Orthomodularity and quadratic transformations in probabilistic theories of physics

Abstract
A characterization of orthomodularity of a set of effects is given in terms of a triangle closedness of this set. This leads to a simple (probabilistic) characterization of observables with a Boolean range. Quadratic transformations on expectation functionals of observables are studied in order to investigate the derivability of the Heisenberg inequality and the form of the lower bound of such an inequality for a pair of observables.