Boole, George (1815–64)

George Boole, a British mathematician, is credited with making a fundamental contribution to modern logic. If Leibniz’s manuscript essays on logic, effectively unknown until the end of the nineteenth century, are excluded, then Boole’s algebra of logic (1847, 1854) was the first successful mathematical treatment of one part of logic. The treatment was mathematical in the broad sense of using a formal language expressed in symbols with definite rules. It was also mathematical in a narrow sense of being closely modelled after numerical algebra, from which it differed by an additional axiom, x2=x. Letter symbols of this algebra were conceived as representing classes, 1 standing for a ‘universe’ of objects and 0 for the empty class. By identifying logical terms with their extensions, that is, with classes, inferences of a much more general character than those of the traditional syllogistic could be carried out. Boole also showed how this algebra could be used in propositional logic, presenting its earliest systematic general formulation.

Effect algebras with a full set of states

It is shown that every effect algebra with a full set of states can be represented as a so-called numerical algebra introduced in the paper. For numerical algebras there are introduced tense operators which indicate dynamical changes of quantum events depending on variability of states. These operators enable to recognize an effect algebra with a full set of states as a temporal logic where events are quantified by these tense operators. The problem of representation of tense operators on a given numerical algebra is solved.