SORITES, ISSN 1135-1349
Issue #11. December 1999. Pp. 82-85.
Benardete's Paradox
Copyright © by SORITES and Michael B. Burke
Benardete's Paradox
by Michael B. Burke
Benardete gave, rather informally, several versions of the paradox. Priest has selected one of those versions and, with five postulates, formalized it. Although Priest has succeeded nicely in sharpening the paradox, the version he chose to formalize has distracting and potentially problematic features that are absent from some of Benardete's other versions. In particular, the selected version involves an infinitude of gods, intentions, and distinct acts performed in a finite time.Foot note 6_4 Suspicion is sure to fall on Priest's fifth postulate, which is the one needed to accommodate those complicating but dispensable features.
I propose to offer a Priestly formalization of a simpler version of the paradox, the one that presents most plainly Benardete's challenge to the spatial continuum. Proposed resolutions of Benardete's paradox should address this version of the paradox as well as the one formalized by Priest.
The version to be formalizedFoot note 6_5 may be stated informally as follows: Point -1 is one meter west of point 0, which is one meter west of point 1. The ground between -1 and 1 is smooth and level. A ball at -1 is rolling eastward with sufficient momentum to reach 1 and beyond, if nothing (other than friction) impedes it. But rising from the ground between 0 and 1 (as they have from all eternity) are infinitely many barriers. Specifically, there are barriers at points ½, ¼, þ, and so on. (The barriers are equal in height and width, but they differ in thickness. The barrier at ½ is one centimeter thick. Each of the other barriers is half as thick as the first barrier to its east.) Each barrier is strong enough to stop the rolling ball. (This does not seem problematic logically. But if it were, we could replace the ball with a massless particle, such as a photon.) Now here is the problem: It seems obvious that the ball cannot progress beyond point 0, since to do so it would have to get past an infinitude of barriers, none of which it is able to get past. But since there is no first barrier, the ball does not reach any barrier (since it can't get past the preceding barriers) and thus is not stopped by any barrier. But there's nothing to stop the ball other than a barrier. And it may be assumed, in accordance with Newton's first law, that the ball will not stop unless something stops it. Thus we arrive at a contradiction -- and a paradox.
In formalizing this version of the paradox, I will use as many of Priest's symbols and postulates as possible (so as to facilitate comparison of the two versions).
First, the symbolization key: x and y range over the set of spatial points belonging to the line segment containing -1 as its westernmost point and 1 as its easternmost point; Bx = there is (the western surface of) a barrier at x; Rx = the (foremost point of the) ball reaches x; Sx = the ball is stopped by the barrier at x (from ever going further than that barrier); x<y = x is west of y.
Four postulates are needed, none of which is an analogue of Priest's fifth postulate. The second and fourth are the same as two of Priest's (except that for Priest, Bx = a barrier is created at x while the moving object is west of x). Like Priest, I have suppressed universal quantifiers.
(1) Bx þ x þ ( ... þ, ¼, ½)(There are barriers at all and only these points: ... þ, ¼, ½.)
(2) (Rx & y<x) þ Ry(The ball reaches a point only if it reaches every point to its west.)
(3) Sx þ (Bx & Rx)(The ball is stopped by a barrier iff the ball reaches the barrier.)
(4) ¬þx(x<y & Sx) þ Ry(The ball reaches a point unless stopped by a barrier to its west.)
Let p be any point east of 0. Given 1, it follows that there is a barrier west of p. But then, given 2, the ball will reach p only if it reaches that barrier. Since, given 3, the ball will be stopped by that barrier if it does reach it, the ball will not reach p. But now consider any barrier b west of p. Given 1, it follows that there is a barrier b» west of b. But then, given 2, the ball will reach b only if it reaches b». Since, given 3, the ball will be stopped by b» if it does reach it, the ball will not reach b. Therefore, given 3, the ball is not stopped by b. It follows, by universal generalization on b, that the ball is not stopped by any barrier west of p. So, given 4, the ball does reach p. And we have reached a contradiction.
Which postulate might we reject? Neither (2) nor (3) seems a promising target. In the unlikely event that we should feel driven to deny the possibility of motion (as per Priest's suggestion), we would reject (4). (But neither (2) nor (3). If motion were impossible, the left side of (2), and both sides of (3), would be necessarily false [on every valuation of «x» and «y»]. In standard logic, that would assure the necessary truth of (2) and (3).) BenardeteFoot note 6_6 suggested an alternative basis for denying (4): that the ball might be stopped, not by any one barrier, but by the infinite sequence of barriers. The ball stops at point 0, despite having encountered no barriers, because it would otherwise have to overcome an infinitude of barriers, none of which it is able to overcome. But as Benardete soon acknowledged,Foot note 6_7 his suggestion doesn't suffice to resolve the paradox. It merely reiterates the proof that the ball will stop; it does not provide a dynamical explanation of its stopping. At least until further ideas are forthcoming, suspicion will fall on (1). And Benardete's paradox will stand as a substantial challenge to a presupposition of (1): the continuity of the spatial continuum.Foot note 6_8
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