Skip to Main content Skip to Navigation
Journal articles

Eigenlogic in the Spirit of George Boole

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.
Document type :
Journal articles
Complete list of metadata

https://hal-centralesupelec.archives-ouvertes.fr/hal-02615451
Contributor : Zeno Toffano Connect in order to contact the contributor
Submitted on : Tuesday, April 27, 2021 - 8:29:10 AM
Last modification on : Tuesday, July 20, 2021 - 3:06:26 AM

File

Eigenlogic_Z-Toffano-2020-post...
Files produced by the author(s)

Identifiers

Citation

Zeno Toffano. Eigenlogic in the Spirit of George Boole. Logica Universalis, Springer Verlag, 2020, 14 (2), pp.175-207. ⟨10.1007/s11787-020-00252-3⟩. ⟨hal-02615451⟩

Share

Metrics

Record views

106

Files downloads

59