Eigenlogic in the spirit of George Boole

TL;DR

This paper introduces an algebraic and geometric framework for logic based on Boole's original ideas, extending to operators in finite vector spaces, with applications in quantum information.

Contribution

It develops a linear algebraic formalization of propositional logic using projection operators, bridging classical Boolean algebra with quantum logic.

Findings

Logical operators as commuting projection operators

Eigenvalues represent truth values 0 and 1

Framework applicable to quantum information

Abstract

This work presents an operational and geometric approach to logic. It starts from the multilinear elective decomposition of binary logical functions in the original form introduced by George Boole. A justification on historical grounds is presented bridging Boole's theory and the use of his arithmetical logical functions with the axioms of Boolean algebra using sets and quantum logic. It is shown that this algebraic polynomial formulation can be naturally extended to operators in finite vector spaces. Logical operators will appear as commuting projection operators and the truth values, which take the binary values {0,1}, are the respective eigenvalues. In this view the solution of a logical proposition resulting from the operation on a combination of arguments will appear as a selection where the outcome can only be one of the eigenvalues. In this way propositional logic can be formalized in linear algebra by using elective developments which correspond here to combinations of tensored elementary projection operators. The original and principal motivation of this work is for applications in the new field of quantum information, differences are outlined with more traditional quantum logic approaches.