Experimental observations are often taken in order to assist in making a choice between relevant hypotheses ∼ H and H. The power of observations in this decision is here metrically defined by information-theoretic concepts and Bayes' theorem. The exact (or maximum power) of a new observation to increase or decrease Pr(H) the prior probability that H is true; the power of that observation to modify the total amount of uncertainty involved in the choice between ∼ H and H: the power (...) of a new observation to reduce uncertainty toward the ideal amount, zero; all these powers are systematically shown to be exact metrical functions of Pr(H) and Pr(o/H)/Pr(o)-1 where the numerator is the likelihood of the new observation given H, and the denominator is the "expectedness" of the observation. (shrink)

Experimental observations are often taken in order to assist in making a choice between relevant hypotheses ~H and H. The power of observations in this decision is here metrically defined by information-theoretic concepts and Bayes' theorem. The exact of a new observation to increase or decrease Pr the prior probability that H is true; the power of that observation to modify the total amount of uncertainty involved in the choice between ~H and H: the power of a new observation to (...) reduce uncertainty toward the ideal amount, zero; all these powers are systematically shown to be exact metrical functions ofwhere the numerator is the likelihood of the new observation given H, and the denominator is the “expectedness” of the observation. (shrink)

Since the time, about a century ago, when DeMorgan, Boole and Jevons, inaugurated the study of the logic of numerically definite reasoning, no one has been concerned to establish the validity rules for a very general type of numerically definite inference which is a strong analogue of the classical syllogism. The reader will readily agree that the traditional rules of syllogistic inference cannot even begin to decide whether the following proportionally quantified syllogism is a valid argument: at most 4/7 p (...) are at most 2/3 not-m and at least 3/5 s are precisely 7/8 m, hence, at most 1/3 s are p. Likely reasons, of broad methodological interest, why this readily definable type of formal inference has been neglected will presently be proposed. Among the masters of twentieth century logic, Hilbert and Bernays have, to be sure, meticulously shown how such numerically definite propositions as “at least 7 individuals have the property p” can be expressed in the predicate calculus with the use of the identity relation. They do not, since their concern with such propositions is indirect, proceed to establish a systematic calculus of classes whose “size” or extent is numerically definite. As we shall always notice, traditional logic and even ordinary Boolean algebra prefer to generalize about the laws governing the interrelationship of classes in inference systems without making any hypotheses concerning class “size” or extent except by the classical quantifier “some”, which quantifier admits the measure or extent of a class only in terms of the proposition that a given class contains or does not contain at least one member. Such a restricted admission of the relevance for inference of class size is among the fundamental reasons for the neglect of proportional quantifiers, i.e., quantifiers where numerical value is any rational number r such that 0/100 ≤ r ≤ 100/100. It can, of course, also be pointed out that the traditional syllogism and ordinary Boolean algebra do not i.e. measure, numerically define, the degree of inclusion of one class in another. Inclusion is taken either as total or partial, with no discrimination of the degree of partial inclusion. (shrink)

Experimental observations are often taken in order to assist in making a choice between relevant hypotheses ∼ H and H. The power of observations in this decision is here metrically defined by information-theoretic concepts and Bayes' theorem. The exact (or maximum power) of a new observation to increase or decrease Pr(H) the prior probability that H is true; the power of that observation to modify the total amount of uncertainty involved in the choice between ∼ H and H: the power (...) of a new observation to reduce uncertainty toward the ideal amount, zero; all these powers are systematically shown to be exact metrical functions of Pr(H) and Pr(o/H)/Pr(o)-1 where the numerator is the likelihood of the new observation given H, and the denominator is the "expectedness" of the observation. (shrink)