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The Form of the Benardete Dichotomy 

Nicholas Shackel 
 

Abstract: Benardete presents a version of Zeno’s dichotomy in which 
an infinite sequence of gods each intends to raise a barrier iff a traveller 
reaches where they intend to raise their barrier. In this paper I exhibit the 
abstract form of the Benardete Dichotomy. I show that the diagnosis 
based on that form can do philosophical work not done by earlier papers 
rejecting Priest’s version of the Benardete Dichotomy, and that the 
diagnosis extends to a paradox not normally classified as a dichotomy. I 
show how the form is exploited to generate paradox.  

 
1 Introduction 
2 The form of the Benardete dichotomy 
3 Applying the unsatisfiable pair diagnosis 
4 Exploiting the form 

1 Introduction 
Benardete ([1964], pp. 259-60) presents a version of Zeno’s dichotomy in which an 

infinite sequence of gods each intends to raise a barrier iff a traveller reaches where 
they intend to raise their barrier. Priest presents a formalisation which replaces 
intentions of gods to raise barriers with a demon who mines the line in such a way 
that ‘passing certain spots brings barriers spontaneously into existence ([1999], p. 2). 
In refuting Priest, Yablo  presents an infinite set of demons each of whom will say 
‘YES if and only if all the earlier-calling demons have called NO’ ([2000], p. 150). 
Perez Laraudogoitia] ([2003]) offers a variant of Benardete’s dichotomy which he 
claims is not refuted by his own earlier ([2000]) refutation  of Priest. Hawthorne 
([2000]) and Angel ([2001]) exploit variants which appear not to be refuted by the 
earlier papers. There is a common contradictory form underlying all of these 
dichotomies. That form is also embedded in some other paradoxes, including Yablo’s 
([1993]) paradox without self reference. 

In the second section I show what the form of the Benardete Dichotomy is. In the 
third section I show that the diagnosis based on the form can do philosophical work 
not done by Yablo ([2000]) and Perez Laraudogoitia ([2000]). In the final section I 
show how the form is exploited to generate paradox in the Benardete Dichotomy. 

2 The form of the Benardete dichotomy 
For the purposes of this discussion, let x range over an infinite set S linearly ordered 

by a relation called ‘before’ (…<<…). Members of  S  may represent or be times or 
places on a line. Let such sets which have no first member be called unbegun sets, and 
the condition of having no first member be called the unbegun condition (∀x ∃y y << 
x).1 Let such sets which have a first member be called begun sets and let b be their 

                                                 
1 Examples of unbegun sets are the positive reals, ℝ+, the sets {1/2n } and {1/n} (i.e. the sets 
{x∈ℝ: x>0}, {x∈ℝ: x=1/2n  for n∈ℕ} and {x∈ℝ: x=1/n  for n∈ℕ}), under their usual 
ordering.  



 

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first member. Let ‘E’ be a predicate symbol, whose interpretation is intended to be a 
property of being instantiated at times and places, such as had by events, particles, 
barriers etc. being at times and places. We have a pair of related predicates: ‘E at …’ 
and ‘E before …’. In general, to be before is ambiguous between being somewhere 
before and being at all before, so we define ‘E somewhere before x’  to mean ∃y 
(y<<x & Ey), ‘E all before x’  to mean ∀y(y<<x → E y) and ‘E nowhere before x’ to 
mean  ¬∃y (y<<x & Ey). 

 
The Benardete dichotomies apply the seemingly innocuous condition that 

something happens at a place or time iff it does not happen anywhere before that 
place or time. We might call this the ‘At iff Nowhere Before condition’: 

Condition ANB:  For all x in S, E at x iff E nowhere before x. 
Condition ANB is not appealed to directly. I claim that the indirection is not innocent 
but, as we shall eventually see, guileful. Benardete says 

A man decides to walk one mile from A to B. A god waits in readiness 
to throw up a wall blocking the man’s further advance when the man has 
travelled ½ a mile. A second god (unknown to the first) waits in 
readiness to throw up a wall of his own blocking the man’s further 
advance when the man has travelled ¼ mile. A third god … etc. ad 
infinitum. (Benardete [1964], pp. 259-60). 

The set S is the unbegun set of places {1/2n }. Implicit is that the nth god doesn’t put 
up his wall unless the man reaches 1/2n . So  

1. for all x in S, the man reaches x iff a god puts up a wall at x.  
Also implicit is ‘that the man does not stop unless a barrier is put in his way’(Yablo 
[2000], p.148). So the man will reach x if there was no barrier before x i.e. no god put 
up a wall before x. Likewise, if the man reaches x there was no barrier before x i.e. no 
god put up a wall before x. So  

2. for all x in S, the man reaches x iff  a god puts up a wall nowhere before x.  
Putting 1 and 2 together we see we have condition ANB embedded in Benardete’s 
dichotomy: 

3. for all x in S, a god puts up a wall at x iff a god puts up a wall nowhere 
before x. 

 
Consider Priest’s premisses for his version of the dichotomy: 

(2) Bx & x < y → ~Ry 2 

(3) ∼∃x ( x < y & Bx) → Ry 

(4) x ≤ 0 → ~Bx 

(5) x > 0 → (Bx ↔ Rx/2) (Priest [1999], p.2) 
Sentence (5) says that there is a barrier at x iff you reach x/2. The effect of sentences 
(2) and (3) is just that you will reach y iff there is no barrier before y.3 Put them 

                                                 
2 I’ve swapped variables from the original to make it easier to follow. 
3 The antecedent of (2) is implicitly existentially quantified, ∃x (Bx & x < y) → ~Ry, so by 
contraposition, Ry → ∼∃x (x < y & Bx), which with (3) gives Ry ↔ ∼∃x (x < y & Bx).  



 

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together, and you get a barrier at x iff there is no barrier before x/2. Perez 
Laraudogoitia derives the same result formally from the same three sentences in his 
derivation of his sentence (10): 

(10) ∀x (x>0 → (Bx ↔ ∼∃y (y < x/2 & By))) (Perez Laraudogoitia 
[2000], p. 154) 

Nothing turns on the minor modification incumbent on the distinction between 
‘before x’ and ‘before x/2’ just because the x is ranging over an unbegun set in which 
x/2 is always before x. Priest could just as well have used ‘x’ instead of ‘x/2’ in (5).  
The set S is the unbegun set, ℝ+, and condition ANB, for all x in S, there is a barrier at 
x iff there is a barrier nowhere before x. So Priest’s sentences (2) (3) and (5) 
effectively disguise an example of condition ANB. 
 

Likewise, we can see that Yablo’s demons, in attempting to say ‘YES if and only if 
all the earlier-calling demons has called NO’ (Yablo [2000], p.150), are attempting to 
conform to condition ANB. The set S is the unbegun set of times {1/2n }. All earlier 
calling demons calling No is logically equivalent to Yes called nowhere before. So we 
have condition ANB, for all t in S, a demon calls YES at time t iff a demon calls YES 
at no time before t.  

 
Condition ANB has a surprising consequence for unbegun sets. Let S be an 

unbegun set, and y be any member of S before x (y<<x). Then for all x in S and for all 
y before x 

1) E at x  iff E nowhere before x (condition ANB)  
which is equivalent to 

2) not E at x iff E somewhere before x   
By substitution in 2) we get 

3) not E at y iff E somewhere before y   
Since y ranges over all members of S which are before x we have  

4) E nowhere before x iff  not E at y   
so from the chain of biconditionals (1), (4) and (3) (in that order) we have 

5) E at x iff  E somewhere before y 
 
Clearly, since y is before x, 

6) E somewhere before y  only if E somewhere before x 
so from (5) and (6) 

7) E at x only if E somewhere before x. 4 
hence from (7) and (2) 

8) E at x only if not E at x 
whence  

9) not E at x   
 

                                                 
4 Comparing lines 1) and 7) shows a contradiction to be lurking. 



 

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So the seemingly innocuous condition ANB, which apparently permits E at x under an 
acceptable bicondition, imposes the impossibility of E at x when x ranges over an 
unbegun set.  
 

A few additional steps gives us an explicit contradiction.  
10) E somewhere before x     2,9 
11) E somewhere before x  iff ∃z before x and E at z 3 
12) E at a    10, 11, unbegun condition, EI 
13) not E at a   9 
14) E at a and  Not E at a 2, 13 
 

So the application of condition ANB to an unbegun set is contradictory.  
 
Condition ANB is satisfiable by a begun set, and forces E at b (b is defined to be 

the first member), for consider the RHS of condition ANB with b substituted for x: E 
nowhere before b. In full, this reads ‘∀ y y << b → ¬Ey’. Since b is the first member 
of a begun set y << b is false for all y and therefore ‘∀ y y << b → ¬Ey’ is true of 
begun sets.5 Hence by condition ANB, E at b. Whence  by condition ANB, not E at x 
for all x after b. 6 

The path to the contradiction is blocked for begun sets. The step to line (3) would 
be invalid because of the restriction on y being a member of S before x. Only unbegun 
sets promise such a member for all x in S. The detail of (3)’s  derivation goes  

2) not E at x   iff   E somewhere before x    
Now since (2) applies to all x in S, if there is a y before x, it will apply to that y. So we 
have  

2b) if there is a y before x then (not E at y iff  E somewhere before y)
  

 From which we get, by the unbegun condition, ∀ x ∃ y y << x, and detachment 
3) not E at y iff E somewhere before y 

So although (2) and (2b) are true of begun sets, we cannot get (3) for begun sets 
because we need the unbegun condition for the detachment. Obviously (3) is false for 
the first member of begun sets, b, since it says not E at b iff there exists a z before b 
such that E at z, but since b is before all other members of S there can be no such z.  

The step to line (12) would also be invalid because it uses the unbegun condition. 

                                                 
5 Whereas for unbegun sets, because for all x in S there exists a y such that y<<x, there is no c 
in S such that for all x, x << c is false, and therefore the RHS of B need not be true for any 
member of S. 
6 Robert Black has pointed out to me that this fact can be shown very shortly as follows: Let 
R be any connected relation on S. Then if condition ANB holds for some property E it can be 
shown to hold for only one member of S and that member will bear R to every other member 
of S. Whence if the connected relation is an ordering, thus transitive, the unique E must be the 
first member, so the set S must be begun.  



 

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The unsatisfiable pair diagnosis 
So now we have the form of the Benardete dichotomy: Embedded in it is a pair of 

conditions which are jointly unsatisfiable: 
1. Condition ANB: For all x in S, E at x iff E nowhere before x. 
2. The set S in condition ANB is an unbegun set. 

Call this the unsatisfiable pair diagnosis. Benardete predicates the execution of 
intentions on the non-execution of prior intentions  where there is no first time or 
place for action or inaction. Priest predicates the existence of barriers on the absence 
of prior barriers with no first place for a barrier. Yablo predicates demons saying yes 
on no previous demons saying yes with no first time for a demon to speak. Each is a 
matter of applying condition ANB where E respectively is the execution of prior 
intentions, the existence of a barrier, the demon speaking yes, whilst x respectively 
ranges over an unbegun set of places, places, and times. Therefore, by so predicating 
they did not allow any such executions, barriers or speaking to happen under a 
condition, but made them contradictory a priori.  

3 Applying the unsatisfiable pair diagnosis 
Perez Laraudogoitia and Yablo both offer diagnoses of the Benardete dichotomy 

which focus on Priest’s version. Perez Laraudogoitia ([2000]) derives a formal 
contradiction from Priest’s premisses. Yablo discusses the coherence of the infinitude 
of intentions of Priest’s demons who ‘“mine” an area of space in accordance with 
[premisses] (4) and (5)’ (Priest [1999], p. 2) by discussing the coherence of an 
analogous infinitude of demons, and concludes that ‘the gauntlet of demons….is 
saved from incoherence only by the assumption that α [the travelling particle in 
Priest’s version] stops at zero’ ([2000], p. 150).  

The unsatisfiable pair diagnosis is compatible with the latter diagnoses, but is more 
general and for that reason can do more philosophical work. First, it applies to a 
variant of Benardete’s dichotomy proposed by Perez Laraudogoitia which he says is 
not resolved by either his earlier paper or by Yablo’s diagnosis  (Perez Laraudogoitia 
[2003], pp. 130-1). Second, Hawthorne ([2000]) and Angel ([2001]) use variants of 
Benardete’s and Zeno’s dichotomies to motivate the rejection of certain plausible 
principles. In neither case is it evident that Perez Laraudogoitia and Yablo’s earlier 
papers can show that those arguments fail, let alone explain why they do. The 
unsatisfiable pair diagnosis, however, illuminates Hawthorne’s and Angel’s uses of 
dichotomies and thereby undermines what they take themselves to have achieved. 
Thirdly, the unsatisfiable pair diagnosis applies to a paradox not normally classified 
with the dichotomies, namely, Yablo’s  paradox without self-reference ([1993]). 
Finally, it lays bare a general mechanism for constructing these paradoxes. 7 

Perez Laraudogoitia 
 Perez Laraudogoitia claims his new variants are not refuted by his or Yablo’s 

earlier paper partly because 

                                                 
7 There may be an argument to be had turning on whether when a paradox uses ‘for all natural 
numbers…’ we are being offered a set of sentences in a formal language specified in a 
metalanguage, and hence on whether we need to address the relation of ω-inconsistency to 
inconsistency and determine whether implicit appeals are being made to ω-rule inferences. 
For the sake of space, I shall not be addressing those arguments. 



 

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from the fact that the particle can only interact with the barriers, and that 
therefore only the barriers can stop it, it does not follow that if it does 
not interact with any barrier it will not stop. This latter is in fact what 
Priest (1999, p. 2 axiom (3)) explicitly assumed, trivially turning 
Benardete’s paradox into a contradiction. …What this paradox shows is 
that the particle can interact with the set of barriers…without interacting 
with any one barrier…[failing to] recognise this fact…commits a 
division fallacy. (Perez Laraudogoitia [2003], p. 126) 

This point makes sense when there are barriers in place, but is more difficult to 
understand if the barriers are non-existent. In Benardete’s dichotomy the man is 
stopped even though no god creates any barrier. Perez Laraudogoitia’s remark seems 
to imply the man is stopped by interacting with the set of non-existent barriers and 
that to derive the contradiction requires committing the division fallacy of thinking 
that in order to interact with a set of non-existent barriers one must interact with one 
of the non-existent barriers. But that doesn’t seem to be the problem at all. Prima 
facie, one would have thought the problem is being stopped by non-existent barriers at 
all, not by how many non-existent barriers get in on the act. Nevertheless, if we grant 
Perez Laraudogoitia his point with respect to non-existent barriers then his claim that 
it is not resolved by his or Yablo’s earlier paper is plausible. I shall now show that the 
condition placed on the gods in Perez Laraudogoitia’s new variants  is in each case 
condition ANB and that the set S in each case is unbegun.  

I will construct a temporal version of the paradox of the gods. Now the 
particle is moving with constant velocity v for t<0. A god decides to 
stop it at t = ½, wherever it is, if it is not at rest already at that instant. A 
second god (unknown to the first) decides to stop it at t = 1/3, wherever 
it is, if it is not at rest already at that instant. A third god… and so on ad 
infinitum. (Perez Laraudogoitia [2003], p. 124) 

Here the set S is the set of times {t = 1/n}. Condition ANB is that for all t in S, a god 
stops the particle at time t iff no god stop the particle before t. The L to R direction 
holds because if a god stops the particle at t it must have been in motion prior to t and 
so no gods stopped it before t. The R to L direction holds because unless the gods stop 
it before t it will not be at rest, so a god will stop it at t. These remarks hold for his 
other variants. Perez Laraudogoitia’s next variant avoids the problem of ‘infinite force 
at one instant’ ([2003], p. 127). 

In general, God-n decides to act as follows: wherever the particle is at t 
= 1/(n+1), he will take it to rest, if it is not at rest already, in the time 
interval [1/(n+1), 1/n]. (Perez Laraudogoitia [2003], p. 127) 

Here the set S is the set {ℑ ⊆ℝ: ℑ= [1/(n+1), 1/n]}. Condition ANB is that for all 
intervals of time ℑ in S, a god will stop the particle during interval ℑ iff a god stops 
the particle during no interval before ℑ. Perez Laraudogoitia’s final variant is 
essentially the same as the latter, only allowing that perhaps some physical processes 
were instant (‘class C– processes’), whilst confining the gods to using processes which 
require intervals of time (‘class C processes’). 

let God-n act like this: wherever the particle is at t = 1/(n+1), it will take 
it to rest by some class C process in the time interval [1/(n+1), 1/n], if it 



 

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is not at rest [already], or if it is at rest but got there by some class C– 
process. (Perez Laraudogoitia [2003], p. 128) 

Here the set S is the set {ℑ ⊆ℝ: ℑ= [1/(n+1), 1/n]}. Condition ANB is that for all 
intervals of time ℑ in S, a god will stop the particle during interval ℑ by a class C 
process iff a god stops the particle during no interval before ℑ by a class C process. 

Hawthorne 
We turn to a paradox for which the division fallacy point is more clearly true. I place 
impenetrable barriers of appropriately diminishing thicknesses at each of {1/n};8 a 
sphere travelling from the left couldn’t pass 0 without passing through infinitely many 
barriers, yet if it stopped at 0 it would do so without being stopped by any particular 
barrier. This sounds odd, but is not contradictory without the assumption Perez 
Laraudogoitia criticised. If we think otherwise, it is because we think that what 
Hawthorne calls the contact principle is generally true: 

The contact principle: If y is the fusion of x’s and z contacts y, then z 
contacts one of the x’s. ([2000], p. 626) 

To think that the contact principle applies to infinitely many barriers is to commit the 
division fallacy: one can be stopped by a fusion of infinitely many barriers without 
being stopped by any single barrier. Hawthorne claims that this diagnosis (rejecting 
the contact principle for infinite fusions) resolves the two dichotomies Benardete 
gives before his dichotomy of the gods.  

Let the peal of a gong be heard in the last half of a minute, a second peal 
in the preceding 1/4 minute… etc. ad infinitum. … Let us assume that 
each peal is so very loud that, upon hearing it, anyone is struck deaf—
totally and permanently. At the end of the minute we shall be 
completely deaf (any one peal being sufficient), but we shall not have 
heard a single peal! .… We are now tempted to coin the barbarous 
neologism of a before effect. 

A man is shot through the heart during the last half of a minute by A. B 
shoots him through the heart during the preceding 1/4 minute …etc. ad 
infinitum. Assuming that each shot kills instantly (if the man were 
alive), the man must be already dead before each shot. (Benardete 
[1964], p. 255-59) 

 Hawthorne says that what deafens is the fusion of sound waves, what kills is the 
fusion of bullets, and that only our adherence to the contact principle leads us to find 
these before effects contradictory. I shall offer a shooting case which Hawthorne’s 
rejection of the contact principle does not resolve, and then show that the unsatisfiable 
pair diagnosis applies to it. 

The plausibility in the case of the fusion of walls rests on the topology of regions of 
space and what should be counted as contact when objects may occupy open, half 
open, or closed regions.9 A definition of contact which will cover all the possible 
combinations Hawthorne considers on page 626 is this: contact occurs iff the 

                                                 
8 Thicknesses of 1/2n would do. 
9 I make use of the standard topological definitions. See, for example, Sutherland [1998]. 



 

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boundaries of the regions occupied by the objects share at least one point.10 Now the 
fusion of walls occupies a region open to the left and its boundary on the left is a 
vertical plane through the origin, O,11 so there is a perfectly good sense in which the 
sphere (whatever sort of region of space it occupies) contacts the fusion when its 
boundary shares the point O with the left hand boundary of the fusion.12   

It is not obvious that there is, and Hawthorne does not provide, a similar spelling 
out of what is happening in the cases of the gong or the shooting. It may be possible 
to do so in the gong case, provided the gong stays still, otherwise it might fall to the 
objection I am about to make about the shooting case.  

In the shooting case, perhaps what Hawthorne has in mind is all the bullets 
travelling the same path to the man’s heart, so that the fusion of bullets occupies a 
region of space with similar properties to that occupied by the region of barriers. 
Analogously to the boundary of the fusion of walls approaching the sphere (if we set 
our inertial frame on the sphere), we have a picture of the boundary of the fusion of 
bullets travelling so as to contact the victim’s heart just as each bullet would, only 
contacting it at time zero and so killing him before any actual bullet does.13  Hence 
the before effect of the fusion of shooters.  

Grant for now the plausibility of that account. It will not suffice. Consider instead 
the infinitude of shooters distributed around the victim so that the last shooter shoots 
from the north and the bearing from north of the shooter who shoots during the time 
interval [1/2n+1, 1/2n] is one radian greater than the shooter who shoots during [1/2n, 
1/2n –1]. Since there are 2π radians in a circle and π is an irrational number there is no 
common path for the bullets.14 I can stipulate that the bullet sizes decrease 
appropriately to avoid overlapping paths at the surface of the heart.15 When we now 
enquire about the boundary of the fusion of these bullets, and in particular about that 
part of the  boundary which is going to do the killing, the analogy collapses. For in the 
wall case the vertical plane running through 0 is a boundary of the fusion because (to 

                                                 
10 The boundary of a region R in a topological space T  is Cl(R)∩Cl(T –R). Cl(R) is the 
closure of R, which is the union of R with all its limit points. Objects are impenetrable so we 
don’t need to have a definition which covers engulfment. 
11 The boundary is part of the y-z plane. I take the x-axis to be horizontal and y-axis to be 
vertical. I won’t burden the reader with three dimensional coordinates. 
12 An alternative not considered by Hawthorne and avoiding the use of fusions is to use 
impenetrability. We can define penetration as follows: 

Object A penetrates D iff there exists a point x on the boundary of the space 
occupied by A for which there exists an open ball B(x) such that B(x) is 
entirely contained in the region of space occupied by D. 

Then rather than having to appeal to contact to explain the power of a barrier to stop an 
object, we appeal to impenetrability. The ball stops at zero because penetration is impossible. 
If it passes zero there will be at least one wall (in fact infinitely many) which it penetrates, but 
penetration is impossible. 
13 Hawthorne supposes ‘if a metal object penetrates ¼ inch into the heart, then the person dies 
at that very moment’ ([2000], p. 627) but we can just as well suppose death is instantaneous 
on contact. 
14 Because for two bullets to share a path would require two shooters to have the same 
bearing, which would required there to be k, m ∈ℕ for which 2kπ = m. But that contradicts 
the irrationality of π. 
15 Or be point sized, if necessary. 



 

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put it very loosely) the walls ‘tend to that plane in the limit’. But in the bullet case 
there is no such ‘limiting’ boundary to hit the heart at time zero. By construction, I 
have eliminated that ‘limiting’ boundary and so the boundary of the fusion is no more 
than the union of all the boundaries of all the individual bullets. None of those 
boundaries contacts the heart at time zero, so the boundary of the fusion can’t do the 
killing at time zero. In the wall case the paradox of the before effect is accounted for 
by rejecting the contact principle and accepting that the sphere contacts the fusion 
without contacting any individual wall by contacting the boundary of the fusion at 0. 
That explanation fails in the bullet case. In the bullet case we can reject the contact 
principle and accept that the heart is contacted if contacted by the boundary of the 
fusion of bullets even if contacted by no individual bullet. But the boundary of the 
fusion of bullets is not in contact with the heart at time zero. Yet there is no time after 
time zero at which he can be alive. So the paradox remains unexplained by a rejection 
of the contact principle.   

Now I have had to speak somewhat loosely in order to keep this objection both 
comprehensible and short. I think it is evidently provable, but it is a non-trivial 
mathematical task to prove it. At the very least, I have made it evident that 
Hawthorne’s picture leaves it quite unclear whether it generalises to the shooting case. 
If it doesn’t then the diagnosis by falsity of the contact principle does not apply in that 
case.  

My diagnosis, however, does apply. In the shooting case, the set S is the set of time 
intervals {ℑ ⊆ℝ: ℑ= [1/2n+1, 1/2n], n ∈ℕ}, which set is unbegun, and condition ANB 
is that for all ℑ in S, a man kills the victim during ℑ iff a man kills the victim during 
no interval before ℑ.  

Hawthorne concedes that falsity of the contact principle does not solve the 
Benardete dichotomy because ‘after all, in that case there is not a wall thrown down 
and so there is no fusion of walls with which to come into contact’ ([2000], p. 627). 
Hawthorne would therefore seem to agree with me that Perez Laraudogoitia’s remarks 
about the division fallacy fall short of the mark. In place of Benardete’s dichotomy, 
Hawthorne considers a variant in which infinitely many assassins are each disposed to 
kill Bob during a time interval should he be alive at the beginning of the time interval 
(see [2000], pp. 627-30). Hawthorne concludes 

The puzzlement resides in the fact that we think of the assassins as 
individually having to do something...yet... the fusion [of assassins] 
causally secures the assassination of Bob without even moving! Nor 
does the fusion need to undergo any other type of change at all in order 
to assassinate Bob. Our puzzlement thus relies, I suggest, on our tacit 
endorsement of the following principle relating fusions to their parts, 
which we can call the “Change Principle” 

If x is the fusion of y’s and y’s are individually capable only of 
producing effect e by undergoing change, then x  cannot, (without the 
addition of some non-supervening causal power), produce effect c 
without undergoing change. 

This principle is mistaken, which I suggest is a big metaphysical 
surprise. By suitably combining things that need to change in order to 



 

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produce a result, we can generate a fusion that can produce that result 
without undergoing change…. 

The Contact Principle, in full generality, could be given up fairly readily 
on reflection. The Change Principle has a rather deeper hold on us. 
([2000], p. 630) 

I think the relevant analogous principles summarised by the Change Principle are 
evident; for example, in Benardete’s dichotomy it would be more naturally expressed 
in terms of heavenly agents and their actions. We could also omit the fusion and 
express this in terms amenable to Perez Laraudogoitia’s criticism of division fallacies.   

I think Hawthorne’s position is incoherent. It is worth pointing out a difference 
between the case of the Contact Principle and that of the Change Principle. In the 
former case, we have an account of contact to replace the contact principle, namely 
that one can contact a fusion by contacting its boundary. The plausibility of this 
account rests on an independent theoretical concern: the necessity for an account of 
contact between open objects (objects occupying open regions of space), which can 
only happen if two objects can be in contact by sharing a boundary point, even though 
neither occupies its boundary. In the case of the Change Principle, the main reason for 
rejection would seem to be avoiding the puzzlement of Benardete’s dichotomy. I 
don’t see, as I do in the case of the Contact Principle, independent theoretical 
concerns which when applied make sense of why the principle should be given up. On 
the contrary, I would think our independent theoretical concerns give substantial 
support to the Change Principle. When Hawthorne says 

getting together enough of the mundane things and suitably arranging 
them is all by itself logically sufficient to entail changeless causation 
[the absence of change is in the thing doing the causing] ([2000], p. 
630). 

to me he has offered a reductio. I suspect that changeless causation is an incoherent 
notion. If they don’t have to undergo any change to collectively kill Bob need they 
even exist in order to kill him? I can arrange for each assassin to be killed at the time 
he would detect and kill Bob iff Bob is already dead at that time. If we grant that the 
Change principle is false and that consequently the mere arrangement of assassins 
must be allowed to be ‘by itself logically sufficient’, Bob cannot be alive at any time 
after 1.00p.m.16 and so every assassin who would kill Bob has gone out of existence 
before he would kill him. We would now seem to have to reject yet another principle: 

If x is a fusion of y’s, and each y is disposed to cause an effect but ceases 
to exist before it causes that effect, x does not cause that effect. 

Rejecting that principle is pretty wild, since to do so seems to be a matter of accepting 
causeless causation. 

Hawthorne carefully avoids stipulating ‘that the laws of nature are such that Bob 
will survive unless he is [killed] by an assassin’  because ‘There is no possible world 
satisfying that description and so the question as to what happens in such a world is 
illegitimate’ ([2000], p. 630). I agree. But he does not avoid impossibility so simply. 
His survival is supposed to be analogous to the continued motion of the man in 
Benardete’s dichotomy. If we are to permit uncaused deaths for Bob we could just as 
                                                 
16 His state at 1.00p.m. is undetermined.  



 

 11

well permit uncaused stops for the man in Benardete’s dichotomy, and the whole 
problem can be forgotten. Hawthorne doesn’t say Bob’s death is uncaused but that he 
is killed by the fusion of assassins. He is committed to this on pain of losing his 
challenge to the Change Principle. So Hawthorne must concede that Bob will survive 
unless he is killed by something. Given only that, we can find the unsatisfiable pair in 
his set up without this stipulation, and that, I think, is the source of the incoherence 
here.  

Hawthorne reconstrues his assassins and sets up the situation in a complicated 
manner, which I will lay out and then simplify in order to locate the unsatisfiable pair.  

Make the assassins A-type point particles. Make Bob a B-type point 
particle….Assassination is the transformation of a B particle into an A 
particle. This occurs by an A particle interpenetrating a B particle at t 
and at that time irradiating x radiation—this being what constitutes 
“attack”. “Survival” here and in what follows is “remaining as a B-
particle”. The causal laws and state of the world are such as to make the 
following three claims true: 

 Particle 1 will not move with respect to the point occupied by Bob 
before 2 p.m. 

If at 2 PM Bob still exists as a B particle, then by 2.30, Bob will have 
been assassinated by particle 1. 

If at 2 PM Bob doesn’t exist as a B particle, particle 1 will not move 
with respect to the point occupied by Bob between 2 PM and 2:30 PM. 
…and so on.  

… If Bob changes into a B-particle he will never change back into an A 
particle …each natural number is assigned to one assassin (Hawthorne 
[2000], pp. 628-9) 

The sequence of time intervals he uses is disjoint, for the first assassin, 2 p.m. to 2.30 
p.m., for the second 1.30 p.m. to 1.45 p.m., and so on. Call 2 p.m. to 2.30 p.m. the 
first time interval, 1.30 p.m. to 1.45 p.m. the second time interval, and so on. Our 
unbegun set is the ordinal ω, with the ‘before’ relation being the reverse of the well 
order on ω (so 6th is before 5th, 5th before 4th, etc.).17 In general, the first conditional 
can be expressed as 

1. if Bob is alive in the nth time interval he is killed by the nth Assassin,  
and the second conditional as  

2. if Bob is not alive in the nth time interval he is not killed by the nth Assassin 
which together give us 

3. Bob is killed by the nth Assassin iff he is alive in the nth time interval. 
We have  

4. Bob is killed by the nth Assassin iff he is killed in the nth time interval.  
Hawthorne will perhaps object to the R to L direction, but this we get because if Bob 
is killed something kills him. In the nth time interval we can’t appeal to a fusion of 

                                                 
17 Strictly, if < is the well order on ω and k, m ∈ω, then k is before m iff m<k. 



 

 12

assassins to do the killing so it’ll have to be the nth Assassin on pain of returning to 
uncaused events. 3 and 4 give us 

5. Bob is alive in the nth time interval iff he is killed by the nth Assassin. 
Because once dead Bob stays dead we have 

6. Bob is alive in the nth time interval iff killed nowhere before the nth time 
interval 

When 5 and 6 give us condition ANB 
for all n in S, Bob is killed in the nth time interval iff he is killed 
nowhere before the nth time interval. 

So whilst Hawthorne seems to put forward a coherent possibility, the causal 
premiss that he needs for the sake of the point he wishes to make about the Change 
Principle proves to be his undoing. It is no help to him to restore coherence by 
abandoning that causal premiss. Abandoning it amounts to nothing more than 
specifying a set of sentences which can be satisfied by a possible world in which an 
infinitude of the A particles sit where they sit, and a B particle spontaneously changes 
into an A particle at 1.00 p.m. Nothing wrong with that, but no interest in it either, and 
certainly no grounds on which to claim that the infinitude of A particles caused the 
change. 

Angel 
Angel offers ‘A Physical Model of Zeno’s Dichotomy’ within ‘Newtonian 

collision mechanics’([2001], p. 347) . We take the x direction to be horizontal, the y 
direction vertical, and ignore the z direction. A spherical particle M of radius 1 is 
travelling from left to right with velocity (1,0). At t0 its centre is at (0,-1). A pair of 
countable infinities of particles of diminishing radii are arranged so that at time t0 
their centres are at (-1/2n , 1) and (1/2n , 1), for all n ∈ℕ, each with velocity (1, -2n ). 
([2001], p. 350 section 3 and figure 1 on p. 352).  

The horizontal velocities of M and the other particles are identical, so we can treat 
the situation in terms of an inertial frame with velocity (1,0) relative to the original 
inertial frame. In that frame M is (initially, at least) stationary and the countable 
infinities of particles ‘rain’ down vertically. The radii of the countable infinities of 
particles have been chosen so that for all n∈ℕ, at time t = 1/2n  a pair of particles 
centred at (-1/2n , 0) and (1/2n , 0) with velocity (0, -2n ) would contact M if M’s 
velocity remains unchanged prior to that time. ([2001], sentence crossing pp. 350-1). 
The particles all have the same mass.  

Use of conservation of momentum and energy laws suffices to show that 
… the particle pairs can never catch up with M on the assumption that M 
has had a collision with at least one higher indexed pair, which is 
required by our assumption being tested, namely that there is no earliest 
collision. (Angel [2001], p. 355) 

The unbegun set, S, is the set of times { t = 1/2n}. The condition ANB is for all t in S, 
a pair of particles will collide with M at time t iff a pair of particles collide with M at 
no time before time t. L to R holds because if it is hit at time t it must have retained its 
initial velocity so not been hit before, and R to L holds because if it is not hit before, 
its initial velocity is unchanged so it is at the place to be hit by the pair of particles set 
up to hit it at that place at that time. 



 

 13

Angel claims that his variant can be resolved by rejecting ‘Principle P, the 
composition of contact interactions does not create a noncontact interaction’([2001], 
p. 349) (or by ruling out infinitudes of particles or placing upper bounds on velocities, 
which solutions are no of interest to us here).  

Rejecting Principle P would avoid my derivation of the R to L direction of ANB. 
However, I can re-write using ‘interacted with’ instead of ‘hit’, where interactions 
include non-contact interactions of this special sort. Now L to R of ANB holds 
because if particles interact with M at time t it must have remained with its initial 
velocity so not been interacted with before, and R to L holds because if it is not 
interacted with before, its initial velocity is unchanged so it is at the place to be 
interacted with by particles. Whence condition ANB, for all t in S, particles interact 
with M at time t iff particles interact with M at no time before t. Consequently we still 
have something contradictory when applied to S (because S is unbegun).  

Angel claims that  
even for some ontologically conservative physics at least as far as 
particle types and the Euclidean spacetime they inhabit are concerned, 
we must think of impact as a spatiotemporal phenomenon in which 
impact between M and a set of particles…may occur at t0 despite there 
being no spatial contact between M and the spatial limit of any subset of 
the…particles.18 ‘Contact’ in a deeper sense is not simply a spatial 
concept. ([2001], p. 357) 

Angel could have made use of Hawthorne’s rejection of the contact principle by 
appealing to the fusion of his particles, whose boundary includes the point (0, 0) at t = 
0 which contacts M. He doesn’t want to, however, because that would undermine his 
desired rejection of Principle P. For that rejection, he needs it to be true that ‘at t0 
there is [no] spatial contact between M and the spatial limit of any subset of 
the…particles’. But this is simply incorrect, and contact with the fusion of particles 
allows us to retain principle P.  

The notion of ‘contact in a deeper sense’ seems to amount to the claim that 
rejecting principle P introduces non-collision causation into collision mechanics. 
Principle P is related to Hawthorne’s Change Principle. Rejecting P and asserting the 
existence of non-contact causal interactions would entail the falsity of the Change 
Principle, since we would have a case of changeless causation in which no change 
occurs to the set of raining particles but they cause a change in M. Rejecting P is 
therefore objectionable for similar reasons to those I gave when discussing the 
Change Principle.  

In claiming that rejection of Principle P avoids the paradox Angel simply helps 
himself to the assumption that a non-contact interaction happens at a convenient time 
to avoid the contradiction (at t = 0). Now of course, we have already proved that 
applying ANB to a begun set such as S∪{0} forces interaction at the first member. 
But the question is, what motivates adding zero to S. We are not being offered well 
motivated constraints for the time and place of these non-contact interactions. As 
presented, the convenience is ad hoc.  

Consequently rejecting principle P leaves it undetermined what happens at time t = 
0. There is an unbounded region in which M cannot be found, but where it can be 

                                                 
18 See footnote 15 again. 



 

 14

found, in other words, what the effect of the non-standard impact of S on M is, is 
undetermined. Indeterminism is not new in Newtonian mechanics,19 and what to do 
about it is much discussed. Earman ([1986], pp. 37-9) explains why he rejects some 
ways of avoiding it, and part of the significance of Perez Laraudogoitia’s Beautiful 
Supertask ([1996]) is that it withstands many plausible evasions of indeterminacy. But 
in those cases which restrict themselves to collision mechanics without gravitation, 
the indeterminacy that arises is very different from that in Angel’s case. The 
indeterminacy arises because of collisions, sometimes with time reversal, and 
whatever causation there is remains a matter of collision.  

So rejecting P does not amount to showing that it is possible within Newtonian 
collision mechanics for the raining particles to cause M to escape to infinity without 
contacting M. It is, rather, a matter of renouncing causation in Newtonian collision 
mechanics and giving a set of sentences which can be satisfied by a possible world in 
which an infinitude of raining particles is accompanied by a particle which 
disappears, or in some other way instantaneously gets out of the way, at time zero. 
Angel is mistaking a logical constraint of satisfiability for a causal constraint. For 
these reasons I reject Angel’s claim that ‘we have a physical model for Zeno’s 
dichotomy’ (Angel [2001], p. 356). We have a model, but rejecting principle P means 
it does not conform to Newtonian collision mechanics and is not a physical model of 
Zeno’s dichotomy in any interesting sense of physical.  

Yablo and Sorensen 

Yablo’s paradox ([1993]) offers the infinite sequence of sentences, for n ∈ℕ 
Sn : For all k > n, Sk  is not true. 

Here the unbegun set S is ℕ, on which once again we take the ‘before’ relation  to be 
the normal ‘greater than’ relation. Condition ANB is only implicit in the paradox, as a 
condition on the truth of the sentences. Let a sentence being true at n be that Sn is true. 
If a sentence is true at n, then since Sn states that all sentence before it are not true, a 
sentence is true nowhere before n. Likewise, if a sentence is true nowhere before n, 
then for all k before n, Sk  is not true, and so Sn, the sentence at n, is true. Thus 
condition ANB is that for all n in ℕ, a sentence is true at n  iff a sentence is true 
nowhere before n.   

In Sorensen’s version of Yablo’s paradox we have an infinite queue of students 
each of whom says ‘Some of the students behind me are now thinking an untruth.’ 
(Sorensen [1998], p. 137). The set S is the queue, which is unbegun when we take 
‘behind’ to be our ‘before’ relation. For convenience we can index the students in the 
queue with the natural numbers (with our now customary reverse ordering). Condition 
ANB is found in the condition on the students’ thoughts being untrue. We take E at x 
to be the student at x in the queue thinking an untruth. Plainly if the nth student 
thinking ‘someone behind me is thinking an untruth’ is thinking an untruth, then no 
one behind them is thinking an untruth (∀n  (En →¬∃k (k<<n & Ek)). Likewise, if no-
one behind the nth student is thinking an untruth then the nth student  is thinking an 
untruth (∀n  (¬∃k (k<<n & Ek)→Fn)). So we have condition ANB: for all n in ℕ, a 

                                                 
19 See, for example, the discussion between Perez Laraudogoitia [1997] and Earman, J  and 
Norton [1998]. 



 

 15

student is thinking an untruth at n iff a student is thinking an untruth nowhere before 
n.  

4 Exploiting the form  
So in each of the variants of Benardete’s dichotomy, and also in Yablo’s paradox, 

we find the jointly unsatisfiable pair to be embedded. Why is this a solution? After all, 
in each case the initial conditions appear acceptable. I think the answer is that the 
appearance is something that has to be carefully managed, and since that management 
amounts to concealing the unsatisfiable pair, in presenting the dichotomies no 
possibility has been specified.  

The mechanism of the dichotomy is to distribute the unsatisfiable pair over three 
inconsistent conditions, any pair of which is consistent: 

1. An ordering on that set which makes it unbegun 
2. A condition on an infinite set which is either 

a)  L to R of Condition ANB or  
b) A biconditional consisting of the LHS of Condition ANB and a 

RHS , Φ.  
3. A condition on an infinite set which is either 

a)  R to L of Condition ANB or  
b) A biconditional consisting of the RHS of Condition ANB and a 

LHS , Ψ.  
The subtlety of the paradoxes is that either 2 or 3 is only implicit in the situation as 
described, and in the (b) cases, that Φ↔Ψ is also implicit. The explicit situation 
offers two are jointly satisfiable conditions (either 1 & 2 or 1 & 3). Because the 
implicit condition is covert and complex, the three conditions appear to be 
independent and so bringing out the consequences of the implicit condition seems 
innocent. Consequently the paradoxical outcomes appear to come from satisfiable 
premisses.  

Perez Laraudogoitia, Hawthorne, and Angel seek to strengthen the paradoxicality 
by rejecting the implicit principle whilst assuming a philosophically significant result 
remains. I think applying the unsatisfiable pair diagnosis shows that rejecting the 
implicit principle is done at the cost of the significant result.  

 We now turn to the matter of guile. Although the unsatisfiable pair are not 
obviously inconsistent, anyone presented bluntly with  

Condition ANB: for all x in S, E at x iff E nowhere before  
will find it easy to bring to mind its equivalent 

Condition ANB*: for all x in S, not E at x iff E somewhere before x.  
The conjunction of ANB and ANB*  looks fishy, and the essence of lines 1 to 9 above 
is quickly seen:  

E at x  iff E nowhere before x  (condition ANB) 
  iff for all y before x, not E at y (what ‘E nowhere before x’ means) 
  iff for all y before x, E somewhere before y  (condition ANB* applied 

to the ys before x, assumes unbegun condition) 
  only if E somewhere before x  (because the ys are before x) 



 

 16

  only if not E at x  (R to L of condition ANB* applied to x) 
Therefore Not E at x. (P only if not-P implies not-P) 

But if all that is quickly seen, the paradox will never get off the ground.  
So distributing the unsatisfiable pair over three conditions is not just a matter of 

being less economical of expression, but necessary for the plausibility of the paradox. 
It is worth noting that Yablo ([2000]), wishing to refute Priest, gives a condition much 
closer to an explicitly stated condition ANB than anyone else.  

Now that we know the form of the paradox we can make up new ones quite easily, 
and to illustrate further the matter of guile I exhibit an overt and a covert use of ANB.  

Suppose a manufacturer claims to have invented a  direct mode of transportation  
from A to any Z (not necessarily instantaneous). It is direct because, unlike normal 
transportation, arriving at Z is not a matter of passing through each of the intervening 
places between A and Z. Their advert says ‘we achieve this feat by arranging space in 
such a way that whenever there is a set of places on a path from A you don’t arrive at, 
you arrive at the first place after it’20  So directness means that if you arrive at Z then 
you didn’t arrive anywhere before Z and this is achieved by ensuring that if you didn’t 
arrive anywhere before Z you will arrive at Z. Since this is an example condition 
ANB, we now know that this machinery must either leave you at A or is logically 
impossible. But even if we didn’t know that, the advertised condition lacks initial 
plausibility, and lacks it, I think because ANB is overt. We can immediately see that 
for any  place Y before Z, it hasn’t placed you at any of the points before that place Y, 
so you have to arrive at Y before Z, not Z, and this applies to all Y before Z, so you 
just stay at A. 21 

With indirection we get something much more beguiling: consider a trolley, 
travelling from left to right on the x axis, equipped with a flag on a small flagpole in 
its centre and a bad tempered elf who hates flags but is asleep. Along its path is a 
sequence of bad tempered goblins with bells who also hate flags. If the trolley reaches 
½  with its flag flying the goblin at that point will ring his bell and wake the elf who 
will lower the flag at 1, if the trolley reaches 1/3 with its flag flying the goblin at that 
point will ring his bell and wake the elf who will lower the flag at ½, and in general  if 
the trolley reaches 1/(n+1) with its flag flying the goblin at that point will ring his bell 
and wake the elf who will lower the flag at 1/n. Then for all x in S= {1/n}, the flag 
will be no longer be flying at x if and only if a goblin rang the bell before x. For every 
member of S there are two earlier members before it, at the first of which the goblin 
must have rung his bell if the flag was flying and at the second of which the elf will 
have lowered the flag.  So for every member of S the trolley will arrive with the flag 
down. But if that is true of every member of S, then no goblin saw a flag flying so no 
bell was rung so the elf remained sleeping and never lowered the flag so the flag must 
fly throughout.  

The natural assumption by which a covert entanglement of ANB has been achieved 
is the assumption that unless lowered by the elf the flag will remain flying and the elf 
stays asleep unless woken by a bell. A bit of digging reveals the unsatisfiable pair. S 
is unbegun.  Since for all x in S, if the flag is lowered at 1/n the goblin must have rung 
the bell at 1/(n +1), so it wasn’t lowered before 1/n. And if it wasn’t lowered before 

                                                 
20 To be strict, they mean that you arrive at the least upper bound of that set. 
21 But perhaps  more plausible examples of overt entanglement could be created. 



 

 17

1/n the goblin at 1/(n +1) will ring the bell and the elf  will lower the flag at 1/n. So 
for all x in S, the flag will be lowered at x if and only if it is lowered nowhere before x 
(condition ANB). Despite appearances, then, we have failed to stipulate a real 
possibility. Inhabitants of this enchanted world will be spared an infinity of goblins 
each of whom both rings and doesn’t ring his bell, an elf who is awake and asleep at 
the same time, and the chilling sight of a trolley passing by with its flag both up and 
down.  

Acknowledgements 
Many thanks to Michael Clark and to two anonymous referees for their helpful 

comments. 
 

Nicholas Shackel  
Department of Philosophy,  
Kings' College 
University of Aberdeen 
Aberdeen 
AB24 3UB 

n.shackel@abdn.ac.uk 

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