Alfred North Whitehead promoted the idea of UNIVERSAL ALGEBRA in his monumental treatise of 1897 [1]. He proposed two candidates for this lofty title, the algebra of Boole and Grassmann’s Algebra of Extension. Boolean algebra has since secured universality status in Set Theory and Symbolic Logic, although only the former is universally known and used by mathematicians. However, Whitehead’s work...

The seeds of the formalization of thought were sown by the French philosopher René Descartes in the 17th century [2]. Descartes argued that thinking is reasoning, and that reason is a chain of simple ideas, linked by applying strict rules of logic. With his ‘cogito ergo sum’, Descartes was the first precursor of cognitive science (even if he was wrong about the dualism of mind and brain). Then,...

1 Polynomials as proof devices Algebraic proof systems based on formal polynomials over algebraically closed fields (the “polynomial ring calculus”) were introduced in [9] (see [10] and [11] for recent developments). Formal polynomials work as a powerful tool for logical derivation in classical and non-classical logics, in particular for propositional many-valued logics, paraconsistent logics a...

We introduce the family of R-based proto-Boolean rings associated with an arbitrary commutative ring R. They generalize the protoBoolean algebra devised by Brown [4] as a tool for expressing in modern language Boole’s research in The Laws of Thought. In fact the algebraic results from [4] are recaptured within the framework of proto-Boolean rings, along with other theorems. The free Boolean alg...

Classical logic is usually interpreted as the logic of propositions. But from Boole’s original development up to modern categorical logic, there has always been the alternative interpretation of classical logic as the logic of subsets of any given (non-empty) universe set. Partitions on a universe set are dual to subsets of a universe set in the sense of the reverse-the-arrows category-theoreti...

Logic should return its focus to valid reasoning in real-world situations. Since classical logic only covers valid reasoning in a highly idealized situation, there is a demand for a new logic for everyday reasoning that is based on more realistic assumptions, while still keeps the general, formal, and normative nature of logic. NAL (Non-Axiomatic Logic) is built for this purpose, which is based...

1 Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia E-mail: [email protected] In the frames of quantum structures, states are a very important notion that model probability on an algebraic structure. Already G. Boole observed that to calculate probability of event of the structure M , it is important to know only which two events A and B we c...

Considering measure theory in which the semifield of positive real numbers is replaced by an idempotent semiring leads to the notion of idempotent measure introduced by Maslov. Then, idempotent measures or integrals with density correspond to supremums of functions for the partial order relation induced by the idempotent structure. In this paper, we give conditions under which an idempotent mea...

This paper proposes an algorithm for deciding consistency of systems of Boolean equations in several variables with co-efficients in the two element Boolean algebra B0 = {0, 1} and find all satisfying assignments. The algorithm is based on the application of a well known generalized Boole-Shannon orthonormal (ON) expansion of Boolean functions. A necessary and sufficient consistency condition f...