I think I have a reference to Curry in my summary link. Anyways, curry-howard is a nice correspondence, about as important to AL as the correspondence between the groups (ℝ, 0, +) and (ℝ \ 0, 1, *); by which I mean, not at all. But type people like bringing it up even when it is not relevant at all.
No, sorry, I really don't have types. Maybe trying to reduce all approaches to logic to curry-howard is the very reductive view here.
If your system encodes invariants, constrains terms, and supports abstraction via formal rules, then you’re doing the work of types whether you like the name or not.
Dismissing Curry–Howard without addressing its foundational and extricable relevance to computation and logic isn’t a rebuttal.
Saying "Curry-Howard, Curry-Howard, Curry-Howard" isn't an argument, either.
I am not saying that types cannot do this work. I am saying that to do this work you don't need types, and AL is the proof for that. Well, first-order logic is already the proof for that, but it doesn't have general binders.
Now, you are saying, whenever this work is done, it is Curry-Howard, but that is just plain wrong. Curry-Howard has a specific meaning, maybe read up on it.
Curry–Howard applies when there’s a computational interpretation of proofs — like in AL, which encodes computation and abstraction in a logic.
You don’t get to do type-like work, then deny the structural analogy just because you renamed the machinery. It’s a type system built while insisting type systems are obsolete.
No, sorry, I really don't have types. Maybe trying to reduce all approaches to logic to curry-howard is the very reductive view here.