It is hard to see how what he says could be true. Suppose that the conclusion of 'A man runs; therefore, an animal runs' did not exist because it was annihilated. That means either the utterance, inscription, etc. was annihilated or else the mental proposition (though it is not clear what it would mean to annihilate a mental proposition, so Buridan is likely considering only the former scenario). Then what of the argument? It seems that it too would not exist. For an argument just is a sequence of propositions where the last follows from the others by valid inference (possibly enthymematic, let us assume). If a member of a sequence is annihilated, so is the sequence. What could Buridan mean by 'argument' so that what he says is both nominalist-friendly and true?
As Klima remarks in his book John Buridan (Chapter 9), Buridan has to make a distinction between true in and true of a scenario. He could say that the conclusion of the argument is true of a situation in which it is annihilated even though it is not true in it, because it does not exist in it to be true. But then what is a merely possible situation and what exactly is an argument?
As a concrete example, he uses 'No proposition is negative'. If it were true, it would exist, but then since it is a negative proposition, it would be false. By reductio, it cannot be true. But it is possible, for there is a situation that is as it signifies, though in such a situation the proposition cannot exist. This is why he thinks the first sophism is valid even though it does not necessarily preserve truth; "for it is necessary that if things are as the [premise] signifies, then things are as the [conclusion] signifies" .
If an argument is valid/invalid, then it is necessarily so. Thus, on the supposition that every man is a donkey, the the third sophism does not become valid. What is valid is the inference that includes the supposition as an additional premise.
Jane utters "I say that a man is a donkey". We can take this two says. The first is that she performs an act with assertoric force whose content is that a man is a donkey, i.e. by uttering this sentence she asserts that a man is a donkey. In other words, an utterance of 'I say that a man is a donkey' is in some sense equivalent to an utterance of 'A man is a donkey'. Then what she utters is false and not true as Buridan claims. The second is that she performs no such act, so that the the truth of teh utterance is just like any other, i.e. it is true just in case she says that a man is a donkey. But on the assumption that she performs no such act, she does not say that a man is a donkey, (she says that she says that a man is a doneky) so what she utters is false, contra Buridan.
It is that the truth or falsity of an utterance depends on its not being embedded in a larger utterance. When Plato utters 'No man is a donkey' he utters every part of that utterance including 'Man is a donkey' which itself happens to be a standalone or complete proposition, at first glance. But Buridan says that the part is itself not a proposition capable of being true or false. When one hears only part of what Plato says, one only believes one hears a false proposition, but one does not actually hear a false proposition.