∀ or ∃?

This paper shows that in some axioms regarding the mixture of random variables, the requirement that the conclusions hold for all values of the mixture parameter can be replaced by requiring the existence of only one non-trivial value of the parameter, which needs not be fixed. This is the case for the independence, betweenness, and the mixture symmetry axioms. Unlike the standard axioms, these weaker versions cannot be refuted by experimental methods.