Three chapters contain the results independent of each other. In the first chapter I present a set of axioms for the propositional calculus which are shorter than the ones known so far, in the second one I give a method of defining all ternary connectives, in the third one, I prove that the probability of propositional functions is preserved under reversible substitutions.

The aim of the paper is to give a new definition of real number. The logical type of any number defined is that of the function B = h(A) which assigns to a class of sets A a class of sets B. I give some conditions which the function h has to fulfill to be considered as number; an intuitive sense of the conditions is as follows: a function, which is number, assigns a class of sets of measure h·m to (...) a class A of sets of equal measure, where m is the measure of A. (shrink)