That reminds me of: https://en.wikipedia.org/wiki/Dialogical_logic https://en.wikipedia.org/wiki/Dialogical_logic also made known as "game semantics" by Hintik

That reminds me of: https://en.wikipedia.org/wiki/Dialogical_logic https://en.wikipedia.org/wiki/Dialogical_logic also made known as "game semantics" by Hintikka, where the two players are "me" and "the Nature", and the two players read alternatively the elements of a quantified mathematical proposition. Each time the Nature finds a counterexample, she wins, and if "me" is able to select the correct OR branches or to find the existing elements of \exists symbols, "me" wins.