¹ The writer wishes to thank Olaf Helmer and John G. Kemeny for constructive criticism.

² P. R. Halmos, “The Foundations of Probability,” American Mathematical Monthly, vol.51 (1944), pp. 493–510.

³ Carnap's Logical Foundations of Probability should be consulted for details and for references to the literature.

⁴ A presumptive factor need not be supporting evidence in the sense to be discussed below; i.e., q may be a presumptive factor for p, although the likelihood of p given q is less than the likelihood of p without q (the a priori likelihood of p). However, it can be shown that this can only happen in the case of statements (p) which are likely a priori, i.e. have a priori likelihood in excess of .5.

⁵ This measure was devised by the writer in the course of a collaborative investigation, with O. Helmer, of evidential reasoning in the social sciences.

⁶ In recent years, considerable attention has been given by logicians to the development of measures of the content of a statement, and of the degree of relevance of one statement to another. Some major contributions to the discussion of content measures are: (1) L. Wittgenstein, Tractatus Logico-Philosophicus (1922), (2) K. R. Popper, Logik der Forschung (1935), K. Hempel and P. Oppenheim, “Studies in the Logic of Confirmation,” Philosophy of Science, vol. 15 (1948), pp. 135–175, (4) R. Carnap, Logical Foundations of Probability (1950), and (5) J. G. Kemeny, “A Logical Measure Function”, The Journal of Symbolic Logic, vol. 18 (1953), pp. 289–308. This investigation of measures of the content of a statement, while closely related and relevant to a discussion of relevance among statements, covers a separate portion of ground.

Major contributions to the discussion of relevance measures include: (1) J. M. Keynes, Treatise on Probability (1921), (2) items number 2 and (3) number 4 above, (4) J. G. Kemeny and P. Oppenheim, “Degree of Factual Support,” Philosophy of Science, vol. 19 (1952), pp. 307–324, (5) K. R. Popper, “Degree of Confirmation,” British Journal for the Philosophy of Science, vol. 5 (1954), pp. 143–149. (See also a review of the last-named paper by Kemeny in the Journal of Symbolic Logic, vol. 20 [1955], pp. 304–305.)

In recent months a lively discussion has broken out in the British Journal for the Philosophy of Science regarding measures of confirmatory relevance among statements (among other tilings). The discussants include Y. Bar-Hillel (vol. 6 [1955], pp. 155–157; vol. 7 [1956], pp. 245–248), K. R. Popper (vol. 6 [1955), pp. 157–163; vol. 7 [1956], pp. 244–245 and 249–256), and R. Carnap (vol. 7 [1956], pp. 243–244). This discussion, insofar as presently relevant, has dealt with the serviceability of alternative relevance measures for various uses in the theory of confirmation.

None of the relevance measures considered in this literature correspond in either intent or substance to the concept of evidential relevance as understood and explicated here. None the less, a comparison among the principal relevance-measures is of interest in revealing relationships and in illustrating the many different forms of the relevance idea. The starting-point of such a comparison is provided by Kemeny in the review cited above, and the upshot is essentially as follows: (A) Carnap is concerned to analyze the measure inherent in the question: “How sure are we of p if we are given q as evidence?” (B) Popper and Kemeny-Oppenheim deal with the question: “How much surer are we of p given q than without q?” (C) The present measure of evidential relevance deals with the question: “How much is our confidence in the truth of p increased or decreased if q is given?”

⁷ Actually, some suitable inequality of the form L(q) ≥ m is also supposed in each of these cases, but no use will be made of this fact. This additional condition may, of course, be of great significance in applications to uses of evidence (e.g., in law) for which it is important to impose the requirement that confirmining evidence be significantly greater than .5. However, the condition will not be needed here, because it has no bearing on the logical form or structure of rules of evidence, this matter alone being presently under discussion.

⁸ In the formulation of rules of evidence, no attempt is made to attain completeness, i.e., to ensure that all rules meeting the respective criterion of acceptability are logical consequences of the given list of basic rules. The formulation of complete axiom systems for the three evidence-relations considered below (the relationships evidential presumption and of confirming and of supporting evidence) is left as an open problem.

⁹ The converse of this proposition is also an REP. This rule is acceptable as a special rule for supporting evidence (and is thus a valid rule for confirming evidence) subject to fulfillment of the additional requirement that L(q ↔ ∼r, p) = L(q ↔ ∼r, ∼p), i.e. that the degree of evidential support for q ↔ ∼r on the basis of p is 0, so that p leaves no reason to think that q and r are inconsistent.

This sums up what can be said on the present approach regarding conjunctive evidence-statements with a given proposition serving as evidence-base.

¹⁰ The question of the mutual or reciprocal evidence of propositions for each other has some interest. RSE 1 settles this matter as regards supporting evidence. With respect to evidential presumption, I offer the following calculation: (1) dep (p, q) = L(p, q), and dep (q p) = L(q, p). By Bayes' Theorem, (2) . By (2) and the rules of “L”, (3) . By (1) and (3), we have (4) dep (p, q) = dep . We are not warranted in saying more regarding the relationship of dep (p, q) and dep (q, p) than is contained in (4), viz. that evidential presumption is a fully reciprocal relationship only among equally likely statements.

¹¹ “A Purely Syntactical Definition of Confirmation,” Journal of Symbolic Logic, vol. 8 (1943), pp. 122–143. “Studies in the Logic of Confirmation,” Mind, vol. 45 (1945), pp. 1–20, 97–121.

¹² Pp. 468–482 give a critique of Hempel's treatment of evidence.

¹³ In particular, all RCE's and all RSE's would be acceptable to Carnap.

¹⁴ Cf. Aristotle, Ethics, I, 3.