Making sense of the Bolzano-Carroll-Wilson paradox

The infinite regress problem involves the identification of a set of foundational problems in logic, deductive reasoning, and the epistemology of understanding through an allegorical dialogue between Achilles and the tortoise. In this paper, the infinite regress problem, traditionally referred to as Carroll's paradox, will be re-identified as the Bolzano-Carroll-Wilson paradox, in recognition of its multiple provenances. Furthermore, we shall investigate the nature and scope of the paradox and attempt to identify the real moral behind what the tortoise said to Achilles. In particular, we shall engage with the Russell-Grice point concerning the need to distinguish between a material implication relation (→) and an inferential relation (⊢), the further need to distinguish between first-order logical statements and higher-order statements (metalogical, meta-metalogical, and so on), and the Ryle-Wittgenstein point concerning the nature of proper inference as knowledge-how and rule-following.