even definitions, such as 'A vacuum is a place not filled with body', can be false if the terms (e.g. the subject term) supposits for nothing. This we have already seen is the reason that the third sophism, 'A chimera is a chimera', is false.