Retortion is the philosophical procedure whereby one seeks to establish a thesis by uncovering a performative inconsistency in anyone who attempts to deny it. If, for example, I were to assert that there are no assertions, the very act of making this assertion would show it to be false: the performance of assertion is 'inconsistent' with the truth of the content asserted. (The scare quotes signal that this 'inconsistency' is not strictly logical since strictly logical inconsistency is a relation that holds between or among propositions. A speech act, however, is not a proposition, though its content is.)

Can a similar retorsive argument be mounted against Zeno's denials of motion and plurality?

The retorsive argument might proceed as follows. For Zeno to convey his Parmenidean thoughts to us he must wag his tongue and draw diagrams in the Eleatic sand. Does he thereby prove the actuality and thus the possibility of motion and fall into performative inconsistency? The answer depends on how we understand the purport of the Zenonian argumentation.

A. If Zeno's arguments are taken to show that motion *conceived in a certain way* does not exist, then the wagging of the tongue, etc. is 'consistent' with the proposition that motion conceived in that way does not exist, and the retorsive argument fails. For example, suppose one maintains that for a particle P to be in motion (relative to a reference-frame) is for P to occupy continuum-many different positions at continuum-many different times (relative to that reference-frame). This popular 'At-At' theory of motion requires the denseness of physical time and physical space. Now if it turns out that motion so conceived does not exist, it may still be the case that motion conceived in some other way does exist, and that Zeno's tongue-wagging and diagram-drawing is motion under that competing conception. The competing conception might, for example, deny the denseness of space or of time, or both.

B. If Zeno's arguments are taken to show that motion *no matter how it is conceived* does not exist, and is a mere illusion bare of all reality, then the retorsive argument refutes him. For then the moving of his tongue is 'inconsistent' with the truth of 'Nothing moves.' By moving his tongue, pulling his beard, flaring his nostrils, adjusting his toga, and pounding the lectern, he demonstrates the empirical reality of motion, a reality that is prior to, and neutral in respect of, all conceptions of this reality.

Although the retorsive argument works against a Zeno so interpreted, this interpretation is uncharitable in the extreme. Read charitably, Zeno is not claiming that motion and plurality are mere subjective illusions, but rather that they are something like Leibnizian well-founded phenomena (*phaenomena bene fundata*) or intersubjectively valid Kantian *Erscheinungen*. They are not mere illusions, but they are not ultimately real either.

C. If Zeno's arguments are taken to show that motion and plurality are intersubjectively valid appearances, but not ultimately real, then they are being taken to show that motion conceived in a certain way, as belonging to ultimate reality, does not exist. This view of motion seems 'consistent' with the motions required to convey Zenonian argument.

My verdict, then, is that retortion does not refute a Zeno charitably interpreted.

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.

A_______1/8_______1/4_______________1/2_________________________________ B

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:

. . . 1/16, 1/8, 1/4, 1/2, 1.

Since there is no first term, I can't get started.

Let us consider the 'calculus solution' to the paradox. Consider the infinite sequence

1/2, 1/4, 1/8, . . . ,1/2^{n}, . . . .

A sequence is said to be *convergent* if it has a limit. Roughly, the *limit* of a sequence is a number L such that the terms of the sequence become and remain arbitrarily close to L as we consider the terms in succession. Thus the limit of the sequence displayed above is 0, since as we run through the sequence, each term gets closer and closer to 0. The larger n becomes, the smaller is the difference between 1/2^{n} and 0. The sequence converges to 0 as its limit.

We can also speak of the convergence of an infinite *series*, where an infinite series is the result of adding all the terms in an infinite sequence. But how do we add the terms in an infinite sequence? What meaning can we attach to 'sum of an infinite sequence of terms'? In standard analysis (calculus), the sum of an infinite sequence is the limit of the sequence of partial sums:

1/2, 1/2 + 1/4 = 3/4, 1/2 + 1/4 + 1/8 = 7/8, . . . .

This sequence of partial sums converges to the limit 1. Thus the sum of a convergent infinite series is a number that can be approximated arbitrarily closely by adding up sufficiently many finite terms.

Now how is this supposed to solve the Regressive Dichotomy paradox? Since the sum of an infinite series has been defined as the limit of the sequence of partial sums, and since this sequence converges to a finite limit (1 in our example), it is intelligible how infinitely many terms in a convergent series can have a finite sum. Note, however, that this intelligibility is a matter of stipulative definition: it is just stipulated that 'sum' in the case of infinite series shall have the meaning 'limit of a convergent sequence.' And since this stipulation is intelligible, it is supposed also to be intelligible how a runner can traverse the infinity of points lying between A and B.

The rub, however, is in applying the mathematical formalism, which I cannot fault, to the physical world. Our convergent infinite series has a sum, but only in the sense that the sequence of partial sums converges to the limit 1. The series does not have a sum in the sense of some number that is actually computed by someone or some computer. But to get from A to B I must actually complete an infinite sequence of runs. There is an infinite number of acts that I must perform assuming that space is composed of points. It is not enough that I converge upon point B; I must actually arrive there. In the mathematical case, however, it is enough that the infinite series converges upon the limit 1. There is no need that one actually compute up to 1.

Although I haven't expressed this as clearly as I ought to, my sense is that the 'calculus solution,' although fine as mathematics, is no solution to the Dichotomy paradox. One can render intelligible how an infinite series of terms can have a finite sum. But this does not render intelligible how an infinite series of acts can be performed in a finite time.

REFERENCE: Wesley C. Salmon, *Space, Time, and Motion*, Chapter 2.