Andrew Ushenko in a Mind article from 1946, "Zeno's Paradoxes," distinguishes five putative ways of refuting Zeno's paradoxes: logical, mathematical, mathematico-physical, physical, and philosophical. Ushenko points out that two logical refutations fail. This post examines one of them. This is of particular interest since a reader floated a similar suggestion. Ushenko states the objection and then answers it cogently:
"Zeno's statement of the conditions of the race [of Achilles and the Tortoise], for example, of the condition that A moves faster than T, is equivalent to the assumption that motion exists, and therefore contradicts his own conclusion that motion is an illusion. Hence Zeno is inconsistent with himself." The falsehood of this accusation can be easily demonstrated. Of course, we must grant that Zeno begins with the assumption that there is motion, and concludes that there is no motion. But this procedure means only that he asserts, on the basis of his "proof", that If there is motion, then there is no motion. And, of course, the underscored conditional statement is true if, and only if, there is no such thing as motion.
Ushenko's reply to the objection is correct. Propositions of the form p --> ~p (where the arrow stands for the Philonian conditional) are none of them contradictory. They are equivalent to propositions of the form ~p v ~p which in turn are equivalent to propositions of the form ~p. It follows that If there is motion, then there is no motion is equivalent to There is no motion.
Consider an analogy. Someone argues on Anselmian grounds that (1) if God exists, then God exists necessarily; but for Humean reasons (2) nothing exists necessarily; ergo (3) if God exists, then God does not exist. There is no logical contradiction here, since the arguer is not affirming the existence of God; he is reasoning from the assumption that God exists, an assumption he does not affirm. Similarly, Zeno is not affirming the existence of motion; he is reasoning from the assumption that motion exists, an assumption he does not affirm.
The Regressive Dichotomy is one of Zeno's paradoxes of motion. How can I get from point A, where I am, to point B, where I want to be? It seems I can't get started.
To get from A to B, I must go halfway. But to travel halfway, I must first traverse half of the halfway distance, and thus 1/4 of the total distance. But to do this I must move 1/8 of the total distance. And so on. The sequence of runs I must complete in order to reach my goal has the form of an infinite regress with no first term: