An Outline of Theory of Semantic Information by Rudolf Carnap and Yeshoua Bar-Hillel

By Rudolf Carnap and Yeshoua Bar-Hillel

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Tll-7. , e)] = Log n. ) Tll-8. For fixed n, est(inf, H i , e) is a minimum for those H i one member of which has the c-value 1 on e (and hence all the other members have the c-value 0 on e). Hence mini[est(inf, Hi, e)]= 0. An expression analogous to ' - E c(hp, e) x Log c(hp, e)' but with degree of confirmation replaced by (statistical) probability, plays a central role in communication theory, as well as in certain formulations of statistical mechanics, where the probability concerned is that of a system being in cell p of its phase space.

Mb') = 0. 7. cont*('Mc'/'Ma. 2 = 0. 1; On the other hand, contD('Mc'/'Ma. Mb') = 0. 125. inf*('Ma. 7370, as against an infD-value of 2. Finally, inf*('Mc'/'Ma. Mb') = Log 02 = Log 3 - Log 2 = 0. 5849 whereas the corresponding relative infD-value is 1. It might perhaps be worthwhile to investigate now another sample language, this time with only one primitive predicate and n distinct individual constants. yields the same values as Laplace's rule of succession. ) Let e be a conjunction of s < n basic sentences with s distinct individual constants, among them s 1 atomic sentences with 'P' and s-sl negations of such.

In our case, '... of the amount of information carried by the observation to be made at t In general, whenever we have a class of sentences H = {hl available evidence e L-implies hlV hV... Vh n .. as well as - (h i. , for all ij, we shall say that H is an exhaustive system relative to e, and the expression n E c(hp, e) x in(hp/e) p= will be called 'the (c-mean) estimate of the amount of information carried by (the members of) H with respect to e', symbolized by 'est(in, H, e)'. So far, our discussion has been proceeding on a partly presystematic, partly systematic level.

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