There is no reason why logic cannot follow various different threads of reasoning, interweave them, merge them, split them again, etc. Logic constitutes a first principle of utmost generality, actually I cannot imagine anything more general. Identity is not equivalent to equivalence, equivalence is a quotient of identity, consisting of two classes: Those values which are identical to True, and those which are not.
When talking about identity/equivalence of types in the context of homotopy type theory, yes. This is literally what the univalence axiom states.
Auggierose, I'm curious about your thoughts on how we can provide more rigor to LLMs when it comes to large-scale program transformations and proof synthesis. Given the complexity and versatility of these systems, what kind of foundational framework do you believe would enable GPT and similar models to synthesize and execute proofs rigorously? How can we ensure that they are both reliable and adaptable while dealing with various mathematical and logical domains?
More importantly, how whould this relate to NLP tasks such as: alright, the story is good, but can you rewrite it in the style of Auggierose ?
