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=                          Sorites paradox                           =
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                            Introduction
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The sorites paradox (; sometimes known as the paradox of the heap) is
a paradox that arises from vague predicates. A typical formulation
involves a heap of sand, from which grains are individually removed.
Under the assumption that removing a single grain does not turn a heap
into a non-heap, the paradox is to consider what happens when the
process is repeated enough times: is a single remaining grain still a
heap? If not, when did it change from a heap to a non-heap?


Paradox of the heap
=====================
The word "sorites" derives from the Greek word for heap. The paradox
is so named because of its original characterization, attributed to
Eubulides of Miletus. The paradox goes as follows: consider a heap of
sand from which grains are individually removed. One might construct
the argument, using premises, as follows:

:' grains of sand is a heap of sand' (Premise 1)
:'A heap of sand minus one grain is still a heap.' (Premise 2)

Repeated applications of Premise 2 (each time starting with one fewer
grain) eventually forces one to accept the conclusion that a heap may
be composed of just one grain of sand.
Read (1995) observes that "the argument is itself a heap, or sorites,
of steps of 'modus ponens'":

:' grains is a heap.'
:'If  grains is a heap then  grains is a heap.'
:'So  grains is a heap.'
:'If  grains is a heap then  grains is a heap.'
:'So  grains is a heap.'
:'If ...'
:'... So  grain is a heap.'


Variations
============
Another formulation is to start with a grain of sand, which is clearly
not a heap, and then assume that adding a single grain of sand to
something that is not a heap does not turn it into a heap.
Inductively, this process can be repeated as much as one wants without
ever constructing a heap. A more natural formulation of this variant
is to assume a set of colored chips exists such that two adjacent
chips vary in color too little for human eyesight to be able to
distinguish between them. Then by induction on this premise, humans
would not be able to distinguish between any colors.

The removal of one drop from the ocean, will not make it 'not an
ocean' (it is still an ocean), but since the volume of water in the
ocean is finite, eventually, after enough removals, even a litre of
water left is still an ocean.

This paradox can be reconstructed for a variety of predicates, for
example, with "tall", "rich", "old", "blue", "bald", and so on.
Bertrand Russell argued that all of natural language, even logical
connectives, is vague; moreover, representations of propositions are
vague.

Other similar paradoxes are:
*Argument of the beard
*The bald man paradox


                        Proposed resolutions
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On the face of it, there are some ways to avoid this conclusion. One
may object to the first premise by denying  grains of sand makes a
heap. But  is just an arbitrarily large number, and the argument will
go through with any such number. So the response must deny outright
that there are such things as heaps. Peter Unger defends this
solution. Alternatively, one may object to the second premise by
stating that it is not true for all heaps of sand that removing one
grain from it still makes a heap.


Setting a fixed boundary
==========================
A common first response to the paradox is to call any set of grains
that has more than a certain number of grains in it a heap. If one
were to set the "fixed boundary" at, say,  grains then one would claim
that for fewer than , it is not a heap; for  or more, then it is a
heap.

However, such solutions are unsatisfactory as there seems little
significance to the difference between  grains and  grains. The
boundary, wherever it may be set, remains as arbitrary and so its
precision is misleading. It is objectionable on both philosophical and
linguistic grounds: the former on account of its arbitrariness, and
the latter on the ground that it is simply not how we use natural
language.

A second response attempts to find a fixed boundary that reflects
common usage of a term. For example, a dictionary may define a "heap"
as "a collection of things thrown together so as to form an
elevation." This requires there to be enough grains that some grains
are supported by other grains. Thus, adding one grain atop a single
layer produces a heap, and removing the last grain above the bottom
layer destroys the heap.


Unknowable boundaries (or epistemicism)
=========================================
Timothy Williamson and Roy Sorensen hold an approach that there are
fixed boundaries but that they are necessarily unknowable.


Supervaluationism
===================
Supervaluationism is a semantics for dealing with irreferential
singular terms and vagueness. It allows one to retain the usual
tautological laws even when dealing with undefined truth values.
As an example for a proposition about an irreferential singular term,
consider the sentence "'Pegasus likes licorice'".
Since the name "'Pegasus'" fails to refer, no truth value can be
assigned to the sentence; there is nothing in the myth that would
justify any such assignment. However, there are some statements about
"'Pegasus'" which have definite truth values nevertheless, such as
"'Pegasus likes licorice or Pegasus doesn't like licorice'". This
sentence is an instance of the tautology "p \vee \neg p", i.e. the
valid schema "'p or not-p'". According to supervaluationism, it should
be true regardless of whether or not its components have a truth
value.

Similarly, "' grains of sand is a heap of sand'" may be considered a
border case having no truth value, but "' grains of sand is a heap of
sand, or  grains of sand is not a heap of sand'" should be true.

Precisely, let v be a classical valuation defined on every atomic
sentence of the language L, and let At(x) be the number of distinct
atomic sentences in x. Then for every sentence x, at most 2^{At(x)}
distinct classical valuations can exist. A supervaluation V is a
function from sentences to truth values such that, a sentence x is
super-true (i.e. V(x) = \text{True}) if and only if v(x) = \text{True}
for every classical valuation v; likewise for super-false. Otherwise,
V(x) is undefined�i.e. exactly when there are two classical valuations
v and v' such that v(x)=\text{True} and v'(x) = \text{False}.

For example, let L \; p be the formal translation of "'Pegasus likes
licorice'". Then there are exactly two classical valuations v and v'
on L \; p, viz. v(L \; p) = \text{True} and v'(L \; p) = \text{False}.
So L \; p is neither super-true nor super-false. However, the
tautology L \; p \lor \lnot L \; p is evaluated to \text{True} by
every classical valuation; it is hence super-true. Similarly, the
formalization of the above heap proposition H \; 1000 is neither
super-true nor super-false, but H \; 1000 \lor \lnot H \; 1000 is
super-true.


Truth gaps, gluts, and multi-valued logics
============================================
Another approach is to use a multi-valued logic. From this point of
view, the problem is with the principle of bivalence: the sand is
either a heap or is not a heap, without any shades of gray. Instead of
two logical states, 'heap' and 'not-heap', a three value system can be
used, for example 'heap', 'indeterminate' and 'not-heap'. However,
three valued systems do not truly resolve the paradox as there is
still a dividing line between 'heap' and 'indeterminate' and also
between 'indeterminate' and 'not-heap'. The third truth-value can be
understood either as a 'truth-value gap' or as a 'truth-value glut'.

Alternatively, fuzzy logic offers a continuous spectrum of logical
states represented in the unit interval of real numbers [0,1]�it is a
many-valued logic with infinitely-many truth-values, and thus the sand
moves smoothly from "definitely heap" to "definitely not heap", with
shades in the intermediate region. Fuzzy hedges are used to divide the
continuum into regions corresponding to classes like 'definitely
heap', 'mostly heap', 'partly heap', 'slightly heap', and 'not heap'.
Though the problem remains of where these borders occur; e.g. at what
number of grains sand starts being 'definitely' a heap.


Hysteresis
============
Another approach, introduced by Raffman, is to use hysteresis, that
is, knowledge of what the collection of sand started as. Equivalent
amounts of sand may be called heaps or not based on how they got
there. If a large heap (indisputably described as a heap) is slowly
diminished, it preserves its "heap status" to a point, even as the
actual amount of sand is reduced to a smaller number of grains. For
example, suppose  grains is a pile and  grains is a heap. There will
be an overlap for these states. So if one is reducing it from a heap
to a pile, it is a heap going down until, say, . At that point one
would stop calling it a heap and start calling it a pile. But if one
replaces one grain, it would not instantly turn back into a heap. When
going up it would remain a pile until, say,  grains. The numbers
picked are arbitrary; the point is, that the same amount can be either
a heap or a pile depending on what it was before the change. A common
use of hysteresis would be the thermostat for air conditioning: the AC
is set at 77 °F and it then cools down to just below 77 °F, but does
not turn on again instantly at 77.001 °F�it waits until almost 78 °F,
to prevent instant change of state over and over again.


Group consensus
=================
One can establish the meaning of the word "heap" by appealing to
consensus. Williamson, in his epistemic solution to the paradox,
assumes that the meaning of vague terms must be determined by group
usage. The consensus approach typically claims that a collection of
grains is as much a "heap" as the proportion of people in a group who
believe it to be so. In other words, the 'probability' that any
collection is considered a heap is the expected value of the
distribution of the group's views.

A group may decide that:
*One grain of sand on its own is not a heap.
*A large collection of grains of sand is a heap.

Between the two extremes, individual members of the group may disagree
with each other over whether any particular collection can be labelled
a "heap". The collection can then not be definitively claimed to 'be'
a "heap" or "not a heap". This can be considered an appeal to
descriptive linguistics rather than prescriptive linguistics, as it
resolves the issue of definition based on how the population uses
natural language. Indeed, if a precise prescriptive definition of
"heap" is available then the group consensus will always be unanimous
and the paradox does not arise.

colspan=7 | Modelling "'X' more or equally red than 'Y'" as
quasitransitive (Q) and as transitive (T) relation
! 'X'**�**'Y'

QT      QT      QTP     QTP     QTP     QTP
Q       QT      QT      QTP     QTP     QTP
Q       QT      QT      QTP     QTP
Q       QT      QT      QTP
Q       QT      QT
Q       QT


Dropping transitivity of the relations involved
=================================================
In the above color example, the argument is tacitly based on
considering the relation "for the human eye, color 'X' is
indistinguishable from 'Y'" as an equivalence relation, in particular
as transitive.  To drop the transitivity assumption is a possibility
to resolve the paradox.

Similarly, the paradox is based on considering the relation "for the
human eye, color 'X' looks more or equally red than 'Y'" as a
reflexive total ordering; again, dropping its transitivity resolves
the paradox.

Instead, the relation between colors can be described as a
quasitransitive relation, employing a concept introduced by
microeconomist Amartya Sen in 1969.
The table shows a simple example, with color differences overdone for
readability. A "Q" and a "T" indicates that the row's color looks more
or equally red than column's color in the quasitransitive and the
transitive version of the relation, respectively. In the
quasitransitive version, e.g. the colors  and  are modelled as
indistinguishable, since a "Q" appears in both their intersection
cells. A "P" indicates the asymmetric part of the quasitransitive
version.

To resolve the original heap variation of the paradox with this
approach, the relation "'X' grains are more a heap than 'Y' grains"
should be considered quasitransitive rather than transitive.


                              See also
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* Ambiguity
* Boiling frog
* Continuum fallacy
* I know it when I see it
* Ring species
* Ship of Theseus
* Slippery slope
* Straw that broke the camel's back


                            Bibliography
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*
*
*
*
*
*Damir D. Dzhafarov, The sorites paradox: a behavioral approach (with
E. N. Dzhafarov), in J. Valsiner and L. Dudolph (eds.).
*
*Kirk Ludwig & Greg Ray, "Vagueness and the Sorites Paradox",
Philosophical Perspectives 16, 2002.
*
*; Sect.3


                           External links
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* by Dominic Hyde.


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