Buridan considers the consequence 'No proposition is negative; therefore, no donkey is running'. The contrapositive is invalid for Buridan; 'Some donkey is running; therefore some proposition is negative'. But then so is the original proposition, assuming premise and conclusion are formed together, and that contraposition is valid. However, if they really are formed together, then the premise ('No proposition is negative') is false, and so we have no counterexample to the original argument, as Buridan remarks in Chapter 8, first sophism. It is a bit strange then that the contrapositive has a clear counterexample that is not a counterexample to the original. Shouldn't logical equivalents stand and fall together relative to the same possible scenarios?
I guess the problem Klima is referring to is the failure of the traditional account of validity due to self-falsifying propositions. But then if propositions are atemporal, abstract entities, the contrapositive of 'No proposition is negative; therefore, no donkey is running', namely, 'Some donkey is running; therefore, some proposition is negative' is valid, for the consequent is necessarily true. So which problem is the same? That there simply exist self-falsifying propositions? First, I don't see why that is a problem if it doesn't affect the traditional account of validity. Second, it is slightly odd on the atemporal view to call a proposition like 'No proposition is negative' self-falsifying since it is (always) made false not only by its own existence but also by the existence of every other negative proposition.
Contraposition follows from modus ponens and (a weak version of) reductio, so giving it up requires giving up at least one of these two plausible principles, which doesn't look very enticing, especially since it is not at all clear how even Buridan could give up either on his token-based semantics. Second, it is clear that contraposition fails given his nominalism about propositions since there is a possible scenario relative to which some donkey runs while no proposition is negative because e.g. no people formed any negative proposition (or to use Buridan's example, God annhilates them all). And his nominalism is more fundamental to his overview methodological view than is the traditional view of validity, so it is more natural to give up the traditional view than to give up e.g. nominalism (or contraposition or reductio or modus ponens).
Consider the liar. If it's true, it's false, so it cannot be true. But that is what it says, so its correspondence conditions are met (its terms (subject and predicate) cosupposit). And yet the liar is not true, for it is only true if things are the way it signifies, and it signifies them to be such that both p and not-p, which is impossible. Given this divergence between truth and correspondence conditions, we can see that Buridan's must reject a plausible principle concerning truth (which some would call a "platitude" even): if it is the case that p, then 'p' is true.
The solution, deviating slightly but importantly from Bradwardine's, is to say that the liar is false because it signifies itself to be both true and false, and that a proposition is true just in case things be in whatever way it signifies them to be. The important underlying assumption here is that every proposition signifies itself to be true. Bradwardine did not hold this general claim but only the weaker and more restrictive claim that propositions that signify themselves to be false also signify themselves to be true. Buridan gives reason to reject the general claim because, e.g, it does not seem that the proposition 'A man is an animal' signifies anything about propositions (e.g. that it is true)--it is about only men and animals. And while this does not affect Bradwardine's solution as Klima notes, that solution still relies on a seemingly ad hoc restriction of the general principle. Why should 'This proposition is false' signify of itself that it is true while 'A man is an animal' does not?
First, Buridan introduces a notion of virtual entailment. A proposition p virtually entails q just in case q signifies or asserts that p is true. Second, he says that a proposition is true just in case things are whatever way the proposition signifies them to be and whatever way its virtually implied propositions signify them to be. It follows that the liar is false, even though things are as it signifies them to be (it's false), since it virtually implies its negation which says that the liar is true.
Klima says "As for the charge of adhockery, one can say that the trivial requirement of Buridan's "virtual implication" for claiming a proposition true is no more ad hoc than the general, trivial requirement that a proposition can only be true if all propositions it validly entails are true as well, as required by modus ponens" . But this isn't quite true. E.g. the proposition 'A man is an animal' virtually implies ''A man is an animal' is true and man is a donkey', since that conjunction signifies the truth of 'A man is an animal' in virtue of its first conjunct. But then things can be the way a proposition signifies without it being the way what it virtually implies signifies, so the situation between the virtual implication requirement is not analogous to the plain implication one. Indeed, probably a better definition of virtual implication that avoids this problem is to say that p virtually implies q just in case q signifies that p is true and nothing else. It is not clear, however, that any proposition can signify only one thing, so its not clear whether this amendment of the view would work.