Descriptive Completeness and Inductive Methods

Descriptive Completeness and Inductive MethodsAuthor(s): Keith LehrerSource: The Journal of Symbolic Logic, Vol. 28, No. 2 (Jun., 1963), pp. 157-160Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2271597 .

Accessed: 16/06/2014 13:54

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].

.

Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to TheJournal of Symbolic Logic.

http://www.jstor.org

This content downloaded from 185.44.77.34 on Mon, 16 Jun 2014 13:54:23 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=asl

http://www.jstor.org/stable/2271597?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


THE JOURNAL OF SYMBOLIC LOGIC

Volume 28, Number 2, June 1963

DESCRIPTIVE COMPLETENESS AND INDUCTIVE METHODS

KEITH LEHRER

In a recent paper Professor Wesley Salmonl has shown that a certain class of c-functions, including the c-function c*, described by Rudolph Carnap2 yield incompatible results when applied to the same sentences in two languages systems which, though they have the same individual constants, do not have the same predicates. Each c-function of the class in question is characterized by a parameter A which is a function of the number of Q-predicates in the language system in which the c-function is used.3 Taking c* as representative of this class of c-functions, I shall argue that Professor Salmon's results do not provide a reasonable basis for rejecting such c-functions in favor of others. More specifically, I shall argue (i) that such c-functions yield incompatible results in two languages because not both of the languages are sufficiently complete, (ii) that for any two languages in which such c-functions yield incompatible results there is a rule that will select either the more complete of the two languages or a language that is more complete than either of the two languages, and (iii) that it is impossible for such c-functions to yield incompatible results in two languages that are equally complete.

Professor Salmon asks us to consider a language 2RWB, which contains three predicates, R, W, and B, that are logically exclusive in pairs and exhaustive, a language QR, which contains only R, a language QB, which contains only B, and a language 2C, which contains only C, where C is equivalent to R v B. As an example, consider a fair die with two sides painted red, two sides painted white, and two sides painted blue. Let R designate that the die turns up a red side on a given toss, W a white side, and B a blue side. We may suppose that all the languages have the same individual constants.4

Now let us consider the c* value of a hypothesis hR that predicates R of an individual constant n that designates the nth toss of the die, hB that

Received January 8, 1963. 1 Salmon, [4], pp. 245-264. 2 Carnap, [2], pp. 44-47. B Carnap, [2], pp. 30-32. Carnap originally restricted the choice of primitive predi-

cates to logically independent predicates. Carnap, [1], pp. 122-130 and [2], pp. 10-11. This restriction may be dropped provided we add meaning postulates. We then can define a Q-predicate of a language as a conjunction in which every primitive predicate occurs either negated or unnegated such that the conjunction is consistent with the meaning postulates. This is the method I shall adopt in this paper. Compare, Carnap, [3], also, Kemeny, [5].

4 Salmon, [4], pp. 247-249.

157



158 KEITH LEHRER

predicates B of n, and he that predicates C of n, when the evidence is tautological. Letting t stand for any tautology,

(a) c*(hR, t) in 2RWB =

(b) C*(hR, t) in 2R =

(c) C*(IB, t) in 2RWB =

(d) C*(hB, t) in 2B =

(e) c*(hc, t) in 2RWB =

(f) c*(hC, t) in 2c =j.

It is obvious that these results are incompatible. Furthermore, it is clear that similar incompatible results would be obtained when non-tautological evidence is taken into consideration.5

Professor Salmon concludes that c* and the other c-functions of the kind in question should be rejected in favor of some other kind of c-function.6 But this conclusion is unjustified. For, while it is clear that the results described in (b), (d), and (f) are counterintuitive, it is equally clear that the results described in (a), (c), and (e) are intuitively correct. The results of using c* to calculate probabilities in QR, 2B, and 2C are counterintuitive, but the results of using c* to calculate probabilities in QRWB are not. Since it is clear that the greater descriptive completeness of 2RWB as compared to PR, 2B, and 5C accounts for the difference in the results, the only conclusion that would seem justified is that the use of c* and other c-functions of the kind in question should be restricted by some rule to languages of greater rather than lesser descriptive completeness.7

I propose the following rule for dealing with such cases: THE RULE OF GREATER COMPLETENESS (RGC). For any two languages

2i and Sk having the same individual constants such that two sentences e and h occur in both languages and c(h, e) in 2i # c(h, e) in 2k, select the language QJk to calculate c(h, e) having the same individual constants as

2f and 2k and having as primitive predicates the primitive predicates of both 2j and 2k with the appropriate meaning postulates.

When c* and other c-functions of the kind in question yield contradictory results when applied to the same sentences in different languages, RGC tells us to select either the more complete of the two languages or a language

5 Ibid. 6 Salmon, [4], pp. 248-249. 7 Such a restriction is contained in Carnap, [1], p. 75. Carnap's requirement of

completeness is that a language system must contain predicates that are sufficient for expressing every qualitative attribute of the individuals of the system. It would

be desirable to use such a language, but it does not seem necessary that only such languages be used. For a language need not satisfy Carnap's requirement to yield

intuitively correct results, as our consideration of 23RWB illustrates.



DESCRIPTIVE COMPLETENESS AND INDUCTIVE METHODS 159

more complete than either of the two languages, and, consequently, the results of applying the c-function in a language of greater completeness. Therefore, if RGC is employed, Professor Salmon's argument does not show that c* and other c-functions of the kind in question should be rejected.

The preceding argument depends on the assumption that two languages that have the same predicates and the same individual constants, even if they do not have the same primitive predicates, will not yield incompatible results for c* and other c-functions of the kind in question. I assume as obvious that if two languages have the same Q-predicates and the same individual constants, then the two languages will not yield incompatible results. If two languages have the same primitive predicates, then they have the same Q-predicates. However, having dropped the restriction of primitive predicates to logically independent predicates (See footnote 3), we can see that two languages that have the same predicates need not have the same primitive predicates. Consequently, it is necessary to prove that any two languages that have the same predicates, even though they do not have the same primitive predicates, have the same Q-predicates.

I shall now prove that any two languages that have the same predicates have the same Q-predicates.

DI Languages 2j and 2k are equicomplete =df the set of predicates of

Q% and Sk are equivalent. The concept of two sets of predicates or two predicates being equivalent is not a fully formalized concept. As I am using this concept, two predicates in the same language are equivalent if and only if they are L-equivalent and two predicates in different languages are equivalent if and only if they are logically equivalent in the usual sense. Two sets of predicates are equivalent if and only if every predicate of each set is equivalent to some predicate of the other set.

D2 D1 is a Q-disjunction of 21 =df Dj is a single Q-predicate of 2j or a disjunction of n Q-predicates of ?3 no two of which are equivalent.

Thus we will speak of 1, 2, ... n-membered Q-disjunctions. From D2 and the true premise P1 Every predicate of a language 2i is equivalent (L-equivalent) to a

Q-disjunction of 2i,8

it follows that C1 If Qy and Sk are equicomplete, then every predicate of Qj is equivalent

to some predicate in Sk that is equivalent (L-equivalent) to a Q- disjunction of Sk and vice versa;

and it follows from CI that C2 If Q1 and 2S are equicomplete, then every Q-predicate of 2a is

equivalent to a Q-disjunction of Sk and vice versa.

8 Carnap, [1], p. 126.



160 KEITH LEHRER

Moreover, since the Q-predicates of a language 2 are logically exclusive in pairs (see footnote 3), it follows that

P2 No two Q-predicates of Sk,, Qt. and Qty, are equivalent to two disjunctions of 2*, Dx and Div, such that there is a Q-predicate of Qju of 2, that is a Q-disjunct of both DX and Dly.

The proof of P2 is that if there were a QJ' that was a Q-disjunct of both Dix and D1y, then D>* and D1y would not be mutually exclusive, and, consequently, Qkx and Qkv would not be mutually exclusive either. Since Qtkz and Qkt are mutually exclusive, P2 is true.

We may now prove that C3 If 2; and Qk are equicomplete, then each Q-predicate of 2j is equi-

valent to one Q-predicate of 2k and vice versa, which is a precise formulation of the proposition that two languages that have the same predicates have the same Q-predicates.

Assume that 2i and ,tk are equicomplete and consider any Q-predicate, Q;j, of 2i. From C2 it follows that Qji is equivalent to some Q-disjunction, Dkt, of n (n 2 1) Q-predicates of 2k. It also follows that each of the n Q-predicates of Dkt is equivalent to some Q-disjunction of 21. Moreover, from C2 and P2 it follows that each of the Q-predicates, Qkl, Qk2, * * . QkJ,

of Dkt is equivalent to a Q-disjunction Dji, DJ2, ..., Din, respectively of 2j such that no Q-predicate of 2j is a Q-disjunct of more than one of the latter. Consequently, the disjunction D1, v D12 ... v D1n is a Q-disjunction of 2j of at least n Q-predicates of 2,. But the Q-disjunction DsI v D12 ... vDI. is equivalent to Qjj. Therefore, if n > 1, then Qj' is equivalent to a Q-disjunction of 2j of n (n > 1) Q-predicates of 21. However, since Q-predicates are logically exclusive in pairs, Q;i is not equivalent to a Q-disjunction of n (n > 1) Q-predicates of 2i. Thus it is not the case that n > 1. This proves that any Q-predicate, Q;j, of 21 is equivalent to one Q-predicate of Qkt. By simply interchanging 'j' and 'k' in the preceding proof we could prove that any Q-predicate, Qti, of Sk is equivalent to one Q-predicate of 2i. This completes the proof of C3.

WAYNE STATE UNIVERSITY

REFERENCES

[1] RUDOLF CARNAP, Logical Foundations of Probability, Chicago, 1950. [2] RUDOLF CARNAP, The Continuum of Inductive Methods, Chicago, 1950. [3] RUDOLF CARNAP, Meaning Postulates, Philosophical Studies, vol. III,

pp. 65-73. [4] WESLEY SALMON, Vindication of Induction, in Current Issues in the Philo-

sophy of Science, edited by Herbert Feigl and Grover Maxwell, New York, 1961. [5] JoHN G. KEMENY, Extension of the Methods of Inductive Logic, Philosophical

Studies, vol. III, pp. 38-42.



Documents

Descriptive Completeness and Inductive Methods