applying two definitions of the union of if-events, p. grzegorzewski gave two generalizations of the inclusion-exclusion principle for if-events.in this paper we prove an inclusion-exclusion principle for if-states based on a method which can also be used to prove grzegorzewski's inclusion-exclusion principle for probabilities on if-events.finally, we give some applications of this principle by...

‎from hilbert's theorem of zeroes‎, ‎and from noether's ideal theory‎, ‎birkhoff derived certain algebraic concepts (as explained by tholen) that have a dual significance in general toposes‎, ‎similar to their role in the original examples of algebraic geometry‎. ‎i will describe a simple example that illustrates some of the aspects of this relationship‎. the dualization from algebra to geometr...

At first, George Boole introduced conditional propositions as the basis for better development of propositional logic. A compound statement composed two atomic statements (antecedent and consequent) by connective 'if—then' is called For example: "If it raining, then roads are wet". Here first this proposition, 'It raining,' antecedent, second, 'the wet,' consequent. The material in which false ...

The well known Boole -Shannon expansion of Boolean functions in several variables (with coefficients in a Boolean algebra B) is also known in more general form in terms of expansion in a set Φ of orthonormal functions. However, unlike the one variable step of this expansion an analogous elimination theorem and consistency is not well known. This article proves such an elimination theorem for a ...

As shown by Pitowsky, the Bell inequalities are related to certain classes of probabilistic inequalities dealt with by George Boole, back in the 1850s. Here a short presentation of this relationship is given. Consequently, the Bell inequalities can be obtained without any assumptions of physical nature, and merely through mathematical argument. The importance of the typically quantum phenomenon...