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Species, Determinates and Natural Kinds Author(s): Richmond H. Thomason Source: Noûs, Vol. 3, No. 1 (Feb., 1969), pp. 95-101 Published by: Wiley Stable URL: http://www.jstor.org/stable/2216160 . Accessed: 14/06/2014 07:10 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Wiley is collaborating with JSTOR to digitize, preserve and extend access to Noûs. http://www.jstor.org This content downloaded from 188.72.127.63 on Sat, 14 Jun 2014 07:10:56 AM All use subject to JSTOR Terms and Conditions

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Page 1: Species, Determinates and Natural Kinds

Species, Determinates and Natural KindsAuthor(s): Richmond H. ThomasonSource: Noûs, Vol. 3, No. 1 (Feb., 1969), pp. 95-101Published by: WileyStable URL: http://www.jstor.org/stable/2216160 .

Accessed: 14/06/2014 07:10

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Wiley is collaborating with JSTOR to digitize, preserve and extend access to Noûs.

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Page 2: Species, Determinates and Natural Kinds

Species, Determinates and 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. Ac- cording 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.4 However, Px H--Gx if

1 This research was supported under National Science Foundation Grant GS1567. Much of what is positive in this paper arose during discussion with Professor B. 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 cal- culus 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. Con- sider 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. More- over, 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 starting- points in California. Relative to a particular formalization, 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.

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to find an "analysis" will suffer from like difficulties. Rather than attempting to characterize philosophically problematic notions by defining them in terms of other notions taken to be less prob- lematic, it has proved more fruitful to search for an abstract, struc- tural characterization of such notions. Cases in which this approach has been especially rewarding are knowledge-claims and the sub- junctive conditional.6

If the notions of species and genus are to be approached in this way, algebraic techniques seem most promising, since the relation of species to genus presents an overt mathematical struc- ture of a sort familiar to algebraists.7 We can thus draw on mathe- matical 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 cannot be any sorts whatsoever; t-hough man and mouse can be species of some genus, man-or-mouse cannot. The term "natural kind" 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 structure representing a system of classification. A taxonomic system 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 classification that if a and b are natural kinds of the system, then there will in general exist a least natural kind 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 categories of 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 condi- tionals", Studies in logical theory (American Philosophical Quarterly supple- mentary 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 transi- tive 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 whatsoever such that a + c and a < c, or b + c and b < c.

In order to obtain a mathematically tidier structure, it is con- venient to assume that every taxonomic system S contains a 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 semilat- tice.9

Taxonomic systems are characterized by a property which is not in general possessed by semilattices:

(D) No natural kinds a and b of a taxonomic system overlap unless a < b or b < a.

The principle D of disjointness holds because the natural kinds of a system of classification may 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 di- vided into disjoint sorts, and so forth.

These divisions need not in general be regarded as exhaustive; everything, for instance, needn't fall under one of the categories animal, vegetable, or mineral. It sometimes happens that things are discovered which can lay claim to membership in sorts supposed to be disjoint: for instance, microbes which appear to be both animal and vegetable. I would prefer to regard such anomalous cases as not falling under the original scheme-e.g., as neither animal nor vegetable-thus preserving the principle D.

If we postulate that every taxonomic system 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 natural kinds 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 taxonomic system as a lattice. It follows

9A standard reference work on semilattices and lattices is Birkhoff's Lattice theory, rev. ed. (Providence, 1948).

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from the principle D that a n b will either be a or b or A . In fact, this is a more formal way of expressing D:

(D') a n b-A or a n b=-a or a n b -b.

Our proposal thus far is that taxonomic systems 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, taxonomic systems cannot in general be regarded as dis- tributive lattices; it is not difficult to find natural kinds 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 taxonomic sys- tems, as characterized above, 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 mathematical investigation of such systems may yield worthwhile information. It would be interesting to 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 promising way 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 no- tions, we can also make sense of this by saying that natural kinds condition the identification of things across possible worlds. In par- ticular, let F be a predicate expressing a natural kind, and suppose that Fx is true in a possible world a and that P is possible with respect to a. Now, since F expresses a natural kind, possession of 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 expressing natural kinds have the property that anything possessing them possesses them neces- sarily. This characteristic of 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 prop- erties 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 property expressing a natural kind. We can do this within the semantics of modal logic by building a taxonomic system into each model of a theory.

Ordinarily, a model structure for first-order modal logic con- sists of a set X( of possible worlds, a relation ?I of relative possibil- ity on (, and a domain D of individuals. (These individuals may or may not exist in the various worlds of the structure; this, however, is not relevant to our present purposes.)10 By a property on such a model structure, we understand a function taking members of q( into subsets of D; a property is a rule which gives, in any situation, the set of individuals satisfying the property. The properties of a model structure constitute a Boolean algebra. In particular, P c Q if and only if for all ad7(, P(a) c Q(a); the universal element 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 property 0 such that for all a e , di 0(a) for all ac e (.

Let S be a taxonomic system whose elements are properties of a model structure. S is said to be embedded in the model struc- ture 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 taxonomic system embedded in this way in a model structure gives a means of telling which prop- erties of the model structure are natural kinds.

By a model structure with taxonomy, we understand a model structure toget-her with a taxonomic system S embedded in the model structure, such t-hat for all d e D, natural kinds P of S, and a c >, if d e P(a) then for all (3 e '(, d e P(P(). In other words, natural kinds are constant functions taking possible worlds into sets of in- dividuals; if P is a natural kind, then P(a) =P(P) for all a, (E3 .

An interpretation of a theory on a model structure with tax- onomy 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).

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a predicate P of the model structure. And we have arranged things so that if I assigns P a predicate which is a natural kind, then (x) (Px Px) wl be made true by L.

7. This account treats natural kinds as semantic rather than syntactic entities; although taxonomic systems play a role in models, there are no correlates of the operators U and n in the formal language. For first-order logic this approach avoids many needless complications and is, I think, much to be preferred. Natural kinds can then be allowed to appear at the syntactic level in second-order logic, as predicates meeting certain conditions.

At the first-order level, however, there remain many refine- ments 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 important role 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 under- standing of these concepts. Framework notions of this kind lie too deep to be exposed by a superficial "analysis".

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