Conditionals and Independence in Many-Valued Logics

Events and possible worlds from physical systems States and observables of physical systems provide a very general source of continuously valued events and possible worlds. Let be a “physical system”. Following [6, pp.362–369], a rigorous account of is given by its associated C -algebra , with the set sa of selfadjoint elements, and the set sa of real-valued normalized positive linear functionals on sa. For anyW and X sa the real numberW X is the expectation value of the observable X when is prepared in mode W Given a set E X1 Xm of nonzero positive elements of sa, W determines, by normalization, the map w : E 0 1 given by w Xi w Xi Xi where Xi is the norm of Xi. Intuitively, the event Xi says “the observable Xi has a high value,” and w evaluates how true Xi is. The set 0 1 E of possible worlds is defined by w W . is a closed nonempty set in the cube 0 1 E 0 1 n.