By Rudolf Carnap and Yeshoua Bar-Hillel

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Purposes of reference, such a table is For given here for some selected integral values of m. -29- __ I _I Table IV m Log m m Log m 1 0. 5849 41 5. 8826 4 2. 9068 5 2. 9541 7 2. 0000 46 5. 5235 64 6. 0000 9 3. 1699 47 5. 5545 100 6. 6438 10 3. 3219 48 5. 0000 49 5. 9657 32 5. 0000 50 5. 0443 51 5. 9772 34 5. 0874 52 5. 9829 35 5. 1292 53 5. 2479 56 5. 9657 Let the E in our m E. 2 3 be E, E E4 m Let C Log m be the conjunction of the first Then contD(Cm) = m/64 (T2a) and infD(Cm) = 6 - Log(64-m) (T3b).

044 (as in E6). Though it may happen that, for some q, sp(in, H, kq, e) is negative, it can be proved that sp(in, H, K, e) is never negative, in other words, that the estimate of the posterior estimate is at most equal to the prior estimate. T11-12. sp(in, H, K, e) >,, 0, with equality holding iff the h's and the k's are induc- tively independent. -42- O Combining T10 and T11, we get Tl1-13. K, e). From T13 follows immediately the following theorem of the symmetricity or mutuality of specification: Tll-14.

2). There is at least one additional requirement for the adequacy of an m-function to serve as an explicatum for (absolute, initial) inductive probability. R9-1. cate. This is Requirement of instantial relevance. Let 'M' be a factual, molecular predi- Let e be any non-L-false molecular sentence. Let i and h be full sentences of 'M' with two distinct individual constants which do not occur in e. -26- Then n m(e. i. i m(e. h) me) ' (This can be formulated more simply in terms of 'c' as c(h, e.