 Reichenbach’s intent with formula I is to say that where it exists, the probability has a unique value. Axiom II is meant to ensure that the probabilities conditional on a nonempty set have values between 0 and 1 inclusive. Axiom III is Reichenbach’s version of the requirement that the probability of the union of mutually exclusive events is the sum of their probabilities. Axiom IV is essentially the chain rule of probability: P(CB | A) = P(C | BA) P(B | A). Axiom I is implicit in the Kolmogorov axioms (see Section 1 (“Kolmogorov’s Probability Calculus”) in the entry on  interpretations of probability)  since probability is taken to be a real valued function. Axiom II corresponds to Kolmogorov’s first and second axioms, that probability values are bounded between 0 and 1 inclusive. Axiom III amounts to finite additivity because Reichenbach’s logic does not have infinite disjunctions: it is a finite restriction of Kolmogorov’s third axiom, which postulates additivity of probabilities for countable, even infinite, disjoint sets. Axiom IV (at least its interpretation in terms of the chain rule) follows from Kolmorogov’s first three axioms. Reichenbach requires it as an additional axiom, because of his mixture of logical and mathematical notation. Without the additional fourth axiom, Reichenbach could not switch between logical conjunction and mathematical multiplication. 