well, in your search for that positive result, i recommend spending some time searching for a critch!simplified alternative to the Y combinator :-p.
not every method of attaining self-reference in the λ-calculus will port over to logic (b/c in the logical setting lots of things need to be quoted), but the quotation sure isn’t making the problem any easier. a solution to the OP would yield a novel self-reference combinator in the λ-calculus, and the latter might be easier to find (b/c you don’t need to juggle quotes).
if you can lay bare the self-referential property that you’re hoping for in the easier setting of λ-calculus, then perhaps others will have an easier time understanding what you want and helping out (and/or you’ll have an easier time noticing why your desires are unsatisfiable).
(and if it’s still not clear that löb’s theorem is tightly connected to the Y combinator, such that any solution to the OP would immediately yield a critch!simplified self-reference combinator in the λ-calculus, then I recommend spending a little time studying the connection between the Y combinator, löb’s theorem, and lawvere’s fixpoint theorem.)
well, in your search for that positive result, i recommend spending some time searching for a critch!simplified alternative to the Y combinator :-p.
not every method of attaining self-reference in the λ-calculus will port over to logic (b/c in the logical setting lots of things need to be quoted), but the quotation sure isn’t making the problem any easier. a solution to the OP would yield a novel self-reference combinator in the λ-calculus, and the latter might be easier to find (b/c you don’t need to juggle quotes).
if you can lay bare the self-referential property that you’re hoping for in the easier setting of λ-calculus, then perhaps others will have an easier time understanding what you want and helping out (and/or you’ll have an easier time noticing why your desires are unsatisfiable).
(and if it’s still not clear that löb’s theorem is tightly connected to the Y combinator, such that any solution to the OP would immediately yield a critch!simplified self-reference combinator in the λ-calculus, then I recommend spending a little time studying the connection between the Y combinator, löb’s theorem, and lawvere’s fixpoint theorem.)