The odd title of this post will become clear I hope in a moment. The first thing to do when trying to come to grips with a concept is to find out the where and why of it. The notion of the counterpositive comes up in relation to perception (c.f. Vedanta Paribhasa on Perception) particularly in relation to erroneous perception i.e. confusion. We take something to be that which it is not. Now the problem arises - what is the ontological status of our confused reality. The mother of pearl is taken to be silver (precious metal). Must we then admit that for the time that we were under that illusion we were experiencing silver and on being disabused are only entitled to say 'I am not seeing silver now’.
The advaitin in response to this simply states that what is denied is not the existence of illusory silver i.e. the existence of a non-existence, which is otiose. You are denying the existence of the counterpositive i.e. real silver, in that locus. Silveriness in truth abides in its own substratum of silver.
The point of the concept of the counterpositive is to preserve the primal substratum ontology. If you did not have a substratum ontology you would be left with a sense data/phenomenalist ontology in which silver existed momentarily as it was experienced as such. The substratum ontology is also vital for the concept of superimposition/adhyasa.
Another translation suggested for pratiyogin (literally ‘opponent’) is absentee, thus my title of the counterpositive shaped hole for the silver in the nacre.