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I’m having difficulty following your comments about adversarial choice as related to the axiom of choice. What you’re saying doesn’t sound well-defined (in the mathematical sense).

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It's actually very well defined (or which part are you confused by). I'm stating these things informally as a formal treatment is right now beyond me and it also doesn't really exist. Look up Chu spaces esp their relationship to linear logic. This paper is pretty informative

https://www.semanticscholar.org/paper/Chu-spaces-as-a-semant...

Fundamentally, think of the minimax algorithm. You have two sides each optimizing for victory.

I don't see how Pratt's paper justifies any of your claims.

Can you give me the summary of the paper?

"It's actually very well defined" and "a formal treatment... doesn't really exist" seem to me to be contradictory statements.

It's a path that unifies several very formally related. The unification itself isn't fully formalized, you need to look at the symmetries in the well formalized subparts.

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