How is apoha supposed to work? ‘Cow’ means ‘not non-cow’ sounds circular to me
I teach mathematics, and when I read that Dignāga's theory of meaning says a word like "cow" works by exclusion, meaning roughly "not non-cow", my first reaction was the one I would have to a student's proof: this is circular. To exclude the non-cows, don't I already need to know which things are cows?
I understand the motivation. Buddhist epistemologists did not want a real universal "cowness" sitting in every cow, which is what the Nyāya realists defended. And they did not want "cow" to be an arbitrary label either, because then why would we all agree on it? Apoha is meant to be a third way.
But I cannot see how it escapes the circle. In set terms, the complement of the set of non-cows is the set of cows, and to define one you need the other.
Could someone explain how apoha is supposed to work, and whether the circularity objection has an answer? I would especially like to know how Dharmakīrti developed it, since I gather he changed it.