The probabilistic foundations of logic

Summary form only given, as follows. A conditional version of the Komolgoroff axioms for probability theory is developed, and it is shown that the resulting theory can serve as a formal semantics for any logic in which the notion of maximally consistent set is definable, provided the probability functions are defined over the power set of the maximally consistent sets. It is shown that the functions can be transformed into functions autonomously defined on the language itself for the case of classical logic and all its extensions. A core confirmation theory is developed whose functions are autonomously defined on arbitrary languages: the theory captures the notions common to all relative frequency schemes and is compatible with the Komolgoroff theory. It is shown that this core confirmation theory can be used as a formal semantics for virtually any finitistic monotonic logic. It is demonstrated that, if rational belief functions are conditionalisable and logical entailment is based on any reasonable universal property of all rational belief functions, then logical entailment must be monotonic.