Abstract
Ever since the famous ‘H2O papers’ of the early forties,1 Carl G. Hempel has been closely associated with the concepts of confirmation and logical probability. His intriguing ‘paradoxes of confirmation’ have brought forth an avalanche of responses, and more keep coming.2 Some of the conditions of adequacy he enunciated, especially the ‘equivalence condition’, are central to current controversies.3 Moreover, he has, in the meantime, continued to make significant contributions to inductive logic,4 as well as to such closely allied topics as the nature of theories and inductive (statistical) explanation.5
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References
Carl G. Hempel, ‘A Purely Syntactical Definition of Confirmation’, The Journal of Symbolic Logic 8 (1943), 122–143; ‘Studies in the Logic of Confirmation’, Mind 54 (1945), 1–26; 97–120 (reprinted, with some changes and with a `Postscript’, in Carl G. Hempel, Aspects of Scientific Explanation, The Free Press, New York, 1965); Carl G. Hempel and Paul Oppenheim, `A Definition of “Degree of Confirmation”’,Philosophy of Science 12 (1945), 98–115; Olaf Helmer and Paul Oppenheim, ‘A Syntactical Definition of Probability and Degree of Confirmation’, The Journal of Symbolic Logic 10 (1945), 25–60.
For citations to some of the more important articles see footnote 1 to the ‘Postscript’ in Aspects of Scientific Explanation, op. cit.,p. 47.
See, for example, John R. Wallace, ‘Goodman, Logic, Induction’, The Journal of Philosophy 63 (1966), 310–328; Marsha P. Hanen, `Goodman, Wallace, and the Equivalence Condition’, The Journal of Philosophy 64 (1967), 271–280; Howard Smokier, `The Equivalence Condition’, American Philosophical Quarterly 5 (1967), 300–307.
Inductive Inconsistencies’, Synthese 12 (1960), 439–469; reprinted with slight changes in Aspects of Scientific Explanation, op. cit.; and `Recent Problems of Induction’, in Mind and Cosmos (ed. by Robert Colodny ), University of Pittsburgh Press, Pittsburgh, 1966.
See all of the papers in Parts III and IV of Aspects of Scientific Explanation, op. cit.,and also ‘Deductive-Nomological vs. Statistical Explanation’, in Minnesota Studies in the Philosophy of Science,vol. III (ed. by Herbert Feigl and Grover Maxwell), University of Minnesota Press, Minneapolis, 1962.
See ‘Postscript’ in Aspects of Scientific Explanation, op. cit.,p. 50.
See ‘On Vindicating Induction’, in Induction: Some Current Issues (ed. by Henry E. Kyburg, Jr., and Ernest Nagel), Wesleyan University Press, Middletown, Conn., 1963, and in Philosophy of Science 30 (1963), 252–261. This article contains a resolution of Goodman’s paradox which I still think to be substantially correct.
This study was initiated by consideration of Mary Shearer, ‘An Investigation of ct’, M.A. dissertation, Wayne State University, 1966. In this thesis, Mrs. Shearer undertook to demonstrate Carnap’s claim that ct does not admit `learning from experience’. In carrying out her proof she used the notion I have introduced below under the name of `complete atomic independence’, but it turns out that essentially the same proposition can be demonstrated very much more simply by using what I call `complete truth-functional independence’. This is a strong motive for introducing the latter as a distinct concept of complete independence.
This point was set out sketchily in my article ‘Carnap’s Inductive Logic’, The Journal of Philosophy 64 (1967), 21.
See ‘Preface to the Second Edition’, Logical Foundations of Probability,University of Chicago Press, Chicago, 1950; 2nd ed., 1962.
A full account can be found in Rudolf Carnap, ibid,Chap. III. I shall sketch a few of the essential points, but the foregoing source should be consulted for greater precision and detail. In one deliberate departure from Carnap’s usage, I shall use the term ‘statement’ in almost all cases in which he would use the term ‘sentence’. I do this to emphasize the fact that we are dealing with interpreted languages and not just uninterpreted strings of symbols. In order to avoid considerable complication of the notation, I frequently use italicized letters and formulas containing italicized letters as names of themselves. I believe the context makes clear in all cases whether such symbols are being mentioned or only used.
See definition D 18–4, ibid.,pp. 78 ff.
Ibid., § 20.
Ibid., § 55B.
Throughout this discussion I shall make free use of substitution under L-equivalence, for all of the L-concepts are invariant with respect to such replacement, and the logical relations of inductive logic as discussed by Hempel and Carnap are likewise invariant as long as we do not switch from one language LN to another.
Carnap, Logical Foundations of Probability, op. cit., § 55B.
I have expressed both of these notions in very elementary terms in my Logic,Prentice-Hall, Inc., Englewood Cliffs, 1963, pp. 14–15; 98–100.
Yehoshua Bar-Hillel and Rudolf Carnap, ‘Semantic Information’, British Journal for the Philosophy of Science 4 (1953–54), 147–57.
Carnap, Logical Foundations of Probability, op. cit., § 55A.
See Karl R. Popper, The Logic of Scientific Discovery, Basic Books, New York, 1959, §§ 31–35, and Rudolf Carnap. Popper, The Logic of Scientific Discovery, Basic Books, New York, 1959, §§ 31–35, and Rudolf Carnap, `Probability and Content Measure’, in Mind, Matter, and Method (ed. by Paul Feyerabend and Grover Maxwell ), University of Minnesota Press, Minneapolis, 1966.
Exception could be taken to the Carnap-Bar-Hillel analysis of content on the ground that L-implication is unsatisfactory as an explication of entailment. Anderson and Belnap, for example, have worked extensively on alternative concepts of entailment; see `Tautological Entailments’, Philosophical Studies 13 (1962), 9–24. Interesting as these investigations are, they do not seem likely to enable us to salvage the idea that completely independent statements have no common content, for the chief obstacle is the logical principle of addition (p entails p v q) which holds for the stronger entailment relations as well as for Camap’s L-implication. My own feeble efforts to cook up an entailment relation that would not have this property produced nothing plausible. I am grateful to Professor Alan Anderson for pointing out a severe difficulty in one proposal.
See Carnap, Logical Foundations of Probability, op. cit., § 21C, for a full and precise statement of the rules.
John G. Kemeny and Paul Oppenheim, `Degree of Factual Support’, Philosophy of Science 19 (1952), 307–324. See especially CA1 1 and Theorem 2. Letting t be a tautology, n(t) becomes the total number of state descriptions. Kemeny and Oppenheim show that zero correlation obtains iff which is shown in footnote 24 to be equivalent to our definition.
Given that we have Adding n(p q) n(p) to both sides yields or, dividing
This fact is proved as follows. Assume Since Subtracting from both sides yields which is equivalent to by commutation of conjunction. Adding Or Dividing by n(q) and n(~ q),neither of which is zero
See Carnap, Logical Foundations of Probability, op. cit.,T 69–1, table lb, p. 378.
We impose this restriction to prevent the denominators of the fractions in the definitions from being zero. This is no serious restriction, since in inductive logic we are concerned with relations among factual statements.
See Carnap, Logical Foundations of Probability, op. cit., §110A,and The Collected Papers of Charles Sanders Peirce,2.744–46.
Chapter VI of Logical Foundations of Probability is devoted to the subject of rele- vance, and it provides considerable illumination for this seriously neglected topic.
Pp. xv-xx.
Though I shall often refer only to inference from past to future, it should be understood that the general problem concerns inference from the observed to the unobserved, regardless of the temporal locations of the events involved.
There are some philosophers, e.g., Anderson and Belnap, op. cit.,who would maintain that we still have not found a fully adequate explication of `entailment’, and that a relation must fulfill additional conditions to deserve the name. That does not matter to the point I am making. They would surely agree that the relation must fulfill at least the conditions I have mentioned, and that is all I need to insist upon.
It would be a mistake, I believe, to conclude that this type of positive relevance must be reflected, in turn, in increased truth frequencies, but that is a separate issue right now. The issue at present concerns inductive relevance where no correlations of truth values or state descriptions obtain.
See Logical Foundations of Probability, op. cit.,110A, for the definition of c*. Although Carnap no longer regards c* as fully adequate, he does maintain that it is a good approximation in many circumstances. In any case, it illustrates nicely the point about relevance under discussion here.
I have discussed the problem of the force of the term ‘rational’ in inductive logic in several papers, especially in the lead paper and reply to S. Barker and H. Kyburg in `Symposium on Inductive Evidence’, American Philosophical Quarterly 2 (1965), 1–16.
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Salmon, W.C. (1969). Partial Entailment as a Basis for Inductive Logic. In: Rescher, N. (eds) Essays in Honor of Carl G. Hempel. Synthese Library, vol 24. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1466-2_4
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