  This paper is part of a project that is based on the notion of dialectical
system, introduced by Magari as a way of capturing trial and error mathematics.
In previous work, we investigated the expressive and computational power of
dialectical systems, and we compared them to a new class of systems, that of
quasidialectical systems, that enrich Magari's systems with a natural mechanism
of revision. In the present paper we consider a third class of systems, that of
$p$-dialectical systems, that naturally combine features coming from the two
other cases. We prove several results about $p$-dialectical systems and the
sets that they represent. Then we focus on the completions of first-order
theories. In doing so, we consider systems with connectives, i.e. systems that
encode the rules of classical logic. We show that any consistent system with
connectives represents the completion of a given theory. We prove that
dialectical and $q$-dialectical systems coincide with respect to the
completions that they can represent. Yet, $p$-dialectical systems are more
powerful: we exhibit a $p$-dialectical system representing a completion of
Peano Arithmetic which is neither dialectical nor $q$-dialectical.
