I hope you don’t mind I asked one of the better AIs to try to grok the specifics of your back-and-forth. Let me know how it did:

– The Curry-Howard correspondence traditionally views logic and proofs as static and “timeless”. Truth values of propositions don’t change.

– Stephen Paul King suggests considering a variant where truth values could oscillate or change over time/interactions, akin to an on/off signal. This brings temporality into the picture.

– Santeri responds that type theory already allows for a type (theorem) to only be inhabited (provable) if it is true.

– But he takes it a step further by proposing logical structures where the matching between types and theorems could change depending on perspective or the direction of reasoning (e.g. whole-to-part vs. part-to-whole).

– This would allow truth values to potentially “flip” when viewed from different angles, capturing an interactive/temporal dimension.

– Any non-matching “residue processes” left over after attempting to pair types and theorems could then be named as new types, expanding the system.

– Stern-Brocot fractions are given as an example type that arises from mereological operations but has no analogue in standard fractions.

– Relating it to Plato’s pre-logical method of diairesis, Santeri suggests it could be seen as a “pre-sheaf” of logics to allow for developmental/evolutionary changes.

– Examples like omega-consistent logics in Haskell and Pratt’s “forgetful” notion are then discussed as further demonstrations of how these ideas might work.

So in summary, they are brainstorming ways to generalize Curry-Howard to better model interactive/temporal aspects by allowing truth values and type-theorem mappings to dynamically change based on context/perspective. New “residue” types would need to be introduced as well.