¹ See Scheffler, Israel and Goodman, Nelson, “Selective Confirmation and the Ravens: A Reply to Foster”, The Journal of Philosophy, vol. 69, pp. 78–83CrossRefGoogle Scholar. Also, see Goodman's, Fact, Fiction, and Forecast, Harvard University Press, Cambridge, Mass., 1955, pp. 71 ffGoogle Scholar. and Scheffier's, The Anatomy of Inquiry, Alfred Knopf, New York. 1963, pp. 286 ff.Google Scholar

² See Hempel, Carl G., “A Purely Syntactical Definition of Confirmation”, The Journal of Symbolic Logic, vol. 8, 1943, pp. 122–143.CrossRefGoogle Scholar

³ See Hempel ibid, and Hempel, “Studies in the Logic of Confirmation”, Mind, vol. 54.(1), pp. 1–26 and (II) pp. 97–121.Google Scholar

⁴ Throughout, sentences of the form ‘All F are G’ called A-form categorical statements, are interpreted in the Boolean way: Thus ‘All F are G’ just means ‘Everything is either nonF or G’. The terms in place of ‘F’ and ‘G’ in an A-form categorical statement are called its subject and predicate terms respectively.

A statement is logically contingent if and only if it is neither logically true nor logically false.

⁵ Throughout, I use ‘R’ and ‘B’ to represent the appropriate grammatical forms of ‘raven’ and ‘black’ respectively.

⁶ The ‘not logically false’ clause is present for the case where ‘disconfirms’; means ‘makes infirmer’ presumably nothing makes a logically false statement infirmer.

⁷ This will happen whenever ‘confirms’ and ‘disconfirms’ satisfy the postulate: For any E, H, and H₁. if E confirms H and H₁ is a contrary of H given E that is not logically false, E disconfirms H₁. Hempel's ‘confirms’ and ‘disconfirms’ satisfy this postulate.

⁸ H₁ is an analogous contrary of H relative to E if and only if H₁ is a contrary of H relative to E and H₁ is as definite as H relative to E. The notion of equal relative definiteness is, of course, unclear. But for hypotheses of a specified form, it can often be clarified in an ad hoc way.

⁹ I am assuming here that the proper way to construe Hempel's explicandum ‘confirms’ would be to take it as conveying ‘rational to believe to at least some small, fixed positive level of rationality of belief’. Scheffler's and Goodman's explicandum for ‘selectively confirms’ can then be read off their explication as is done in D6 above. (In D6, I intend ‘rational to believe’ to be short for ‘rational to believe to at least some small, fixed level of rationality of belief’.)

¹⁰ It is not obvious that Scheffler and Goodman are right. That is, it is not clear that there is a difference in the scientific reasonableness of predicting ‘Bb’ on the basis of ‘Not-Ra and not-Ba and Rb’ and predicting ‘Not-Rh’ on the basis of ‘Not-Ra and not-Ba and not-Bb’.

To say this is to go against a piece of conventional wisdom. For it is commonly assumed, in the absence of background assumptions, that predictions based on observed cases must be like the cases observed to be reasonable predictions given such cases. And indeed no one would say, in the absence of background assumptions, that ‘Not-Rc’ on the basis of ‘Ra and Rb’ is more reasonable than ‘Re’ on the same basis. Scheffler and Goodman, however, think that ‘Bb’ on the basis of ‘Not-Ra and not-Ba and Rb’ is not reasonable while ‘Not- Rb’ on the basis of ‘Not-Ra and not-Ba and not-Bfe’ is. But this is not clear. Consider a different but parallel case. Again no background assumptions are to be made — a stricture presumably compatible with the testing of ‘All R are B’

It is clear that one can test ‘All R are B’ by selecting non Bs and checking each for R or not. The class of non Bs contains all the refuting objects there are, if any, for ‘All R are B’. Selecting a nonB and checking it for R or not thus exposes ‘All R are B’ to falsification. Suppose n (n > 1) nonB are checked and each is nonR. This would constitute a test of ‘All R are B’ that ‘All R are B’ passed. Moreover, since by hypothesis nothing is known of the comparative sizes of the class of Rs and the class of nonBs—classes that contain all the refuting objects, if any, for ‘All R are B’ — and since nothing is known as to which proper sub-classes of these classes contain refuting objects, if any, for ‘All R are B’, it is plain that given no background assumptions, the above test is as severe a test of ‘All R are B’ as any other that could be devised involving n test subjects. If n were very large, one ought to be prepared to say that ‘All R are B’ is firm given the test results, for the test results point to the non-existence of refuting objects for ‘All R are B’ and hence point to the truth of ‘All R are B’ and all logical equivalents.

Now suppose we should come across an object b, not one of the n objects examined as part of the test. We ought to be prepared on the basis of test results to predict that b too is not a refuting object for ‘AH R are B’. But if b happened to be observed to be a R (nonB), we ought to be prepared to predict nonetheless that b is not a refuting object for ‘All R are B’ and so predict that b is B (nonR) on the basis of the test-results and the new observation. In all of this, we should not think that one of these predictions is not reasonable given its basis while the other is reasonable given its. We would assess the reasonableness of the predictions on their respective bases in much the same way.

If now we turn to the situation dealt with by Scheffler and Goodman, we can view ‘Not-Ra and not-Ba’ as resulting from a test of ‘All R are B’ wherein we selected a nonB and checked it for R or not — a test of ‘All R are B’ that ‘All R are B’ passed and a test of ‘All R are B’ as severe as any other involving just one test object that can be devised given what is known. As in the preceding example, why would not ‘Bb’ on the basis of ‘Not-Ra and not-Ba and Rb’ be as reasonable as ‘Not-Rft’ on the basis of ‘Not-Ra and not-Ba and not-Bb’? Would reducing the sample from n to I, and hence reducing the bases at issue, likely create a differential in relative reasonableness of prediction? The answer ‘Yes’ does not look very promising.

¹¹ I am here using quotation marks as substitutes for Quine's corners. See Quine's, Mathematical Logic, Harvard University Press, Cambridge, Mass., revised edition, 1951.Google Scholar

¹² These remarks on unwanted interpretations of ‘All R are B’ will strike some as arbitrary. But it is futile to try to determine the truth or falsity of (2:), (6), (7) or (8) unless some definite interpretation of A-form categorical hypotheses is assumed. There is no reason to suppose that what confirms an A-form categorical hypothesis construed one way also confirms it construed another way. The Boolean interpretation is a customary one, and one that is clear and has a well understood logic. Besides, Hempel posed the problems associated with the paradoxes of the ravens for A-form categorical hypotheses under the Boolean interpretation. More to the point, however, is the fact that the paradoxes arise only for A-form categorical hypotheses construed in such a way that they are logically equivalent to their contrapositives and ‘universal subject transforms’. Thus ‘All R are B’ must be logically equivalent to ‘All nonB are nonR’ and ‘All R or nonR are either nonR or B’. Such logical equivalences partially fix the interpretation of the A-form categoricals that generate them. Indeed, only the Boolean and lawlike interpretations, among the many customary or generally contemplated interpretations of A-form categoricals, generate the requisite logical equivalences. Other interpretations don't, and so don't give rise to the paradoxes. In this sense, the other interpretations are innocent of the paradoxes. Moreover, such interpretations always lead to trivial solutions of the paradoxes, for given any such interpretation the derivation of the paradoxical results from (1') and (2') turns out to be invalid, i.e., based on mistaken assumptions as to what is logically equivalent to the A-form categoricals thus interpreted. Moreover, the main problems are left unsolved by appealing to any such interpretation, for there are customary interpretations of A-form categoricals that generate the requisite logical equivalences, and so the paradoxes are validly derivable from (1') and (2') for A-form categoricals interpreted in some customary ways.

¹³ That ‘confirms’ has the two senses ‘makes firm’ and ‘makes firmer’ was emphasized by Carnap, in the second edition of his Logical Foundations of Probability, University of Chicago Press. Chicago, 1960Google Scholar. The truth functions referred to are mainly conjunctions and disjunctions. See Rescher, Nicholas, “Theory of Evidence”, Philosophy of Science, vol. 25. 1958, pp. 83–94CrossRefGoogle Scholar and Vincent, R. H., “Corroboration and Probability”, Dialogue, vol. 2. 1963, pp. 194–205.CrossRefGoogle Scholar

¹⁴ For, if H₁ is logically equivalent to H₂, then P(H₁ or not-H₁/E) = P(H₂ or not-H₁/E) = 1 by the implication axiom. Hence, P(H₁/E) = P(H₂/E) by the special disjunction and implication axioms.

¹⁵ If E1 is equivalent to E2 and if T is a logical truth, then by the equivalence condition for hypotheses and the general conjunction axiom, P(E1 and H/T) = P(E2 and H/T) = P(E1/T) x P(H/T and E1) = P(E2/T) x P(H/T and E2). Hence, the equivalence condition for evidence statements, given merely that P(H/T and E) = P(H/E) for any E and H and logical truth T.

¹⁶ The view here contemplated reflects one interpretation only of the potency or agency dimension of ‘makes’ in ‘E makes H firm’. Another possible, though less plausible, interpretation would support D9. For. even if P(H/T) > n, still the addition of E to T could result in P(H/E) < n. Since P(H/E) > n, E is ‘responsible’ for the fact that P(H/E) is not equal to or less than n and so may be said to make H firm.

¹⁷ The expression ‘E1 is logically independent of E2’ has as part of its meaning ‘Neither E1 nor E2 logically implies the other’ but is logically stronger than this. The basic idea underlying ‘logical independence’ as here used is the following: E1 is logically independent of E2 if and only if E1 and E2 have no content in common. An explication of ‘logical independence’ in this sense can be given following Kemeny. Where the function m, from statements in a language with a finite universe of discourse to numbers, measures the ratio of the number of models for the language in which the statement holds to the total number of models for the language (i.e., m is Carnap's m function that assigns equal numerical values to all state-descriptions), E1 can be said to be logically independent of E2 if and only if m(E1 and E2) = m(E1) x m(E2). See Kemeny, J. G., “A Logical Measure Function”, The Journal of Symbolic Logic, vol. 18, 1953, pp. 289–308.CrossRefGoogle Scholar

¹⁸ I omit quotation marks within expressions of the form ‘P(…/---)’.

¹⁹ See footnote 17 for remarks on ‘logical independence’. In applying D12 in the case of the observation reports ‘Ra and Ba’, ‘Not-Ra and Ba’, etc., I assume that where E is ‘Not-Rfl or Ba’, ‘Not-Ra’ or ‘Ba’, there are no R and A as per D12. Where E is ‘Ra and Ba’, Not-Ra and Ba’, or ‘Not-Ra and not-Ba’, I assume there are. In the case of ‘Ra and Ba’, R and A could be ‘Ra’ and ‘Ba’ respectively or ‘Ba’ and Ra’ respectively. I also assume, ignoring logical equivalences, that for ‘Ra and Ba’, these are the only R and A as per D12. Where E is ‘Not-Ra and Ba’ or ‘Not-Ra and not-Ba’, I make analogous assumptions.

²⁰ A direct test of ‘All R are B’ with a as test object is an arrangement featuring (i) a test antecedent (condition or property) A that is not logically empty and is possibly logically universal and that is such that any refuting object (nonblack raven) there might be could in logic satisfy A, (ii) a set O of two test outcomes that meet the condition of being pairwise logically contradictory given A and individually not logically implied or logically excluded by A, and such that in 0 there is one “refuting” outcome that in conjunction with A logically implies the refuting condition of being a nonblack raven and one “nonrefuting” outcome that in conjunction with A logically implies the nonrefuting condition of being a nonraven or black, (iii) the selection of the object a meeting the test antecedent A, and (iv) the observing of a to see which member of 0 a satisfies. The test observation report is given by a statement to the effect that a is both A and 0', where 0' is the test outcome from 0 that is satisfied by a. The statement saying that a is 0' is the test result. One further condition is essential, (v) If the test occurs at time T and K is the logically contingent total knowledge at T, the conjunction of K and a statement to the effect that a satisfies A must not logically exclude that a satisfies either test outcome in 0 and must not logically imply or logically exclude ‘All R are B’ either. The limitation in (ii) to two test outcomes is to avoid certain complications that arise if the class of test outcomes contains two or more nonrefuting outcomes.

²¹ J. W. N. Watkins has claimed, however, that “…if we suppose no background information at all… the notion of a test becomes empty. Without any indications about where to look for likely counter-evidence all undirected looking equally “tests” a scientific hypothesis: anything, anywhere may be a raven which is not black, and the discovery that something is, after all, a non-raven will always constitute the favourable outcome of a pseudotest.” See Watkin's, “Confirmation, the Paradoxes, and Positivism”, in Bunge, M., editor. The Critical Approach to Science and Philosophy. New York. The Free Press, 1964, pp. 92–115.Google Scholar

Watkin's view here is surprising. Why should not an alleged test conducted in the presence of no logically contingent background information at all, with a logically universal test antecedent and test outcomes as indicated above in footnote 20. one of which is a refuting condition for the hypothesis, be a weak but genuine test, rather than an empty or pseudo test? After all, selecting an object a that is self-identical and checking it for R-and-nonB or not does expose ‘All R are B’ to falsification. If I select three things, a, b, and c, each, naturally, self-identical, and check the first for the presence or absence of kidneys, and the second and third for the presence or absence of R-and-nonB, with the results that a has kidneys, h is R and nonB, and c is nonR or B, it is not unreasonable to suppose that in the last two cases ‘All R are B’ has been genuinely tested and in the first case not. Moreover, ‘All R are B’ has failed one of the two tests while passing the other. Not all undirected looking equally tests ‘All R are B’. Examining a for the presence or absence of kidneys certainly does not. But that any examination of any object, given no logically contingent background information and a logically universal test antecedent, for the presence or absence of R-and-nonB does genuinely, if weakly, test ‘All R are B’ seems simply incontrovertible.

²² See Hempel's, listing of those who have put forward a class-size explanation in his Aspects of Scientific Explanation, The Free Press, New York, 1965Google Scholar. There are many different variations of the class-size explanation. The variations differ in point of the content of the background assumption invoked, which paradoxical confirmation statements are to be explained, how general the explanation is intended to be, and whether or not further paradoxes are generated by the explanation. (See “Relevance and the Ravens” by Hooker, C.A. and Stove, D.C. in The British Journal for the Philosophy of Science, vol. 18, 1967, pp. 305–315CrossRefGoogle Scholar for a critique of the H.G. Alexander — J.L. Mackie class-size explanation along the line that further paradox is generated by the explanation)

²³ ‘Ra and Ba’ cannot be viewed as resulting from a nontest of ‘All R are B’. For even if being B is the test antecedent and being R and being non-R the test outcomes, still if a is B, i.e., a satisfies the test antecedent, this by itself means that ‘All R are B’ has in effect passed another test, viz., one with a logically universal test antecedent and test outcomes of being R-and-nonB and being nonR-or-B.

²⁴ That is, if T 1 and T2 are two tests of ‘All R are B’, both with a as test object, and with test antecedents Ai and A2 and sets, O1 and O2, of two outcomes with one refuting condition Oil in O1 and one, 0r2 in O2, Ti is more severe than T2 given K if and only if the probability that a satisfies Ori given that a is A 1 and K and some Rs are not Bs is greater than the probability that a satisfies 0r2 given that a is A 2 and K and some Rs are not Bs.

²⁵ The qualifications needed are four in number. First, ‘confirms’ is to be construed as a generalization of D13. Second, the test results must say that a is 0', where 0' is the nonrefuting test outcome of the test, not something logically stronger, and so test observation reports must say that a is both A and 0'. Moreover, if the statement that a is A is not logically true, the statements that a is A and that a is 0' must be logically independent. Third, dicta (ii) requires the preamble ‘If P(H/K and a is A1) and P(H/K and a is A2) are small and close to one another’, where A1 is the test antecedent of T1 and A2 of T2. Fourth, extent of confirmation is to be measured as follows: If the statement that a is logically true, the extent to which E confirms H given K is the quotient P(H/K and E)/P(H/K). If not, the extent to which E confirms H given K is the larger of: The preceding quotient or the quotient P(H/K and R and A)/P(H/K and A) where A is the statement that a is A and R the statement that a is 0'. Given these qualifications, it is easy to show that dicta (i) — (iii) are demonstrable within the elementary theory of sentential probability.

The demonstration makes use of the symmetry or inverse theorem given which P(H/K and E)/P(H/K) = P(E/K and H)/P(E/K) if the conjunctions of K. and H and of K and E are not logically false. Dicta (i)is true, for the conjunction of K, H, and the statement that a is A logically implies the statement that a is 0', whereas K and the statement that a is A by themselves do not. Dicta (ii) and (iii) are true, for the quotients that determine the extent to which E confirms H given K. transformed via the symmetry or inverse theorem, can be shown to have the form:

where, as indicated above, 0' is the nonrefuting test outcome of the test in question. Since the severity of a test equals P(a is not 0'/K and not-H and a is A), dicta (iii) is obvious. So is dicta (ii) if the above quotient is instantiated for the cases of E1 and E2 and the resulting two quotients compared.