
Sorites Paradox - ikeboy
http://plato.stanford.edu/entries/sorites-paradox/
======
n0us
I wrote my senior thesis on this topic when I was completing my philosophy
major. One of the best texts on the subject is "Vagueness" by Williamson.

It's an arcane but super interesting topic that actually has wide
implications. It's also IMO one of the most thorny problems in contemporary
philosophy.

edit: I'm currently busy but can come back later to answer questions if anyone
is interested.

[http://www.amazon.com/Vagueness-Problems-Philosophy-
Timothy-...](http://www.amazon.com/Vagueness-Problems-Philosophy-Timothy-
Williamson/dp/0415139805)

~~~
random28345
I'm not sure what the big deal here is. This has an obvious solution.

"Heap" is a word. Words have meaning to people. Different people ascribe
different meanings to the same word.

Each grain added to a pile of grain measurably changes the probability that an
observer will classify the resulting collection as a heap. While it's not a
linear function, it's generally a positive correlation, more grains means a
higher probability of any individual calling it a heap.

That is a collection of grain of size n+1 has a probability of being
classified as a heap that is usually greater than the probability of a
collection of grain of size n being classified as a heap.

Where's the paradox?

~~~
hderms
It's just interesting when you have discrete entities and you can't show
inductively that there is any point where the classification changes from non-
heap to heap, but clearly that point must exist. It's interesting from the
same perspective as what makes the ship of Theseus interesting.

the probability of something being viewed as a heap could only be measured
statistically as the probability doesn't derive from some direct process. The
issue can be viewed in both an axiomatic and a empirical way and both views
lead to interesting conclusions.

~~~
empath75
That point may not exist though, if it just gains more 'heapness' over time.
Even if 99% of people agree that a collection of grains of sand is a 'heap'
there might be some people who still insist that it is not.

~~~
coldtea
After a point, 100% will agree (anybody not agreeing that an obvious heap of
say 100kg of sand is not a heap but just grains, can be discarded as a
madman).

But still the exact point can't be specified, we only know when it's over-
surpassed.

It's like in the famous trial, when they asked someone to specify is a film is
"porn" or "erotic art", and he said:

"I shall not today attempt further to define the kinds of material I
understand to be embraced within that shorthand description ["hard-core
pornography"], and perhaps I could never succeed in intelligibly doing so. But
I know it when I see it, and the motion picture involved in this case is not
that."

------
SilasX
I love this paradox! It actually transforms into an interesting general
heuristic for life[1]: basically, any time reality throws a categorization
boundary at you (you observe f(a) = X and f(b) = not-X), gradually transform
it (a to b) until you see exactly where the category flips.

For example, if you have a bug in case A but not case B, gradually transform A
into B and see where it disappears.

If a law says that you can do X in condition A but not condition B, see if
it's legal midway between A and B.

[1] which I call the "continuity heuristic" since it depends on reality
obeying a rule that if a is close to b, f(a) is close to f(b).

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stcredzero
_Given then that one grain of wheat does not make a heap, it would seem to
follow that two do not, thus three do not, and so on._

The weasel words in this case are "and so on." Four grains of wheat can make a
heap. Put 3 on the bottom and one on top. If it's too windy to work where you
are, slather them in honey. A sticky heap is still a heap.

~~~
dsg42
You're right, that isn't really the best statement of the paradox. Try this:

Imagine you have a heap of sand. Now, if you remove a single grain of sand
from a heap, it's still a heap, right? But if you remove every grain of sand,
then eventually, you're left without any sand, which is definitely not a heap.
So somewhere along the way, when removing individual grains of sand, the heap
stopped being a heap. But where, exactly, did that happen?

~~~
chongli
It happened at the exact moment when none of the grains were supported by any
of the others. A flat, single grain layer of sand on the ground is not a heap
no matter how many grains are in it.

~~~
cgio
Soros in Greek does not have the structural connotation of heap in English
(which I infer from your reply, given English is not my mother tongue).

------
jbandela1
For the HN population, translating this into security can be useful.

Here is a Sorites argument for AES key strength.

A key of length 1 bit is not secure

A key of length 2 bit is not secure

A key of length 3 bit is not secure

\------------------------------------

A key of length 1024 bits is not secure

We would consider the conclusion absurd. Why?

Because we have a definition of security as such: The expense of breaking the
key is not worth the value of the secret it protects.

For 1 bit, the expense of breaking the key is so low, it is very easy to come
up with secrets that are greater than the value.

For 256 bits, the expense of breaking the key is so (literally) astronomically
high, that it would be very hard to come up with secrets that have greater
value than the expense of breaking the key.

In addition, note that for a given secret 1 bit can make the difference
between secure and not secure.

Notice as we increase the key length bit by bit, more and more secrets are
having values that are less than the expense of breaking the key.

Now let us look at a heap. First let us come up with a working definition. Let
us say that we define a heap as an amount of grain that is larger than can be
held in a closed fist.

Notice how this varies from person to person. Also notice for a given person,
there can easily be the case where 1 extra grain makes the difference between
heap and not.

Also notice, that as we increase the number of grains by one, more and more
people would consider the collection of grains a heap.

------
Dove
So, mathematics has this concept of an expression being "well-defined". It
means you can always tell if something meets the definition, exactly, in every
case.

Almost no definitions in the real world are well-defined. Heaps are just the
tip of the iceberg. Any practical thing, a cat, a tree, a table, a car -- if
you mutilate it enough, it turns into something else, and the transition point
is rarely sharp. And the reason is very simple; it's that in practical life,
we don't define things propositionally, we define them inductively. This is a
cat, and that's a cat, and that's another cat, and cats are things with this
constellation of properties that you learn over time, so when you come across
an item where some of those properties are true and some aren't, you say,
"Well, that's sort of a weird cat".

It is such an incredibly common thing in daily life to run up against fuzzy
definition boundaries. "Can you get milk from the store for me if it's open?"
"Well, there's some sort of demonstration going on right outside -- how bad do
you need it?" We handle it just fine. We recognize when "open" may not be
well-defined in every circumstance and valid in the same way for every intent,
and we're able to discard the simple definition for something more complex as
needed. It's not even weird when you consider that definitions are just models
you are using to describe the world, and those models can be made simple or
complex. We rarely describe the world as it IS, down to the atom. We simplify.

Sure, when you throw inductively defined practical models intended for
communication into a rigorous logical framework, they don't have the sharp
edges such a framework expects. So? That really shouldn't be unexpected. :)

Everything we have to say about the real world is an abstraction. Some ideas
are leakier abstractions than others, but it's all simplification and modeling
in the end.

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trhway
"Vagueness" is one side where the other side of the same coin is abstraction -
instead of counting number of sand grains in any given case and using "set of
N grains" we abstracted such cases - where N is large enough for precise value
of N not being important - into "heap".

Abstraction is a convenient tool, and trying to put abstracted out details
back in - like the precise number of grains - makes the tool more of a
unnecessary liability, i.e "heap" abstraction disappears the moment you ask
the precise number of grains, and instead "set of N grains" appears back.

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pjdorrell
A related question (I think), is what counts as "impossible"?

According to quantum mechanics, anything that is not prohibited by a
conservation law is possible, with some non-zero probability.

So we have to pragmatically redefine "impossible" to mean "the probability is
so small that it won't happen".

But then where do we draw the line between "too small" and "not too small"?

(We can also qualify "won't happen" by saying "cannot have happened in the
history of the observable universe", and take into account an upper bound on
the number of "events" that can ever happen in the history of the observable
universe, eg ~ 2^380. So, for example, an event with probability 2^-500 on one
occasion has probability < 2^-120 of ever having happened that we could know
about.)

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Xcelerate
It might not be apparent from the examples provided in the article, but I
would argue that this paradox is the primary polarizing force behind most
people's morality disagreements.

A lot of the disagreements come down to the question "Well where do we draw
the line?"

~~~
tunesmith
This is why I feel like I'm transitioning, over time, from value ethics to
deontology. Value ethics are really susceptible to this paradox. But if I can
induce what appear to be my principles from my value-ethics instincts, and
then, as an exercise, try to reason out from my principles and act
accordingly, then perhaps I am more consistent.

That's if you see being consistent as an axiom to try to always meet. A lot of
people don't, and I don't know how to argue with them.

------
chris_va
Why is it necessary to reduce language to set theory?

I say it is a heap when doing so improves the odds of my preferred outcome in
a conversation. Being labeled a heap doesn't have to be a binary truism. Or
even a probability. I guess I reject the entire field of study :).

~~~
coldtea
> _Being labeled a heap doesn 't have to be a binary truism. Or even a
> probability._

That in itself is the whole point of the original observation.

That there are things, and not fleeting things like notions and feelings, but
actual tangible things, that are nevertheless not in a specific binary state,
but in some non-specified intermediate state.

~~~
chris_va
The semantics are difficult here, apologies. I can't tell if we are agreeing
:)

Why do things have to be in any state, even a "non-specific intermediate" one?
The original observation seems to have devolved into an argument on the
fundamental nature of language expressing "state". State here implying some
sort of formalism, like sets.

Can't we just assert that language doesn't assign or derive from a formal
state? Rather, the use of language is highly correlated with
outcomes/observations, but not intrinsically meaningful.

~~~
coldtea
> _Why do things have to be in any state, even a "non-specific intermediate"
> one?_

Because being in a state is just another name of being a thing (or plain
"being"). If you "are", by definition you are in some state (at any given
point in time).

> _Can 't we just assert that language doesn't assign or derive from a formal
> state? Rather, the use of language is highly correlated with
> outcomes/observations, but not intrinsically meaningful._

Well, even without using language, we experience things as being in particular
states -- we see them having color, making a noise, being there and not here,
etc.

~~~
chris_va
I'm enjoying this too much :)

Let us say "the sky is blue". With formalism, you'd argue that blue is a
state, and the sky is a thing, and they have a relationship.

I'd argue that language does not actually assign strict identities/state.
There isn't some platonic "blue" or "heap". Rather "sky" is a word highly
correlated with a set of observations, as is "blue". When we combine the words
into the sentence above, we end up with a sequence that results in a useful
outcome from our perspective, so we say it.

The original link shows how people really try to assign boundaries, as if one
"thing" really "is blue" and another isn't. And then it becomes self
contradictory, and people invent yet more explanations. I just question the
base assumption that we need to assign logic/boundaries/formalism to language.

------
zw123456
This is related to the Continuum Fallacy
[https://en.wikipedia.org/wiki/Continuum_fallacy](https://en.wikipedia.org/wiki/Continuum_fallacy)
Or another really fun one, Loki's wager.

------
mrpoptart
Why is it defined that a single grain of wheat isn't a very small heap?

~~~
n0us
If a single grain is a heap then any number of grains is a heap and the
property of being a heap is meaningless. A single grain as a "small" heap also
runs into a problem called higher order vagueness. Where do you draw the line
between a small heap and a regular heap or a large heap? Determining the
cutoff seems to be another heap problem.

~~~
empath75
That problem applies with any arbitrary category. If you look at any color,
you're going to describe it as whatever color category its closest to. If it's
more red than yellow, you might call it orange, I might call it red, someone
else might call it orange-red, or I dunno 'pumpkin'. Language is imprecise.

~~~
n0us
Is it categories themselves that are arbitrary or the set of items belonging
to a category are arbitrary? Do you mean that the meaning of a statement is
imprecise in that it is vague or ambiguous? Or do you mean that the way that
humans employ language to convey concepts is imprecise?

You are thinking like a psychologist or a linguist. I'm not denying that many
people view this is a tedious and seemingly silly thing to argue about but you
can't metaphorically just throw up you hands and claim that it doesn't matter
as a valid counter argument when you are doing philosophy.

~~~
mannykannot
You make philosophy sound like a form of OCD, obsessing about putting
everything in its place.

The answer to your questions is 'all of the above, to some degree'.

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GunboatDiplomat
This is simply a failure to sufficiently define a word.

~~~
cbd1984
> This is simply a failure to sufficiently define a word.

That's the point: How do you "sufficiently define" the word involved here?

Also, how do you "sufficiently define" the word 'ethical'?

