  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.
