Note on Probability Implication

In a recently published paper J. C. C. McKinsey has pointed out some difficulties which arise from Axiom I of my theory of probability implication. This axiom states the unambiguity of the degree p of a given probability implication (03VP) for the case that the class 0 is not empty, a condition formulated by (o), but postulates ambiguity of p in case of an empty class 0, this condition being formulated by (0). The latter ambiguity is necessary for probability implication because of the relation to Russell's material implication. From the proof published by McKinsey we can infer that this ambiguity has to be restricted to values of p between 0 and 1, limits included, in correspondence with the same restriction holding for the unambiguous degree p of probability in cases of a non-empty class 0, formulated by me in (8, §13). That this general restriction is derivable from Axiom II, 2 is obvious as this axiom contains 0and p as free variables and therefore states the restriction for all classes 0 and all values p. A further objection, which was already indicated in a footnote of McKinsey's paper, has been presented to me in a letter by the referee of this journal, Mr. S. C. Kleene. This objection shows that if the ambiguity of degrees of probability for empty classes 0 is assumed, it can be proved that this ambiguity cannot be restricted to the limits O t o l . This proof is connected with the theorem of addition (Axiom III) which reads III. (03PP).(03qQ).(0.PDQ)3(Br)(03rPVQ)-(r=p+q).