  The paper aims at studying, in full generality, logics defined by imposing a
variable inclusion condition on a given logic $\vdash$. It turns out that the
algebraic counterpart of the variable inclusion companion of a given logic
$\vdash$ is obtained by constructing the Plonka sum of the matrix models of
$\vdash$. This association allows to obtain a Hilbert-style axiomatization of
the logics of variable inclusion and to describe the structure of their reduced
models.
