** **

**On
the Circularity in the Sorites Paradox**

Paul
Franceschi

University
of Corsica

revised November 2003

p.franceschi@univ-corse.fr

http://www.univ-corse.fr/~franceschi

Abstract: I begin by highlighting the importance of the step size in the
induction step of the sorites paradox. A careful analysis reveals that the step
size can be characterised as a proper instance of the concept *very small*. After having accurately
described the structure of sorites-susceptible predicates, I argue that the
structure of the induction step in the Sorites Paradox is inherently circular.
This circularity emerges in the structure of Wang's paradox and also of the
classical variations of the paradox with the *young*, *bald*, etc. predicates.

**1.
Elements of the Paradox**

The *sorites paradox* (thereafter, SP) is one of the most ancient and
unresolved paradoxes, which is attributed to Eubulides. It runs as follows:

(B1) a collection with 100000 grains of sand is a heap

(I1+) if a collection with *n* grains of sand is a heap then a collection with *n* - 1 grains of sands is a heap

(C1) \ a collection
with 1 grain of sand is a heap

Given the two prima facie
indisputable premises, one is then led, after a seemingly legitimate repetition
of modus ponens, to the unpalatable conclusion that a collection with 1 grain
of sand is a heap. Other variations of SP involve vague predicates such as *young*, *small*, *bald*, etc.

Let us proceed,
for the sake of accuracy, to highlight some elements of the internal structure
of SP. To begin with, it is worth drawing a distinction between the *incremental* and *decremental* versions of SP. P being a vague predicate, we have then
the incremental version (SP+) of SP:

(B2) P(1)

(I2+) if P(*n*)
then P(*n* + 1)

(C2) \ P(10^{20})

Let also Q be a vague
predicate such as Q = ~P. We have then the decremental version (SP-):

(B3) Q(10^{20})

(I3-) if Q(*n*)
then Q(*n* - 1)

(C3) \ Q(1)

On the other
hand, it is worth considering the *step
size* notion within the induction step (I). Let us denote this step size by *s*. Indeed, the induction step has the
following structure:[1]

(I4+) if P(*n*)
then P(*n* + *s*)

This leads to consider a
hierarchy of versions of the induction step:

(I_{1}) if P(*n*)
then P(*n* + 1)

(I_{2}) if P(*n*)
then P(*n* + 2)

...

(I_{50}) if P(*n*)
then P(*n* + 50)

...

(I_{1000}) if P(*n*)
then P(*n* + 1000)

Here, one has intuitively: *Tr*(I_{1}), *I*(I_{50}), *Fa*(I_{1000}).[2]
Hence, it proves that the choice of the value of *s* is essential for the truth-value of the induction step. Because
one only notes the emergence of SP under the condition that (I) is true.
However such is the case when *s* = 1.
One has then *Tr*(I_{1}). But
the choice of a step size equal to 1 appears as one of the possible choices
within a hierarchy. In SP, one classically only takes into account (I_{1}).
Such an attitude appears as a partial and incomplete view. Rather, all the
semantic facets of the concept *step size*
should be apprehended. Thus, even if it is legitimate to choose for SP a step
size equal to 1, one should apprehend simultaneously (I_{1})-(I_{1000})
and in particular (I_{1}), (I_{50}) and (I_{1000}).[3]

To what thus does
lead the fact of taking into account simultaneously (I_{1}), (I_{50})
and (I_{1000})? This amounts to considering *s* in (I_{1}), (I_{50}) and (I_{1000})
respectively as a proper instance, a borderline case and a proper
counter-instance of the concept *a very
small number*.[4] Thus, the
choices of *s* = 1, *s* = 50 and *s* = 1000 correspond respectively to a proper instance, a borderline
case and a proper counter-instance of the concept *a very small number*. It
ensues that the concept *a very small
number* satisfies all the criteria of vagueness. Consequently, the fact of
taking into account simultaneously (I_{1}), (I_{50}) and (I_{1000})
leads to consider the *step size* in SP
as a proper instance of the intrinsically vague concept *a very small number*. Let us denote it by e. This leads to the more
general formulation of (I) corresponding respectively to (SP+) and (SP-), that
better captures the essence of SP:

(I5+) if P(*n*)
then P(*n* + e)

(I6-) if P(*n*)
then P(*n* - e)

In light of this analysis,
the classical induction step:

(I2+) if P(*n*)
then P(*n* + 1)

can be regarded as a mere
instance of the more general formulation (I5+). In this context, the numerical value '1' in (I2+) can be considered as a proper instance of e.

In fact several
authors[5]
refer explicitly to a version of SP based on the concept *indiscriminable* (or *indistinguishable*).
Indeed, such a predicate presents the characteristics of vagueness. These
authors thus consider veritably a version of SP explicitly based on a concept
of vague step size e. The
corresponding version of SP is based on the *tolerance
principle*. Such a version is in particular mentioned by Koons (1994) and
Read (1995).[6] In this
context, the induction step is formulated as follows:

(I7) if P(*n*) then P(*m*) (*n*
» *m*)

where *n* » *m* denotes the fact that *m*
is indiscriminable from *n*. An
alternative formulation is also:

(I8) if P(*n*) then P(*n* ±* *e)

that leads directly to the
above formulations (I5+) and (I6-). Just as previously, such an analysis leads to
consider as the core of SP a version where the concept *step size* identifies itself as e.

**2.
The Structure of Sorites-susceptible Predicates**

Before considering the
paradox itself, it proves necessary to examine preliminarily several elements
of the structure of the vague predicates involved in SP. I will limit the
present analysis to sorites predicates,[7]
given that it is controversial whether all vague concepts are
sorites-susceptible.[8]
In what follows, my concern will be with showing that any sorites-susceptible
concept can be reduced to a component of vague degree and one (or several)
precise component(s). In particular, I shall argue that a sorites predicate P
can be reduced to a vague degree in {*small*,
*large*} and to a certain number of
precise elements.

Let us consider
thus several paradigmatic sorites predicates: *young*, *small*, *bald*, etc. Uncontroversially, we have
the following definitions: is *young* =
has a *small* number of *years* in [1, 120]; is *small* = has a *small* number of *centimetres*
in [1, 230]; is *bald* = has a *small* number of *hairs* in [1, 10^{6}].[9]
It should be observed that with these latter definitions, the concepts *age*, *size*,
*bald*, etc. can be reduced precisely
to a numerical criterion of number of years, centimetres, hairs, accurately
defined with regard to a discrete interval.

Taking into
account these elements, one is then in a position to highlight the structure of
sorites-susceptible predicates. P being a sorites predicate, we have the
following general structure: P(x) = d[*m*-*M*]X(x)
where d denotes a given
vague degree in {*small*, *large*} of a criterion X, with regard to
an interval [*m*-*M*].[10] If one
denotes *small* by *e* and *large* by *ē*, one obtains the two following
types of definitions for P(x):

The structure of these
definitions thus comprises explicitly the vague concept *small*. And this last occurs in a *direct* form in (D1) and in an *indirect*
one in (D2).

For simplicity,
we have been until now solely concerned with vague partitions comprising only
two taxa: {*small*, *large*}. However, uncontroversially, a
vague duality {*e*, *ē*} leads to the emergence of a
hierarchy of *higher-order vagueness*.
We have then classically the corresponding construction of the hierarchy: {*e*, *ē*},
{*e*, *e*_{1}, *ē*}, {*e*, *e*_{21},
*e*_{1}, *e*_{22}, *ē*},
{*e*, *e*_{31}, *e*_{21},
*e*_{32}, *e*_{1}, *e*_{33},
*e*_{22}, *e*_{34}, *ē*},
etc. Within this construction, the primitive concept is that of *small*. We have then the following
definitions: *ē* = *large* = *contrary of small*; *e*_{1}
= *intermediate* = *neither small nor large* = Ø (*e* Ú *ē*), etc. In general, the other taxa are defined with the help
of the primitive taxon *small* and of the
logical connectors of *negation* and *conjunction*. The various concepts
inherent to the hierarchy of higher-order vagueness can thus be simply built
with the help of the primitive *small*,
and the connectors Ø and Ú.

Indeed, the
present analysis does not apply only to the duality {*small*, *large*} but also to
other vague taxonomies such as {*small*,
*intermediate*, *large*}, etc. In general, one can take into account a taxonomy
comprising *n* taxa. Consequently,
sorites predicates can be defined with the help of a vague degree within a
taxonomy of vague degrees {*e*, *e*_{1}, ..., *e*_{n}} comprising *n*
taxa and a criterion X, defined with regard to an interval *m*-*M*. The corresponding
definitions are as follows: *e*_{1}[*m*-*M*]X,
*e*_{2}[*m*-*M*]X,..., *e*_{n}[*m*-*M*]X.[11]
And the structure of these definitions is thus based, directly or indirectly,
on the vague concept *small*.

**3.
Highlighting the Circularity**

Given the foregoing
definitions, we are now in a position to highlight the semantic structure of
the induction step. It is worth noting first that the following consequence
ensues from what precedes. If I regard the induction step as true, I am
committed to the following claim: P being a sorites predicate, the things that
are P are the things that are small (relative to a certain interval), plus the
things that differ by a very small amount from the small things.[12]
Consider, first, the predicate *young*.
Accordingly, the things that are *young*
are the things that have a small number of years in [1, 120] plus the things
that differ from the things that have a small number of years in [1, 120] by a
very small number. Consider, second, the predicate *small* applied to numbers. It follows that the small numbers are the
small numbers plus the numbers that differ from the small numbers by a very
small number. More generally, it appears that these last two formulations
instantiate the general structure of the induction step: the things that are P
are the things that have a small number of X in [*m*-*M*] plus the things that
differ from the things that have a small number of X in [*m*-*M*] by a very small
number in [*m*-*M*].

At this step, it
should be apparent that the formulation which has been just highlighted reveals
the presence of a *circularity*. For
the sake of accuracy, let us proceed now to formalise the preceding informal
considerations. Consider, to begin with, a mere numerical variation of the
above definition. Accordingly, the small numbers are the numbers that are
small, plus the numbers that differ by a small number from the small numbers.
The corresponding structure of the induction step is then as follows (where *e* denotes *small*):

(I9+) if Is[*e*](*n*) then Is[*e*](*n* + *e*)

that leads to the following
structure of SP:

(B4) Is[*e*](1)

(I9+) if Is[*e*](*n*)
then Is[*e*](*n* + *e*)

(C4) \ Is[*e*](10^{80})

Now it appears
that this last version describes straightforwardly the structure of the
following numerical variation of the sorites (known as Wang's paradox[13]),
which incorporates the predicate *small*:

(B5) 1 is *small*

(I10+) if *n*
is *small* then *n* + 1 is *small*

(C5) \ 10^{80}
is *small*

At this step, it
could be pointed out that the step size should be best rendered with the
predicate *very small* rather than *small*. However, this appears as a minor
qualm. In effect, it suffices to consider that *e* denotes *very* *small* instead of *small* to obtain the following equally acceptable variation of the
sorites:

(B6) 1 is *very small*

(I11+) if *n*
is *very small* then *n* + 1 is *very small*

(C6) \ 10^{80}
is *very small*

Alternatively, one could
also use *e*^{2} (this last
notation is borrowed from fuzzy logic) to denote a *very small number*. The preceding formulation could then be rendered
as:

(I12+) if Is[*e*](*n*) then Is[*e*](*n* + *e*^{2})

that still gives
rise to the circularity.

Let us proceed
now a bit further. For we also need to take into account the fact that numbers
are small *relative to a given interval*.
The following definition then ensues: the small numbers relative to a given
interval are the numbers that are small relative to a given interval, plus the
numbers that differ by a small number relative to a given interval, from the
small numbers relative to a given interval. To take an example: the small
numbers relative to [1, 10^{80}] are the numbers that are small
relative to [1, 10^{80}], plus the numbers that differ by a small
number relative to [1, 10^{80}], from the small numbers relative to [1,
10^{80}]. The corresponding version of the induction step is then as
follows:

(I13+) if *n*
is a small number relative to [1, 10^{80}] then *n* + a small number relative to [1, 10^{80}] is a small
number relative to [1, 10^{80}]

Now restoring
this latter interval in the structure of the induction step, we get the more
general formulation:

(I14+) if Is[*e*][*m*-*M*](*n*) then Is[*e*][*m*-*M*](*n* + *e*[*m*-*M*])

where *m* and *M* denote
respectively the minimum and the maximum value of the interval.

At this step, we
are in a position to handle the more standard version of the paradox. Let us
restore first the criterion X in the formula. The corresponding informal
definition is then: the things that are P are the things that have a small
number of X in [*m*-*M*] plus the things that differ from the
things that have a small number of X in [*m*-*M*] by a very small number in [*m*-*M*].
To take an example: the men that are *young*
are the men who have a small number of *years*
in [0-120] plus the men who differ from the men that have a small number of *years* in [0-120] by a very small number
in [0-120]. We then get the general formulation:

(I15+) if Is[*e*][*m*-*M*]X(*n*) then Is[*e*][*m*-*M*]X(*n* + *e*[*m*-*M*])

In this context, an instance
of SP with the predicate *young* goes
as follows:

(B7) 1 is a small number of years in [0-120]

(I16+) if *n*
is a small number of years in [0-120] then *n*
+ a very small number in [0-120] is a small number of years in [0-120]

(C7) \ 100 is a small
number of years in [0-120]

from which we can derive
straightforwardly, according to the preceding definitions:

(B8) a man who has 1 year is *young*

(I17+) if a man who has *n* years is *young* then a
man who has *n* + 1 years is *young*

(C8) \ a man who has
100 years is *young*

From the above,
it can be concluded that the structure of the induction step in SP is
inherently circular. At this step, it is worth analysing the circularity which
has just been highlighted. Let us then delve more deeply into the structure of
(I9+). Consider first the *antecedent* Is[*e*](*n*) of (I9+). This latter ('*n*
is a small number') appears quite correct. On the other hand, the *consequent* Is[*e*](*n* + *e*) appears basically circular. For one
has an identity of concepts between the primitive constituent (i.e. *small number*, *e*) of the predicate P(x) and the step size (*e*). What appears fundamental here is that at a given time, one has
an identity of concepts between the vague degree *e* presented by a given object, and the step size expressing the *tolerance principle*. This identity can
be diagnosed as the cause of the circularity.

On the other
hand, it is worth pointing out that the semantic content of the predicate P(x)
is described more accurately by Is[*e*](x)
in the above formulations. This best renders the fact that the concept *small *is the primitive concept here,
which leads to the derived predicate *IsSmall*.
Thus, one observes indeed the double occurrence of *e* that emerges in both Is[*e*]
and (*n* + *e*), corresponding semantically to the concept *small*.

Lastly, it is
useful to characterise this last type of circularity. For circularity is
usually regarded as a property of arguments or definitions. In effect, from a
certain viewpoint, the induction step can be regarded as a definitional rule,
which aims at describing what other objects are P, given that some objets are
P. However, from a weaker standpoint, the induction step can be viewed as
making a claim relative to the properties of some objects. And it seems that
such a property of circularity can reasonably be extended to some similar
ontological rules, which aim at reporting the properties of certain objects.
Hence, it seems legitimate that this last type of propositions be also
vulnerable to circularity.

At this stage, it
could perhaps be objected that in the present framework, the circular
definition should disappear when one uses a *precise*
characterisation of the step size *s*,
i.e. by replacing the step size with a very small precise value such as 1.
Because a precise characterisation, this type of objection goes, annihilates
the identity that results from the association of Is[*e*] and (*n* + *e*), by replacing the consequent with Is[*e*] and (*n* + 1)* *(*e* being vague and 1 being precise) thus
making disappear the circularity. However, such objection does not take into
account the above-mentioned fact that the step size can be characterised as a
proper instance of the concept *very small*.
The consequence is that by merely replacing the step size with 1, one does not
handle all variations of SP. In particular, one fails to take into account the
variation of SP which is based on the concept *indiscriminable*. Thus, by narrowly interpreting the step size as a
precise value, one fails to capture the semantic content of the induction step.
By contrast, the present account is capable of handling both variations of the
sorites: those based on an explicitly vague step size (*indiscriminable*) and those based on a precise value that
constitutes a proper instance of the concept *small*.

I should mention
lastly that the present account should be cautiously distinguished from the
claim that by highlighting the circularity in SP one hereby solves the paradox[14].
Rather, the scope of the present paper should be narrowly limited to the claim
that the induction step in SP is inherently circular. To demonstrate that this
circularity is eventually the cause of the paradox would necessitate some
further steps, with which I have not been concerned here. For the paradox and
the circularity could well share a common cause. Finally, I hope that the
current claim will be of interest not only *per
se*, but also for those engaged in the debate[15]
concerning self-referential paradoxes.[16]

**References**

Cargile,
J. 1969, 'The Sorites Paradox', |

Dummett,
M. 1975, 'Wang's Paradox', |

Fine,
K. 1975, 'Vagueness, Truth and Logic', |

Goguen,
J. A. 1969, 'The Logic of Inexact Concepts', |

Hyde,
D. 2002, 'Sorites Paradox', The Stanford Encyclopedia of Philosophy (Fall
2002 Edition), E. N. Zalta (ed.), http://plato.stanford.edu/archives/fall2002/entries/sorites-paradox. |

Keefe,
R. 1998, 'Vagueness by Numbers', |

Koons,
R. 1994, 'A New Solution to the Sorites Problem', |

Priest,
G. 1994, 'The Structure of the Paradoxes of Self-Reference', |

Read,
S. 1995, |

Smith,
N. 2000, 'The Principle of Uniform Solution (of the Paradoxes of
Self-Reference)', |

Soames,
S. 1999, |

Sorensen,
R. 1988, |

Tye,
M. 1990, 'Vague Objects', |

Williamson,
T. (1994), |

Yablo,
S. 1993, 'Paradox without self-reference', |

[1] One considers here (SP+). Concerning (SP-), one has then: (I-) if
P(*n*) then P(*n* - *s*).

[2] Where *Tr*, *I*, *Fa*
denote respectively the predicates *True*,
*Indeterminate*, *False*.

[3] The reasoning that follows is also worth for others *n*-valued logic (*n* > 3), and the current analysis does not require the
preferential choice of a 3-valued logic.

[4] One can choose, alternatively, the concept *a small number*. It leads to a conclusion of the same nature.

[5] Cf. Koons (1994), Read (1995).

[6] Read (1995, p. 199) notably considers a version of the induction
step based on the concept *indiscriminable*:
'(...) the major premiss (...) in its most virulent form, 'if *a* is F and *b* is indiscriminable from *a*
then *b* is F'.'.

[7] Following a terminology from Scott Soames (1999, p. 217).

[8] I thank Nicholas Smith for pointing out this point to me. In
particular it is dubious, as Soames argues (1999, p. 217-8) whether *vehicle* is sorites-susceptible. Rosanna
Keefe (1998, p. 569-70) argues along the same lines. Keefe (1998, p. 569-70) draws
a distinction between one-dimensional and multidimensional vague predicates.
Multidimensional vague concepts such as *heap*
depend on their number of grains of sand *and*
their arrangement. Keefe points out that vague predicates like *nice* or *intelligent* do not present a clear-cut number of dimensions. I do
not address this question here, but there is room for discussion. In
particular, a predicate such as *intelligent*
seems to be sorites-susceptible. It suffices to begin with the following base
step: a person with an IQ of 155 is *intelligent*,
etc., to get the soritical argument moving decrementally.

[9] For simplicity, I have only considered one-dimensional vague predicates. But the current analysis applies straightforwardly to multidimensional vague predicates.

[10] Where *m* and *M* respectively denote the minimum and
the maximum element.

[11] This schema notably allows taking into account the different
taxonomies of *n* colours defined with
regard to the wavelength of the light.

[12] I owe this last informal formulation to Nicholas Smith.

[13] Cf. Dummett (1975).

[14] It should be also noted that the present claim does not involve the
preferential choice of such or such multi-valued logic (to the difference of
the accounts based on *n*-valued logic
(*n* > 2), suggested in particular
by Goguen (1969) or Tye (1990).) Neither does it entail the obligation to
choose between an epistemic (cf. Cargile (1969), Sorensen (1988), Williamson
(1994)) or semantic (cf. Dummett (1975), Fine (1975)) conception of vagueness.

[15] I think notably to Yablo (1993) and related papers, but also to Priest (1994) and Smith (2000).

[16] I thank Nicholas Smith for detailed and extremely useful comments. I am also very grateful to Dominic Hyde for accurate and very helpful criticisms on an earlier draft.