On ring-like structures induced by Mackeyʹs probability function

The aim of this paper is to show that the structure of the set L of experimental propositions induced by Mackeyʹs probability function p(A, α, E) can be conveniently described and investigated in the framework of the theory of ring-like structures called generalized Boolean quasirings. With the help of only one axiom (the so-called extension axiom) a representation theorem for L is stated and proved. The operations · and + generalizing the classical “and”-operation and the classical “exclusive or”-operation, respectively, are investigated. It is shown that under some natural assumptions the associativity of + implies the distributivity of L and that the unique solvability of the equations a + x = b is equivalent to the classicality of L. A physical interpretation for the obtained results is given.