It seems you haven't read my references? As your [2] is my [3] from above!
> The existence of a different kind of CPU isn't a meaningful distinction at the level of discussing paradigms.
Well, that was my point above: You can't really lump quantum programming together with probabilistic programming, as they are paradigms on different "levels".
> practical papers like [2] cite it wistfully as a dream of what could be achieved
Are you sure about that? I skimmed [1] as I wasn't hadn't read it and it seems to describe a rather restricted set of functions ("types are interpreted as vector spaces and terms as functions defined by power series on these spaces"), as there are many differentiable functions that cannot be defined as power series.
Moreover, in [2] it is only claimed: "Ehrhard and Regnier {i.e. your reference [1]} do not give an operational semantics but they do give rules for symbolic differentiation and it should not be too difficult to use them to give an operational semantics. However their language with its convenient vector space semantics only supports total functions. It therefore cannot be extended to include recursive function definitions or conditionals (even with total predicates, as continuous functions from R^n to the booleans are constant)." So I would not say they cite [1] as a wistful dream ...
> There are only a few slight axiom differences that separate differential, logic, probabilistic, and quantum programming though.
Give me the axioms and their differences and I believe you. :) (Honestly, I'm not even sure if the discussion on which axiomatization captures the existing developments has been settled; it seems to me you have some kind of category theoretic approach in mind where you just change the category and get a new paradigm - I'd be happy to accept this as well, if there is a clear reference, though I'm doubtful one exists ...)
> The existence of a different kind of CPU isn't a meaningful distinction at the level of discussing paradigms.
Well, that was my point above: You can't really lump quantum programming together with probabilistic programming, as they are paradigms on different "levels".
> practical papers like [2] cite it wistfully as a dream of what could be achieved
Are you sure about that? I skimmed [1] as I wasn't hadn't read it and it seems to describe a rather restricted set of functions ("types are interpreted as vector spaces and terms as functions defined by power series on these spaces"), as there are many differentiable functions that cannot be defined as power series.
Moreover, in [2] it is only claimed: "Ehrhard and Regnier {i.e. your reference [1]} do not give an operational semantics but they do give rules for symbolic differentiation and it should not be too difficult to use them to give an operational semantics. However their language with its convenient vector space semantics only supports total functions. It therefore cannot be extended to include recursive function definitions or conditionals (even with total predicates, as continuous functions from R^n to the booleans are constant)." So I would not say they cite [1] as a wistful dream ...
> There are only a few slight axiom differences that separate differential, logic, probabilistic, and quantum programming though.
Give me the axioms and their differences and I believe you. :) (Honestly, I'm not even sure if the discussion on which axiomatization captures the existing developments has been settled; it seems to me you have some kind of category theoretic approach in mind where you just change the category and get a new paradigm - I'd be happy to accept this as well, if there is a clear reference, though I'm doubtful one exists ...)