In the hope of soliciting an expert opinion rather than offering one (I am neither a probabilist nor a logician), let me bring up here the topic of logical probability. Jan Lukasiewicz came up with infinitely valued propositional logic (with values in the interval $[0,1]$), which he also viewed as a way of formalizing probability, addressing the problem of interpreting conditional probabilities as well. According to Wikipedia, Richard Threlkeld Cox later showed that any extension of Aristotelian logic to incorporate truth values between 0 and 1, in order to be consistent, must be equivalent to Bayesian probability (http://en.wikipedia.org/wiki/Cox%27s_theorem). Cox's axioms deal with the notion of plausibility of a proposition A given a proposition B.
Added: this is the relevant paper by Lukasiewicz: Die logischen Grundlagen der Wahrscheinlichkeitsrechnung, Kraków, Polska Akademia Umiejętności, 1913 English translation in Selected Works, ed. by L. Borkowski, Amsterdam-London, North- Holland Publishing Company/Warsaw, PWN, 1970, pp. 16-63