Species, Determinates and Natural Kinds
Author(s): Richmond H. Thomason
Source: Noûs, Vol. 3, No. 1 (Feb., 1969), pp. 95-101
Published by: Blackwell Publishing
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Species,Determinatesand Natural Kinds?
RICHMOND H. THOMASON
YALE UNIVERSITY
1. In an article appearing in this Journal,2 John Woods offers
an account of certain classificatory notions. His analysis uses a
familiar technique: presentation of a truth-functional definition,
which is then modified in an ad hoc manner to meet various
counterexamples. In recent years, the fruits of this technique have
been subjected to criticism of the most devastating sort. Woods'
proposals are no exception to this, and in fact abound in crippling
flaws. Below, I will list some of these, paying most attention to his
account of the relation of species to genus. In a concluding section
I will make some suggestions concerning a more suitable approach
to these problems.
2. Woods' analysis of the determinate-determinable relation
suffers from the defect that if Fx is a determinate of Gx, then Gx
is a theorem of the predicate calculus. This follows at once from his
condition 2*), which is built into all his later reformulations. According to 2*), if Fx is a determinate of Gx then "if there is a third
term Px, distinct from Fx and Gx, such that (Gx Px) entails Fx,
then Px entails Gx and Px entails Fx".3 Where Fx and Gx are any
terms, let Px be ( .-Gx V Fx). Clearly, (.. Gx V Fx) is distinct
from both Gx and Fx, and (Gx Px) F--Fx.4However, Px H--Gxif
1 This research was supported under National Science Foundation Grant
GS1567. Much of what is positive in this paper arose during discussion with
ProfessorB. van Fraassen.
2 THIS JOURNAL, vol. 1 (1967), pp. 243-254.
3 I have used Woods' notation without clarifying it, though it is vague
on many points; e.g., it is not clear whether 'Fx differs from Gx' means that
Fx and Gx differ semantically or syntactically. Niceties such as this, however,
do not affect our counterexamples.
4 That is, Fx is deducible from Gx Px in the two-valued predicate calculus of first order. Woods makes it clear (p. 250) that this is the sense in
95
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and only if P-Gx. This establshes that if Fx is a determinate of
Gx, then F--Gx.
3. Similar reasoning exposes many flaws in Woods' treatment
of species and genus. For instance his second and third conditions
for Fx being a species of Gx are, under ordinary circumstances,
equivalent to the first. In particular, 2) is equivalent to 1) if not
[--Gx and not tGx [ Fx; and 3) is equivalent to 2) unless F--Gx.
Condition 4), however, is not equivalent to 1). It follows from
4) that Fx is a species of Gx only if F- Gx. According to 4), if Fx is a
species of Gx then for any conjunction Hlx . . . Hnx such that
1- Fx
(H1x . . . Hn,x) it is the case that Hix F-Gx for all i. Consider the conjunction ( (Fx V Gx) *(Fx V .Gx) ); 1 Fx
((Fx V
Gx) - (Fx V Gx) ), but if (Fx V Gx) F-Gx, then F- Gx.
Condition 5), as it stands, is satisfied by no terms Fx and Gx
whatsoever, unless Fx is inconsistent. For, if Fx is consistent, then
according to 5) no conjuncts of any conjunction to which Fx is
equivalent can stand in proper entailments. But, where Hx and Jx
are any atomic terms, Fx is equivalent to (Fx- (Fx V Hx)- (Fx V
Hx V Ix)); and (Fx V Hx) I-(Fx V Hx V Jx), and not (Fx V
-
.
Hx V Jx) - (Fx V Hx).
Conditions 6) and 7) are subject to similar objections. Moreover, 7) is phrased in such a way that it never is the case that Fx
is a species of Gx. 7) implies that Ex is a species of Gx only if
whenever F-Fx
(Hx V ... - V Hnx), it is not the case for any i
that both Hix F-Gx and Gx does not entail Hix. But consider (Fx.
Gx) V (Fx-, Gx); F-Fx ((Fx -Gx) V (Fx- ..-Gx)), but (Fx
Gx) F-Gx. Thus, if Fx is a species of Gx, then Gx entails Fx, which
is impossible, since 7) also implies that if Fx is a species of Gx then
Gx does not entail that Fx.
4. I think many philosophers will agree with me that the
remedy for symptoms such as this is not further patching of the
original account.5 There is of course no proof of this, but after
enough counterexamples one begins to suspect that any attempt
which he is using "entails";however, our counterexamples apply even if 'Fx
entails Gx' is taken to mean that Fx D Gx is a necessary truth.
5 Nor would it do to say that only primitive terms of English can be
used in conditions 1) -7)
and 1*) - 7* ), since it is as inappropriate to
speak of the primitive terms of a natural language as to talk about the startingpoints in California.Relative to a particularformalization,English may be said
to possess primitive terms, just as relative to a particular journey California
may be said to have starting-points. But specifying a formalization which will
give reasonable results in connection with Woods' conditions is, to say the
least, an unpromising task.
SPECIES, DETERMINATES
AND NATURAL
KINDS
97
to find an "analysis"will suffer from like difficulties.Rather than
attempting to characterizephilosophicallyproblematic notions by
defining them in terms of other notions taken to be less problematic, it has proved more fruitful to search for an abstract,structural characterizationof such notions. Cases in which this approach
has been especially rewarding are knowledge-claimsand the subjunctive conditional.6
If the notions of species and genus are to be approachedin
this way, algebraic techniques seem most promising, since the
relation of species to genus presents an overt mathematicalstructure of a sort familiarto algebraists.7We can thus draw on mathematical material in presenting an abstract account of this relation.
5. If a and b are sorts standing in the relation of species to
genus, they cannotbe any sortswhatsoever;t-houghman and mouse
can be species of some genus, man-or-mousecannot. The term
"naturalkind"has been used to refer to sorts which can be species
or genera; using this terminology,we will say that if a is a species
of b, then both a and b are natural kinds.
By a taxonomtc system S, we mean a structurerepresentinga
system of classification.A taxonomicsystem S will consist of a set
of elements-the natural kinds of S-and a relation < on these
elements-the relation of species to genus. We will stipulate that
< is reflexive,transitive,and antisymmetric,so that for all natural
kinds a, b, and c of S, a < a, if a < b and b < c then a < c, and
if a < b and b < a then a -b.
We have a right to expect of any system of classificationthat
if a and b are naturalkinds of the system,then there will in general
exist a least naturalkind a U b of the system such that a < a U b
and b < a U b. For instance, if a is man and b is clam, a U b will
be animal; if a is man and b is porpotse, a U b will be mammal.
This will hold unless a and b are ultimate categoriesof the system,
6 For the former, see Hintikka's Knowledge and belief (Ithaca: Cornell
University Press, 1963); for the latter, see Stalnaker's "A theory of conditionals", Studies in logical theory (American Philosophical Quarterly supplementary volume), N. Rescher, ed. (1968).
7A less abstract, but equally valid, approach would use the theory of
trees to explain taxonomic structures. On this approach a taxonomic system
would be a tree, and a would be a successor of b in such a tree if and only if
a is a direct species of b.
8 Some people may prefer to start with the relation aSb which holds if
and only if a is a direct species of b. (Sapiens is a direct species of homo, but
not of anthropoidea.) To obtain our relation < from S let R be the least transitive relation containing S, and let a < b if and only if aRb or a = b.
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so that there is no c whatsoeversuch that a + c and a < c, or b + c
and b < c.
In order to obtain a mathematicallytidier structure,it is convenient to assume that every taxonomicsystem S containsa largest
element V, so that for every natural kind a of S, a < V. (This
does not mean that we need to abandon the notion of an ultimate
category of S; these will now be those natural kinds a such that
a+V and there is no b such that a+b and b+V and a <
b < V.)
Having postulated the existence of a universal kind V, we
can accept in full generality the principle that for all a and b in
S, there is a least upper bound a U b of a and b in S. In algebraic
terms, this means that any taxonomic system is an upper semilattice.9
Taxonomicsystems are characterizedby a propertywhich is
not in general possessed by semilattices:
(D) No naturalkinds a and b of a taxonomicsystem overlap
unless a < b or b < a.
The principle D of disjointnessholds because the natural kinds of
a system of classificationmay be conceived of as obtained by a
process of division. The universe is first divided into disjoint sorts
(e.g., animal, vegetable, and mineral), then these are further divided into disjoint sorts, and so forth.
These divisionsneed not in generalbe regardedas exhaustive;
everything, for instance, needn't fall under one of the categories
animal,vegetable, or mineral.It sometimeshappens that things are
discoveredwhich can lay claim to membershipin sorts supposed to
be disjoint:for instance, microbes which appear to be both animal
and vegetable. I would prefer to regard such anomalouscases as
not falling under the original scheme-e.g., as neither animal nor
vegetable-thus preservingthe principle D.
If we postulatethat every taxonomicsystem should contain an
empty element A such that for all a, A < a, we can introduce a
greatest lower bound operator;the greatest lower bound a n b of
naturalkinds a and b is the most inclusive natural kind contained
in both a and b. In view of the presence of both operations U and
nf, we can regard every taxonomicsystem as a lattice. It follows
9A standard reference work on semilattices and lattices is Birkhoff's
Lattice theory, rev. ed. (Providence, 1948).
SPECIIES, DETERMINATES
AND NATURAL
KINDS
99
from the principle D that a n b will either be a or b or A. In fact,
this is a moreformalway of expressingD:
(D') a n b-A
or a n b=-a or a n b -b.
Our proposal thus far is that taxonomicsystems be regarded
as lattices with universal and empty elements, and satisfying D'.
It is easy to show that any such lattice will be modular: i.e., will
satisfy the rule
If a<b
then a U(c n b)
=
(a U c)fn b.
However, taxonomicsystems cannot in general be regarded as distributive lattices; it is not difficultto find naturalkinds a, b, and c
which do not satisfy the identity
a U(b n c) - (a U b) n (a U c).
For instance, man U (mouse n beetle) = man. But (man U mouse)
n (man u beetle) - mammal n animal - mammal.
If these suggestions are correct, the study of taxonomicsystems, as characterizedabove, may help to shed light on the notion
of a natural kind, and on the relation of species to genus. It may
be that there are further properties (e.g., finite chain conditions)
that can plausibly be added to the ones listed above. And further
mathematicalinvestigation of such systems may yield worthwhile
information.It would be interestingto know, for instance, whether
any taxonomic system can be represented as a scheme of classes
obtained by a method of division of the sort indicated above.
6. From a philosophic point of view, however, perhaps the
most promisingway of developing these ideas would be to study
the relationships of taxonomic systems with familiar systems of
logic. To indicate how this would work, I will give one example.
For Aristotle, natural kinds enter into the essence of things
and so give rise to necessary truths. Using modern semantical notions, we can also make sense of this by saying that natural kinds
condition the identificationof things acrosspossible worlds. In particular,let F be a predicate expressinga naturalkind, and suppose
that Fx is true in a possible world a and that P is possible with
respect to a. Now, since F expressesa naturalkind, possessionof F
will be used as a means of identifying a thing in other situations;
for instance, what is a man in one situation cannot be a zebra in
others. Thus, Fx must be true in P as well as a, since otherwise x
would not denote the same thing in a as in P3.This means that
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(x) (Fx D [IFx) is true; predicates expressingnaturalkinds have
the property that anything possessing them possesses them necessarily. This characteristicof natural kinds is one which helps to
differentiate them from arbitrary properties; it is not true, for
instance, that if anything is red it is necessarily red. Some properties having this characteristic,however, are not natural kinds:
the property of being a man or a mouse, for instance.
In order to make these ideas precise, it is necessary to make
sense of the notion of a propertyexpressinga naturalkind. We can
do this within the semanticsof modal logic by building a taxonomic
system into each model of a theory.
Ordinarily,a model structurefor first-ordermodal logic consists of a set X( of possible worlds, a relation ?I of relative possibility on (, and a domainD of individuals.(These individualsmay or
may not exist in the various worlds of the structure;this, however,
is not relevant to our present purposes.)10By a propertyon such a
model structure,we understanda function taking members of q(
into subsets of D; a propertyis a rule which gives, in any situation,
the set of individuals satisfying the property. The properties of a
model structureconstitute a Boolean algebra. In particular,P c Q
if and only if for all ad7(, P(a) c Q(a); the universalelement of
the algebra is the property 1 such that for all aEl, d 1(a) for
all d e D, and the empty element is the property0 such that for all
a e , di 0(a) for all ace (.
Let S be a taxonomic system whose elements are properties
of a model structure.S is said to be embedded in the model structure in case V=l, A=0, and P< Q if and only if P C Q for
all natural kinds P and Q of S. A taxonomicsystem embedded in
this way in a model structuregives a means of telling which properties of the model structureare naturalkinds.
By a model structurewith taxonomy,we understanda model
structure toget-herwith a taxonomic system S embedded in the
model structure,such t-hatfor all d e D, natural kinds P of S, and
a c >,if d e P(a) then for all (3e '(, d e P(P(). In other words,natural
kinds are constant functions taking possible worlds into sets of individuals;if P is a naturalkind, then P(a) =P(P) for all a, (E3 .
An interpretationof a theory on a model structurewith taxonomy will assign each singulary predicate letter P of the theory
10For background on the semantics of first-order modal logic, see the
author's "Modal logic and metaphysics," The logical way of doing things,
K. Lambert, ed. (New Haven: Yale University Press, 1969).
SPECIES,
DETERMINATES
AND NATURAL
KINDS
101
a predicateP of the model structure.And we have arrangedthings
so that if I assigns P a predicate which is a natural kind, then
(x) (Px
Px) wl be made trueby L.
7. This account treats natural kinds as semantic rather than
syntactic entities;althoughtaxonomicsystemsplay a role in models,
there are no correlates of the operators U and n in the formal
language. For first-orderlogic this approachavoids many needless
complicationsand is, I think, much to be preferred.Natural kinds
can then be allowed to appearat the syntacticlevel in second-order
logic, as predicates meeting certain conditions.
At the first-orderlevel, however, there remain many refinements of our semantic approach. For instance, natural kinds may
be helpful in explaining how analogies are supported; also, it is
likely that they play an importantrole in inductive reasoning.
I suspect that only by investigating in this way the part
played by notions such as species and genus in various areas of
reasoning, will we begin to obtain a robust philosophic understanding of these concepts. Frameworknotions of this kind lie too
deep to be exposed by a superficial"analysis".