To me (but I might be wrong) Rice's theorem asserts that it's not possible to formalise the demonstration of a non-trivial property of a recursively enumerable language within the same given language. The modus ponens being the basis for demonstrating, Rice therefore says in substance that you will necessarily need a metalanguage to define it.
What I call Carroll's paradox is referred in Wikipedia under What the Tortoise Said to Achilles.
Wikipedia explains the following:
The Wittgensteinian philosopher Peter Winch discussed the paradox in The Idea of a Social Science and its Relation to Philosophy (1958), where he argued that the paradox showed that "the actual process of drawing an inference, which is after all at the heart of logic, is something which cannot be represented as a logical formula ... Learning to infer is not just a matter of being taught about explicit logical relations between propositions; it is learning to do something" (p. 57). Winch goes on to suggest that the moral of the dialogue is a particular case of a general lesson, to the effect that the proper application of rules governing a form of human activity cannot itself be summed up with a set of further rules, and so that "a form of human activity can never be summed up in a set of explicit precepts" (p. 53).
In other words, Carroll's paradox shows that it's not formally possible to prove a property of a language purely with propositions of this language. You will need to admit a rule (or a principle) that "the modus ponens means X" with X using everyday words. I.e. you will need a meta language.
I will concede that it is a philosophical way to ask the question. But Carroll's paradox can be described in a formal (mathematical) way (done here).
Hence am I mistaken in principle to say that Caroll's paradox implies Rice's theorem?