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(1020)

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

(B3)            Q(1020)

(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:

(I1)       if P(n) then P(n + 1)

(I2)       if P(n) then P(n + 2)

...

(I50)      if P(n) then P(n + 50)

...

(I1000)   if P(n) then P(n + 1000)

Here, one has intuitively: Tr(I1), I(I50), Fa(I1000).[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(I1). 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 (I1). 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 (I1)-(I1000) and in particular (I1), (I50) and (I1000).[3]

To what thus does lead the fact of taking into account simultaneously (I1), (I50) and (I1000)? This amounts to considering s in (I1), (I50) and (I1000) 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 (I1), (I50) and (I1000) 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, 106].[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):

(D1)     e[m-M]X(x)

(D2)     ē[m-M]X(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, e1, ē}, {e, e21, e1, e22, ē}, {e, e31, e21, e32, e1, e33, e22, e34, ē}, etc. Within this construction, the primitive concept is that of small. We have then the following definitions: ē = large = contrary of small; e1 = 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, e1, ..., en} comprising n taxa and a criterion X, defined with regard to an interval m-M. The corresponding definitions are as follows: e1[m-M]X, e2[m-M]X,..., en[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](1080)

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)     \ 1080 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)     \ 1080 is very small

Alternatively, one could also use e2 (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 + e2)

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, 1080] are the numbers that are small relative to [1, 1080], plus the numbers that differ by a small number relative to [1, 1080], from the small numbers relative to [1, 1080]. The corresponding version of the induction step is then as follows:

(I13+)   if n is a small number relative to [1, 1080] then n + a small number relative to [1, 1080] is a small number relative to [1, 1080]

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', British Journal for the Philosophy of Science, 20, 193-202 Dummett, M. 1975, 'Wang's Paradox', Synthese, 30, 301-324. Fine, K. 1975, 'Vagueness, Truth and Logic', Synthese, 30, 265-300. Goguen, J. A. 1969, 'The Logic of Inexact Concepts', Synthese, 19, 325-733. 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', Mind, 107, 565-579. Koons, R. 1994, 'A New Solution to the Sorites Problem', Mind, 103, 439-449. Priest, G. 1994, 'The Structure of the Paradoxes of Self-Reference', Mind, 103, 25-34. Read, S. 1995, Thinking About Logic, Oxford & New York: Oxford University Press. Smith, N. 2000, 'The Principle of Uniform Solution (of the Paradoxes of Self-Reference)', Mind, 109, 117-122. Soames, S. 1999, Understanding truth, Oxford & New York: Oxford University Press. Sorensen, R. 1988, Blindspots, Oxford: Oxford University Press. Tye, M. 1990, 'Vague Objects', Mind, 99, 535-557. Williamson, T. (1994), Vagueness, London: Routledge. Yablo, S. 1993, 'Paradox without self-reference', Analysis, 53, 251-252.

[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.