The paper provides a method for a uniform complete Hilbert-style axiomatisation of Post's -conditionals and Post's negation, where m is the number of truth values and u is the number of designated truth values . The main feature of the technique which we employ in this proof generalises the well-known Kalmár Lemma which was used by its author in his completeness argument for the ordinary, two-valued logic.

. The connection presented in this paper mirror-links two metamathematical structures, the finitary closure operators, and the compact consistency properties, in such a way that a specification of one structure induces a provably equivalent specification of the other.

The method based on the idea of Galois connection is well known. It facilitates investigations into similarities between mathematical structures, including isomorphisms between these structures, the highest degree of similarity. This idea is employed here and adapted so as to get to the core of aspects of the relationship between some metamathematical structures. The focus is put on the relation between traditional methodological orthodoxy based on the idea of proof , on the one hand, and on some alternative methodological set-ups (...) based on other ideas such as consistency or some forms of maximality, on the other hand. (shrink)