The "informal logical path" proves the informal implication, but what counts as a proof of this implication here can be agent dependent. If we talk about proving formal implication, then we have still the problem of deduction[1][2][3]. In short, natural language doesn't distinguish between "inference rules" and expressing this rule in a sentence and using it as a premiss. Which leads to an infinite regress, unless the agent just stops at some point, unlike the tortoise in Carroll's story.