Concept

Boolean algebra (structure)

Overview

In mathematics, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution).

Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

Possible passages

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Any such select set is called the set of postulates, or premises, from which the reasoning proceeds. The reasoning is nothing else than the exhibition of the whole pattern of general conditions involved in the pattern derived from the selected postulates.
Alfred North Whitehead, Science and the modern world
States what Boolean algebra is; the author is tied to Boolean algebra in the knowledge graph.
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The idea of a number is also enlarged. These letters, standing for any numbers, are called sometimes variables and sometimes parameters. Their essential characteristic is that they are undetermined, unless, indeed, the algebraic conditions which they satisfy implicitly determine them. Then they are sometimes called unknowns.
Alfred North Whitehead, The organisation of thought, educational and scientific
States what Boolean algebra is; the author is tied to Boolean algebra in the knowledge graph.
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The chief business of the logician is to classify arguments; for all testing clearly depends on classification. The classes of the logicians are defined by certain typical forms called syllogisms. For example, the syllogism called _Barbara_ is as follows: S is M; M is P: Hence, S is P. Or, to put words for letters— Enoch and Elijah were men; all men die: Hence, Enoch and Elijah must have died. The “is P” of the…
Charles S. Peirce, Chance, Love, and Logic
States what Boolean algebra is.
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Philosophers who discussed Boolean algebra (structure)

George Boole

Related ideas

Place in the tradition

Part of Classical logic.

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