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39th Mid-south Philosophy Association Presentation Wai San, Ng Rice University, Ph.D. Student

Is Vagueness Friend or Foe of Natural Kind?

Vagueness has long been a foe of natural kind theorists. Proponents of the Sorites argument, despite

having no consensus on which ontology they should hold, agree that our commonsensical understanding

of the world as filled with concrete, discrete, multifarious objects classified into kinds is, if not

fundamentally flawed, at least spurious. For example, Hogan and Potrč hold that all discourse involving

these mundane objects is, strictly speaking, false, while Ludwig and Ray, using the similar line of

argument, maintain that these discourses are all non-truth-apt. Though these attacks never target only on

natural kind terms, it is only a step forward to see that these terms lies within the ambit of the fire - a lot

of natural kind terms lack a sharp boundary as required by the Sorites argument (e.g. biological kinds are

subject to indeterminacy in case of mutation, or on the boundary of species separation), and thus run the

risk of being wiped out from our ontology. Such a theoretical consequence, however, is far from

conclusive. In this paper I will show that bitterness between vagueness and natural kind can be resolved,

and friendship can be established.

I. The Bitterness

Let me start by offering some characterizations to what I mean by “natural kind”. These characterizations

are by no means complete or unquestionable, but as a starting point for discussion they are

uncontroversial enough to do to the task here:

a) Naturalness: genuinely natural divisions and distinctions are independent of psychological or

social attitude of human beings, “carve nature at its joint” (Bird, 2015; Hacking, 1990)

b) Kindness: things which are similar to each other and only to each other, such that they allow for

classification or systemization. Members of a kind share a “cluster of properties”, the knowledge

of some of which enables us to tell other properties of the kind under question. (Boyd, 1988, 2011;

Quine, 1969; Bird, 2015; Rosch, 1976)

c) Support Induction: Recognition and the use of such natural kind terms plays a significant role in

the growth of human knowledge, for they can take up subject position in predications, and can be

quantified over, enabling the inductive knowledge of a group without investigating into particular

members. (Lowe 2006:29; Quine, 1969; Boyd, 1999; Bird, 2015; Hacking, 1990)

d) Indeterminacy of membership: Natural kind is generally understood with reference to other

properties, but membership in borderline cases can be indeterminate, especially when the

property cluster characterizing a kind is not completely possessed by a particular instance. (Boyd,

1988, 2011; Bird, 2015; Rosch, 1976)


Given (b) and (d) it is not difficult to see why (at least some) natural kind terms are subject to the

criticism of being vague; for two things: (i) the package of properties which jointly characterize a natural

kind term may contain vague predicates which do not have clear boundary for applications; (ii) even if all

the properties characterizing a natural kind term are non-vague, it is unclear whether all of them are

necessary, and how many properties are sufficient for membership. 1 Now, recall the standard form of the

Sorites argument:

[1] x0 is K.

[2] For all n (0≤n<k), if xn is K, then xn+1 is K.

[3] x100,000,000,000 is K.

Plug in a mundane natural kind term in K (be it mammals, amphibians, fish) and its prototypical member

in x (be it homo sapiens, frogs…), the argument will generate a counterintuitive result at [3], for there is

no sharp dividing line between K and non-K, and the denial of [2] is “the miracle of metaphysical illusion”

(as Unger puts it) (Unger, 1979). Of course, the argument does not immediately show that [1] is false. But

if we hold fix two-value logic and modus ponens, denial of [1] seems to be the only viable option.

Anyone who finds the argument troubling may hope for some ways to deny [2]. But there are also some

philosophers who choose to accept the consequence. Ludwig and Ray, for example, push the theoretical

consequence of vagueness further by an elaborate and rigorous argument (Ludwig and Ray, 2016). So

construed, all sentences with vague predicates are not truth-apt.

[4] Exclusion Principle: for all x and y, if x applies to y then x does not fail-to-apply to y, and for all

x and y if x fails-to-apply to y then x does not apply to y.

[5] Higher Order Vagueness: There are no first or higher-order borderlines for vague predicates.

(Assumption)

[6] HOV implies that for every n (0 ≤ n< k), vague predicate „p‟ applies to n iff „p‟ applies to n+1

[7] HOV implies that for every n (0 ≤ n< k), vague predicate „p‟ fails-to-apply to n iff „p‟ fails-to-

apply to n+1

[8] For every n (0 ≤ n< k), vague predicate „p‟ applies to n iff „p‟ applies to n+1 (5,6)

[9] For every n (0 ≤ n< k), vague predicate „p‟ fails-to-apply to n iff „p‟ fails-to-apply to n+1 (5,7)

[10] There is some n to which „p‟ does not apply (Assumption)

[11] There is some n to which „p‟ does not fail-to-apply (Assumption)

1 Of course, this is not to say that all natural kind terms must be vague. It is well recognizable that some natural kind

terms are clearly definable in non-vague simples, for example, chemical elements are sharply defined kinds as are

the kinds of subatomic particles. The point here is simply that a lot of natural kind terms are vague.


[12] „p‟ does not apply to all n+1, and so all n (10, repeated application to 8)

[13] „p‟ does not fail-to-apply to all n+1, and so all n (11, repeated application to 9)

[14] „p‟ does not apply to any n and „p‟ does not fail-to-apply to any n. (12,13)

Horgan and Potrč, motivated also by the Sorites argument, come to a slightly different but equally

disastrous conclusion – all the sentences containing vague predicates are, in the strictest sense, false.

(Horgan and Potrč, forthcoming) Below is my formulation of their central argument.

[15] Ontological vagueness requires the factual realization of logical incoherence.

[16] Factual realization of logical incoherence is impossible.

[17] Ontological vagueness is impossible. (15,16)

[18] All sentences involving vague predicates do not correspond to the fact, ontologically conceived.

[19] Only sentences that correspond to the fact can be true.

[20] All sentences involving vague predicates are false.

Now since some natural kind terms do not have clearly definable boundaries, all inductive generalizations

involving them are either non-truth-apt, or simply false.

II. Rapprochement

My first move is to scrutinize [17] and [18]. The disparity between the ontological fact (which is non-

vague) and our description of it (which is vague) is where the force of the argument lies. My claim is that

[17] is true – ontological vagueness is indeed impossible, not because the ITEMs in the world are sharply

distinguished from each other, but because “vague” is not a predicate applicable to facts.

Claim 1: A fact itself is neither vague nor non-vague, for “vague” is a predicate of the language, of

thoughts, of representations, of descriptions of fact, but not of the fact as such.

I hold this claim to be highly intuitive and linguistic evidence is abundant. A parallel of this predicate is

“true”, which is a predicate concerning a sentence‟s relationship with the world, but not the world as such.

Another way to motivate this point is to think of the adjective “sharp”. If vague predicate cannot

correspond to the world just because the world cannot be not vague, then by the same line of thought a

sharp predicate cannot correspond to the world as well - the world is not sharp either. Just because a

predicate is vague and the world is not vague is not a good reason to hold that the vague predicate cannot

correspond to the world. If one agrees with me on this point, then it is obvious that [18] does not hold.


Therefore, I argue that [20] is false. Vague predicate can also correspond to the world, though in a vague

way. And thus whether a description of the world is true or not does not depend on its vagueness or

sharpness, but on whether it succeed in transmitting correct information about the world.

One may wonder how there can be vague correspondence: after all, the vague predicate is not quite

exactly the THING. But this attitude to truth is over-pessimistic and ill-founded. Consider the question

“what is the ratio of the circumference to radius”, is it wrong to answer “~3.14159”? In physics, answers

sometime are given in terms of light year (radius of visible universe is 45.7 billion light years), and they

are thought to be correct if they hit upon the right order of magnitude. Are these knowledge claims all

wrong? Here I would like to employ Daniel Dennett‟s idea on reality, which accepts all compressible

description (which he calls pattern) of a fact (or data) as real, however noisy the pattern is. If a true

description of the world has to be exact, then almost all our knowledge, even those in physics, is not

possible.

But how does it related to the Sorites argument and Ludwig and Ray‟s non-truth apt challenge? Here

comes my second move. Obviously, the force of these arguments lie on the premise [2], [6] and [7]. I

argue that these three premises are true only when we interpret the phrase “+1” correctly. In both

arguments the “+1” are crucial, for it is where the repeated application of an undeniable principle can

generate an unbelievable conclusion. But what does it mean by “+1”exactly?

Claim 2: Given a universe U with n variables (x), with “+1” means “next variable”, in other words, “next

x”, whatever x is.

This is an uninformative answer; however, for Claim 2 does not tell us what x is. In fact, this is not

something that could be told before the universe or domain is chosen. But how can we choose a universe

or a domain in which we found our x? Isn‟t it obvious that we are in a universe in which all objects are

composed of atoms or quarks or strings or some physically simple thing? Isn‟t it that “+1” just mean the

next atom or quark or string or some physically simple thing? This is exactly the point I want to reject.


Claim 3: the “next x” depends on the values of the variables present in a universe, and the variables of a

universe depend on the Dennettian stance chosen. Such that (for example):

“next x” in physicist stance: next atom or quark or string or some physical simples

“next x” in biological kind stance: next discrete living organism (let‟s assume so)

“next x” in daily stance: next discrete sensible/visible/identifiable object (let‟s assume so)

Now I have enough resource to refute [2], [6] and [7]. I argue that they are judged as intuitive just because

we have confused the stances, the universes/domains, the “x”, and the “+1”. These premises are

misleadingly false. Consider the following two instances of the Sorites argument:

[21] Here is a table T.

[22] For all x (0≤x<k), if Tx is a table, then Tx+1 is a table.

[23] Tk is a table.

[24] Here is an atom A.

[25] For all x (0≤x<k), if Ax is an atom, then Ax+1 is an atom.

[26] Ak is an atom.

Unger allows for the rejection of [25] but not [22]. He thus accepts that there are atoms (or some other

clearly definable simples) in the world, but not tables. (Unger, 1979) Horgan & Potrč too, give no

argument concerning vagueness against the rejection of [25]. But what licenses this blatantly unfair

treatment to two sentences with identical form? Nothing licenses. The discrepancy between the treatments

to the two instances of Sorites argument above is a result of the equivocation of the order/scale/stance – in

evaluating [24] & [25] one uses the same stance, but in evaluating [21] & [22] one uses different stances.

[21] is a description under of the daily stance, but the “x” in [22] is some amount of atoms or quarks or

strings or some physical simples, whereas they do not belong to the universe in the daily stance. If one is

uniform in taking the daily stance in [21] and [22], then the “x” in [22] would be some visible/identifiable

discrete objects, and one can judge, unabashedly, at some point that T stops to be a table, or at least not

clear whether it is a table or not (higher order divide line). In other words, if we hit the order/scale

correctly and consistently, [22] could be rejected, just as we reject [25].


What if we take the “x” in [22] to be some atoms or quarks or strings or some physical simples? In that

case [21] is false and [22] is true. Why? This is because in the physicist stance there is no table at all;

table is a description of the world under the daily stance. In the physicist stance there are a lot of atoms

but no table, just as in the daily stance there are no atoms but discrete objects. So, [21] is false on

physicalist stance. And since [22]‟s antecedent contains the predicate “table”, the antecedent is false and

the whole conditional is true. But isn‟t it that n atoms (n is a very large number) equal to a table? Nobel

Prize winner physicist Eddington says no, for there are no easy ways of reducing our mundane objects to

physicist‟s object. If a reduction is ever possible, then a complete knowledge of our consciousness must

have already been presupposed (Eddington 1928) – the reduction is never an easy transaction, and

sometimes the transaction cost is forbiddingly high.

Claim 4: The first and the second premises of the standard Sorites argument cannot both be true at the

same stance. The affirmation of the first leads to the rejection of the second, vice versa.

But why we have to be bound to the same stance in evaluating the propositions? Let‟s assume we don‟t.

In that case, premise [21] is true on the daily stance. And since multiply stances could be employed, the

antecedent of [22] - Tx is a table - is true. However, is the consequence – Tx+1 is a table– true? No! This is

because in that case the domain/universe in which we found the “x” will then include both the physical

simples (observable only in physicalist stance), and mundane objects (observable only in daily stance),

which are heterogeneous to each other – they are of different quality. But the mathematical operation “+1”

requires a homogeneous relation. A “plus” operation can only be done with numbers with the same unit.

The following figure may help to explain, and explain away our intuition that Tx+1 is a table:


Under all the situations shown above, the equivocation of stance troubles the Sorites argument. It should

be noted that the moral of my strategy is not that in evaluating [21] and [22] we must be bound to the

same stance, but that only under the same stance can the mathematical operation “+1” could be carried

out. Given such constraint, we now come to a two-way dead end – either we allow for trans-stance

evaluation of [21] and [22], or not. If we do not allow for the employment of different stance in

evaluating [21] and [22], then one of them is false. If we allow for the trans-stance evaluation of the two

premises, then the crucial step “+1” is prohibited.

One may wonder: isn‟t that the classification of different stances a blatant surrender to pragmatism/

instrumentalism or even anthropocentricism? How can the “x” depend on one‟s stance? What is stance

actually? To answer these questions I would like to invoke Don Ross‟s materialistic formulation of stance

to show in what sense “stance” can be understood as real and objective. In his paper “Rainforest Realism”

Ross define stance as is (i) a possible physical measurement, be it a measurement of the computer, or a

human cognitive faculty, (ii) under which the encoding of information about the structure of at least one

event or entity that is more efficient than the bit-map encoding. (Ross, 2000) So construed, stance is just

like a physicalist frame of reference within which objective results could be measured. I should also stress

that my view can be made compatible with pragmatism or even anthropocentricism, but it need not be

committed to any of those. If a pragmatist wants to employ my strategy to argue for a pragmatist world

view, he has to bring with his private machinery.

If stances are some possible physical measurements, and “a stance” refer to a particular group of them, it

should be understandable to us why atoms could not be added to a table – in daily stance our detector is

our sensory faculty, which is far too crude for an atom, and in a physicalist stance our detectors are some

scientific instruments, fine enough to detect an atom but too fine for a table – they are of totally different


format. How can we add the result of two different measurements when we do not know the conversion

formula between two units of measurement?

One may still ask: is the daily stance really a stance? I am not sure. In the above I am talking as if there is

such a stance, but I am not really committed to it. 2 What I am insisting here is that the strategy works for

natural kind discourse. The difference between natural kind discourse and daily discourse can be boiled

down to one thing: there is uniform “+1” or “x” in a natural-kind discourse, but there is no uniform “+1”

in daily discourse. (It should also be noted that by natural kind stance I do not mean that all natural kind is

covered under just one stance. In fact, there are many.) There is agreement on what candidates are

counted as “x” in the question “is x a horse?”, but such agreement in daily discourse may be barred by the

fact that our daily discourses include everything like atoms, planet, molecules, and a buck of water…..

The “x” in daily discourse is never uniform, and our daily predicates may or may not be threatened by the

Sorites argument.

III. The friendship

In this session I argue that the concept of vagueness as construed above can provide us with an argument

for the irreducibility of natural kinds.

The ontological status of natural kinds is a perennial debate, and the center of which lies in the possibility

of reduction of natural kind to basic-level properties. Few people nowadays accept that natural kind terms

denote merely arbitrary or cultural classifications, although how to understand the reality of these natural

kinds remains a puzzle. To the reductionist natural kinds are redundant because their realities are nothing

2 One reason that holds me from committing to the daily stance is that there are studies showing that a lot of

sentences without meaning can be widely recognized as profound, and they may be unhelpful or incapable of

transmitting information. In a study released in early November, researchers from the Canadian University of

Waterloo asked nearly 300 university undergraduates to evaluate a series of statements containing randomly selected,

vague buzzwords on a scale of 1 to 5 as to their profundity. Many of the evaluated statements for the study were

generated by the New Age Bullshit Detector, operated by writer Seb Pearce (http://sebpearce.com/bullshit/). They

are syntactically correct, but are mundane and ambiguous. It turns out that students rate those bullshits as having

great profundity. This at least shows that not everything in the “daily stance” should remain unthreatened by the

Sorties argument. (See Pennycook, 2015)


but the reality of the simples and so these kinds should not occupy an additional seat in our ontological

garden. Irreducibility, therefore, is the desideratum of natural kind realists. But what can vagueness offer?

If what has been said in part II is accepted, this argument is just a corollary:

1. A natural kind predicate “PNK” is reducible to a set of lower-level predicates “PLLs” only if “PNK”

is definable in terms of “PLLs”.

2. For all predicate P and for all possible level L1-Ln, PLn is definable in terms of PL<n iff whether PLn

can apply to an object O is decidable in terms of PL<n .

3. If whether PLn can apply to an object O is decidable in terms of PL<n , then PLn is not vague with

respect to PL<n.

4. There exist some natural kind predicates “PNK” such that “PNK” is vague with respect to “PLLs”.

5. There exist some natural kind predicates “PNK” such that whether “PNK” can apply to an object O

in terms of “PLLs” is undecidable.

6. There exist some natural kind predicates “PNK” such that “PNK” is not definable to “PLLs”.

7. There exist some natural kind predicates “PNK” such that “PNK” is not reducible to “PLLs”.

One may object that if vagueness suffices for a seat in the ontological garden, then the vagueness in our

everyday discourse will commit us to infinitely many daily objects in our ontology, which is absurd. But

the claim here is not that vagueness is sufficient for reality, but that vagueness is a sufficient part for a

necessary condition for something being real, namely irreducibility. Irreducibility is never sufficient for

ontological reality: evil demon is irreducible to any physical matters, but does not show that this is real.

IV. Conclusion

I should end by leaving some caveats: firstly, I talked as if there are physicalist stance, biological kind

stance, and daily stance, but stances do not have to be such individuated. In fact, my individuation is only

to facilitate discussion. Roughly speaking, by “a stance” I refer to a bundle of physical points of

view/measurement. A detailed explanation of stance individuation is out of the scope of this paper.

Secondly, the term “natural kind” is itself vague and needs clarification. Very probably, different natural

kinds will require many different stances to cope with the Sorites argument. If these caveats are taken

seriously, I can declare: the state of war between vagueness and natural kind has ended.

(Word Count: 3495)


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