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the probability of a propostion X given a proposition Y is the "degree to which Y logically entails X". Problems:
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'According to the logical interpretation, associated with Carnap (see also Logical Foundations of Probability, 1950; and Continuum of Inductive Methods, 1952), the probability of a propostion X given a proposition Y is the "degree to which Y logically entails X." Carnap developed an ingenious and elaborate set of systems of logical probability, including, e.g., separate systems depending on the degree to which one happens to be, logically and rationally, sensitive to new information in the reevaluation of probabilities.'
Audi (1999)
"The probability of theories has been treated in terms of a logical relation between the theory and a class of evidence. J. M. KEYNES and C. D. BROAD, followed by CARNAP, Kneale, and others, have attempted to define the essential logical relation, but the enterprise comes up against the problems of CONFIRMATION theory: it also shares the problem bedevilling the classical theory, that without an a priori input of probability judgments no final assessments ensue."
Flew and Priest (2002)
'The logical theory identifies probability with degree of rational belief. It is assumed that given the same evidence, all rational human beings will entertain the same degree of belief in a hypothesis or prediction.'
Gillies (2000)
"The logical relation theory makes probability a logical relation between evidence and a conclusion, rather like entailment (see IMPLICATION) only weaker (cf. CONFIRMATION). Probability is therefore always relative to evidence. Apart from the difficulty of finding such a relation, one defect of this theory as an analysis of ‘probably’ is that if we know a true proposition, p, which entails another, q, we can ‘detach’ q, i.e. assert it on its own, but if p only makes q probable, we can at best say ‘Probably q’, which leaves ‘probably’ unanalysed – and even that we cannot say if we know there is another true proposition which makes q improbable."
Lacey (1996)