Probabilistic Substitutivity at a Reduced Price

Abstract

One of the many intriguing features of the axiomatic systems of probability investigated in Popper (1959), appendices _iv, _v, is the different status of the two arguments of the probability functor with regard to the laws of replacement and commutation. The laws for the first argument, (rep1) and (comm1), follow from much simpler axioms, whilst (rep2) and (comm2) are independent of them, and have to be incorporated only when most of the important deductions have been accomplished. It is plain that, in the presence of (comm1), the principle (sub), which says that terms that are intersubstitutable in the first argument are intersubstitutable also in the second argument, implies (comm2), and in Popper’s systems the converse implication obtains. It is naturally asked what is needed in an axiomatic theory of probability in order to enforce this equivalence. Leblanc (1981) offered a rather weak set of axioms, containing (comm1) and (comm2), that suffice for the derivation of (sub). In this paper Leblanc’s result is improved in a number of different ways. Three weaker systems, one of which is incomparable with the other two, are shown to admit the same implication.