Probability as a Measure of Information Added
 Peter Milne
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Some propositions add more information to bodies of propositions than do others. We start with intuitive considerations on qualitative comparisons of information added. Central to these are considerations bearing on conjunctions and on negations. We find that we can discern two distinct, incompatible, notions of information added. From the comparative notions we pass to quantitative measurement of information added. In this we borrow heavily from the literature on quantitative representations of qualitative, comparative conditional probability. We look at two ways to obtain a quantitative conception of information added. One, the most direct, mirrors Bernard Koopman’s construction of conditional probability: by making a strong structural assumption, it leads to a measure that is, transparently, some function of a function P which is, formally, an assignment of conditional probability (in fact, a Popper function). P reverses the information added order and mislocates the natural zero of the scale so some transformation of this scale is needed but the derivation of P falls out so readily that no particular transformation suggests itself. The Cox–Good–Aczél method assumes the existence of a quantitative measure matching the qualitative relation, and builds on the structural constraints to obtain a measure of information that can be rescaled as, formally, an assignment of conditional probability. A classical result of Cantor’s, subsequently strengthened by Debreu, goes some way towards justifying the assumption of the existence of a quantitative scale. What the two approaches give us is a pointer towards a novel interpretation of probability as a rescaling of a measure of information added.
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 Title
 Probability as a Measure of Information Added
 Journal

Journal of Logic, Language and Information
Volume 21, Issue 2 , pp 163188
 Cover Date
 20120401
 DOI
 10.1007/s1084901191420
 Print ISSN
 09258531
 Online ISSN
 15729583
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Information
 Probability
 Comparative probability
 Koopman
 Cox
 Authors

 Peter Milne ^{(1)}
 Author Affiliations

 1. University of Stirling, Stirling, United Kingdom