Alternative axiomatizations of elementary probability theory

1 Outline In this paper I offer some new axiomatizations of elementary (finite) probability theory. The new axiomatizations have the following distinctive features. First, both conditional and unconditional probabilities are simultaneously axiomatized. Second, the axiomatizations are extensions of Kolmogorov probability theory, and have the usual definitions linking conditional and unconditional probabilities as theorems. Third, while both probabilities are simultaneously axiomatized, the resultant axioms are about as simple as the usual axiomatizations of conditional probability alone. Fourth, the axioms adopted have strong and direct justifications independently of the interpretation of the probability function (axiomatization L however is only a permissible extension of intuitions). The reader who wishes to review the axiomatizations before their rationale can at this point go to section 5 (the main axiomatization is system KK).