  We extend classical Propositional Logic (PL) by adding a new primitive binary
connective $\varphi|\psi$, intended to represent the "superposition" of
sentences $\varphi$ and $\psi$, an operation motivated by the corresponding
notion of quantum mechanics, but not intended to capture all aspects of the
latter as they appear in physics. To interpret the new connective, we extend
the classical Boolean semantics by employing models of the form $\langle
M,f\rangle$, where $M$ is an ordinary two-valued assignment for the sentences
of PL and $f$ is a choice function for all pairs of classical sentences. In the
new semantics $\varphi|\psi$ is strictly interpolated between
$\varphi\wedge\psi$ and $\varphi\vee\psi$. By imposing several constraints on
the choice functions we obtain corresponding notions of logical consequence
relations and corresponding systems of tautologies, with respect to which $|$
satisfies some natural algebraic properties such as associativity, closedness
under logical equivalence and distributivity over its dual connective. Thus
various systems of Propositional Superposition Logic (PLS) arise as extensions
of PL. Axiomatizations for these systems of tautologies are presented and
soundness is shown for all of them. Completeness is proved for the weakest of
these systems. For the other systems completeness holds if and only if every
consistent set of sentences is extendible to a consistent and complete one, a
condition whose truth is closely related to the validity of the deduction
theorem.
