Two’s company, …

by John Q on September 28, 2004

We have a pretty clear division of labour as regards paradoxes here at CT. Brian, Daniel and occasionally Chris set them up, while I, along with the commenters, try to knock them down. Following Chris’ discussion of the paradox of rational voting, I found myself wondering about the sorites paradox[1]. Once I got the thing into a form where I felt confident about the truth-value of the premises, I came to the conclusion that the argument fails, at least in my language, at n = 2.

Here’s my general version of the paradox

1. I would never call a single item “a heap”
2. If I would never call n items “a heap”, I would never call n+1 items “a heap”

Hence, by repeated induction, I would never call n items “a heap” for any integer n

Introspection tells me that I would under at least some circumstances, call three items “a heap”, but that[2] I would never call two items “a heap”. And if I replace “heap” by “crowd”, I don’t have to rely on introspection. Premise 2 is proverbially false in this case, for n = 2.

So, the paradox comes down to the fact that, in ordinary English usage, three items are sometimes a heap and sometimes not.

fn1. The problem is that, as the Florida 2000 election showed, no single voter is ever decisive. But if Gore had received, say, 5 000 more votes, he would almost certainly have taken office.

fn2. There are some formal contexts, for example in computing and game theory, where a heap might have two elements, or one, or zero.



Enzo 09.28.04 at 10:58 am

Ho about second order vagueness in this case?


Enzo 09.28.04 at 10:58 am

How about second order vagueness?


bad Jim 09.28.04 at 11:07 am

Thus speaks a man without a yard, a garage or an attic, and utterly unfamiliar with Pascal or the 8086 family.


Nick Simmonds 09.28.04 at 11:47 am

There are probably no cases wherein I would call one person a “crowd”, or one object a “heap” (in the sense of quantity), but I would hypothetically be willing to call one person a “heap”.


des von bladet 09.28.04 at 11:57 am

In my capacity as an occasional mathematician, I would have no qualms about a heap of zero items. (The trivial heap.)

But in any case I mostly deal with sorites paradoxes by waving my paw and saying “Bah!”. It is worth noting that (half of) the reasons I consider it perfectly sensible to play the (Euro) lottery is that, unlike Rationaliste economistes, I do not consider that the concept of “rich” is undermined by sorites issues. (The other half is the Finite Lifetime Hypothesis, of course.)


abb1 09.28.04 at 12:22 pm

When you glance at a group of homogeneous objects, there is a threshold where you can’t immediately tell how many objects there are in the set.

When you glance at a set of three objects, for example, you know immediately that there are three objects. But when it’s a set of, say, eleven objects – you can’t tell right away how many there are; you have to do some mental grouping (4-4-3, or something like that) first.

I want to suggest that the minimum number of objects that you have to make a conscious effort to count is a ‘crowd’. It’ll be a different number for different people, for an ordinary person probably between 6 and 9.


des von bladet 09.28.04 at 12:33 pm

All hail the Magic Number 7±2!

The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information, George A. Miller. (The section on “subitizing”.)


Brian Weatherson 09.28.04 at 3:16 pm

I’ve never understood the paradox concerning ‘heap’, it part because I never use the word in my natural language. I think the paradox works much better for other words, like ‘smart’ or ‘beautiful’, where it is clear you don’t want to include the more-or-less degenerate cases (e.g. Bush, Cheney.)


Kieran Healy 09.28.04 at 3:26 pm

I think the paradox works much better for other words, like ‘smart’ or ‘beautiful’,

Really? I thought it’s supposed to be strongest when there’s a definitely measurable micro-unit that eventually creates the macro-object — like grains of sand and a pile of sand, or (decrementing) hairs on a head and a bald head. “Beautiful” and “Smart” are vague around the edges but don’t have that definitely unit-increment, so they don’t seem as clear-cut cases of sorites to me.


eudoxis 09.28.04 at 4:06 pm

#2 doesn’t look right. if n is any integer, #2 fails as a premise beyond the vagueness level.


abb1 09.28.04 at 4:18 pm

‘smart’ could be quantified in IQ units.


cafl 09.28.04 at 5:19 pm

If the first item is a car just sold to the junk yard, I would call it a heap.


cafl 09.28.04 at 5:20 pm

If the first item is a car just sold to the junk yard, I would call it a heap.


Njorl 09.28.04 at 6:28 pm

I think there is a flaw in the presentation of the alleged paradox that hides the real solution.

The paradox relies on human perceptions. A person sees some number of objects and judges whether or not it is a heap. As each object is added, the pile is still noticeably a collection of discrete things.

But human percetion is not static. Just as an observer walking from a well lit room into a dark room does not see what an observer who has acclimated to the darkness sees, an observer who watches a collection grow does not see the same thing that an observer who is presented with that collection all at once.

The observer who watches the collection grow is NEVER going to see a heap. It will always be a collection of discrete items. What he will do is make a judgement of when he would consider the collection a heap, if he were presented with it in different circumstances. Such a judgement is made with an eye jaundiced from the very experience of watching the pile grow though.


Jack William Bell 09.28.04 at 6:46 pm

Yeesh. What a lot of long-winded over-analyzation (with emphasis on the ‘anal’ part)!

Perhaps it is because I am a programmer, but I am perfectly capable of stepping up to a second order of abstraction and saying “A heap is something you put things in.” Which means I can have an *empty* heap, a heap with one item and a heap with several billion items. All are heaps. End of story.

Is that so hard? So many of these kinds of paradoxes are built on weak structures of argument, butressed only by restricting the terminology and meanings of same. And they usuall fall like a house of cards when a bit of simple common sense is applied.


John Quiggin 09.28.04 at 8:13 pm

Jack, read footnote 2.


dsquared 09.28.04 at 8:24 pm

John; if you got bored one Saturday and started pulling out the hairs on your chin with tweezers, at what point would you cease to have “a beard”? My guess is that three whiskers wouldn’t be a beard, and so on … :-)


Matt Austern 09.28.04 at 8:35 pm

There’s a similar paradox that I like better:

If I can afford something that costs n cents, then I can afford something that costs n+1 cents. I can afford something that costs $0.01. By mathematical induction, I can afford anything.


rilkefan 09.28.04 at 8:56 pm

I don’t think there are heaps. Just because a term is usually convenient doesn’t mean it represents something actual.


John Quiggin 09.28.04 at 9:25 pm

Matt, I think it’s easy to spot points where this induction breaks. If the item is of modest importance, then I won’t go to the bank to get money to pay for it (or borrow money from a friend etc). So I can afford it if it costs less than or equal to the amount in my pocket, and not otherwise.

DD, why does philosophy always involve such gruesome examples? I don’t even want to think about that idea.


Jack William Bell 09.28.04 at 10:17 pm

John, my point was that there really isn’t a paradox in the first place; not there there wasn’t a paradox when using the definitions of computer programming.

The only reason I bring up computer programming is the fact the discipline (of programming)requires you to use abstractions to solve real-world problems in such a way that that the real-world mappings are not broken. Whereas the kind of word-play used in the original discussion creates false paradoxes by doing the exact opposite.


Anthony 09.29.04 at 4:36 am

The paradox exists because “ordinary language” allows vagueness. As soon as precision is required, for logical or commercial reasons, a definition will be propounded. We will then be able to speak of 1.37 heaps of leaves in your front yard, as that is the quantity the garbage company will bill you for the removal of.


Jay 09.29.04 at 5:33 pm

Interestingly, transfinite numbers fit this definition. (i.e. 1 is not transfinite, and for any finite N, N+1 is not transfinite.) Perhaps aggregate terms such as heap, crowd, etc. represent a sort of practical/verbal infinity.


eudoxis 09.29.04 at 7:22 pm

I’m still at a loss to see a connection between voting and this “paradox” (why is it a paradox that something not clearly defined doesn’t fit a clear definition?). When the transition from a loss to a win is defined at one vote, there is no more vaguery to a gain of 1 vote than a gain of 5000 votes.

I do see a parallel between voting for national candidates, home recycling to help the environment, and heatedly discussing world affairs.


John Quiggin 09.29.04 at 8:26 pm

eudoxis, the Florida election showed that if the initial count produces a very small majority, there will be (with high probability) a recount producing a different outcome. So a majority of 1 is not decisive.


Jack William Bell 09.29.04 at 9:45 pm

John; you just described the whole problem I had with the Florida recounts and subsequent court battles. As usual I need to preface this by saying “I am not a Bush supporter.” But…

But, I felt at them time (and do now) that Gore was the one perverting the system by trying to force recounts until they showed that he won, with every intention of then trying to stop recounts and declare victory.

Anyone with a little brains, modicum of math and an understanding of human engineering (like hand counted voting systems) would know that in cases where the margin of victory is *less* than the margin of error sucessive recounts will produce random results. And Gore and his people were not stupid.

Given that Bush won in the long run (after the Supreme Court settled it) it seem clear to me that Gore did this country a huge disservice by forcing an issue that could only have led to that result. A better man would have conceded defeat with the appropriate caveats.

My biggest fear is that we will see a similar situation in the next election. The last thing we want is court-decided political battles becoming the norm. No matter who wins, democracy loses…


Tom Runnacles 09.29.04 at 11:11 pm

How’s about ‘is close to [your city of choice]’?


Anarch 09.30.04 at 4:41 am

I’ve always maintained that the problem with “heap” (or, for example, the “child of a monkey is a monkey” variant) is that it presumes a bivalent truth value where bivalency is not applicable. The problem, then, is not so much with the language but rather the error in our modelling of truth and falsity in this instance.


Anarch 09.30.04 at 4:59 am

Perhaps it is because I am a programmer, but I am perfectly capable of stepping up to a second order of abstraction and saying “A heap is something you put things in.”

But you would be wrong.

More precisely, you would be “restricting the terminology and meanings” of the word “heap” in order to avoid the sorites paradox. Augmenting the sense of “heap” by creating such an additional technical meaning — functionally equivalent to “set” in your post, although I suspect the more usual technical definition of heap has an inherent notion of (well-)ordering, too — doesn’t solve the problem which is that in the original meaning of the word the paradox remains.

[If you don’t like the use of the sometimes-technical word “heap”, you can try this variant of the sorites paradox: the child of a monkey is a monkey, not a human. According to Darwin, one of your ancestors was a monkey, not a human. Therefore, by a sorites-style induction, you are a monkey, not a human.]

A similar problem emerges, incidentally, with the paradox of the smallest undefinable number. If you code that into a precise mathematical statement the paradox is avoided because, crudely speaking, “definable” is not definable and therefore we cannot diagonalize against the requisite property. The paradox remains in English, though, where “definability” presumably means “uniquely specified” in some metatheoretical sense.

FWIW, the distinction to me is that a heap refers to an actual physical reality, so we have to check whether typical bivalent logics are the correct logics to model the reality of the situation. The paradox of the least definable ordinal, however, lives entirely within bivalent mathematics (specifically {0,1}-valued set theory) and, as such, we have no choice over which logics to use.


Tracy 10.01.04 at 9:09 am

Surely part of the solution to the paradox is that we need not only look at the 1 by 1 level, but can step back and occasionally look at how far we’ve come from the starting state. So, for the example of a balding man, as he loses hairs we can occasionally step back from the hair-by-hair view and say, “well, he’s lost enough hairs, he’s definitely got a receding hair line”, even if we’re not able to identify the exact point that he changes from “full head of hair” to “receding hair line”. What we’re talking about here is a situation where our ability to discriminate is less precise than the unit of measurement.

For another way of looking at it, imagine that Strider of LOTR fame is walking a thousand miles. Obviously each step is tiny compared to the whole journey. Also, one step must take him from slightly less than half-way to his goal to slightly more than half-way. Now, since there are no pedometers on Middle Earth, Strider doesn’t know exactly when that step will be. However, if he calculates that the half-way point is somewhere between Hobbiton and Bree, and on leaving Hobbiton announces that he is not yet half-way, and on reaching Bree that he is over half-way there, would any philosopher accuse him of being illogical? And if not, how could it be illogical to say that someone has definitely shifted from hirsute to balding, even if you are unable to pinpoint the loss of the particular hair that created the shift.

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