  The class of involutive bisemilattices plays the role of the algebraic
counterpart of paraconsistent weak Kleene logic. Involutive bisemilattices can
be represented as Plonka sums of Boolean algebras, that is semilattice direct
systems of Boolean algebras. In this paper we exploit the Plonka sum
representation with the aim of counting, up to isomorphism, finite involutive
bisemilattices whose direct system is given by totally ordered semilattices.
