By Rudolf Carnap and Yeshoua Bar-Hillel
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Extra resources for An Outline of Theory of Semantic Information
Since m* seems to have a special status among the various mI-functions, cont* and inf* are deemed to be of special importance. Various computations and tables regarding these functions are presented. We shall now define and investigate two special I-functions that might turn out to be -31- ___1_1 I __I_ I__ of special importance. ] §110 essen- tially as a proper m-function which has the same values for all structure-descriptions, that is, disjunctions of isomorphic state-descriptions. Recalling the definition of 'isomorphic sentences' given in §6, the reader will easily 2 23 has exactly 20 structure-descriptions.
K, e) - 15 and est(inf, H. 792. 918. We verify that est(inf*, H. K, e) < est(inf*, H, e) + est(inf*, K, e), the h's and the k's not being inductively independent on this e with respect to m*. They are, however, independent with respect to m D , and indeed est(infD, H. K, e) = 2 = est(inf D , H, e) + est(infD, K, e). -40- 1 _ In general, est(in, H, e) will be different from est(in, H, e. k), where k is a sentence that has been added to the prior evidence e. Since 'est(in, H, e. k)' and similar expres- sions are of great importance, it is worthwhile to give it a special name.
Then est(cont, K, e) > est(cont, H, e). Tll-3. For fixed n, est(cont, H i , e) is a maximum for those Hi all whose members have the same c-values on e. Hence maxi[est(cont, H i,e)] = m(e) x nn (This is, of course, also the cont-value of each hi belonging to these Hi. ) p T11-4. For fixed n, est(cont, 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 the c-value 0 on e); hence mini[est(cont, H i , e)] = 0. Theorems similar to T2, T3, and T4 can be obtained for the second explicatum inf.
An Outline of Theory of Semantic Information by Rudolf Carnap and Yeshoua Bar-Hillel