  We shall settle the completeness of some classical positive propositional
calculi (positive propositional calculi in which the so-called Peirce's law
holds) by resorting to a close adaptation of Kalmar's completeness proof
procedure. First of all, we shall employ this adaptation to establish the
completeness of the most familiar axiomatic characterization of the classical
logic of material implication and disjunction. Next, we shall demonstrate that
the completeness of the so-called classical positive propositional calculus can
be proved either through essentially the same adaptation of Kalmar's procedure
or as an immediate consequence of the conjunction of our first result and the
circumstance of each positive formula being reducible to a syntactically and
semantically equivalent one exhibiting a particular normal form. Finally, by
defining disjunction in terms of material implication and so mirroring the
classical calculus of material implication and disjunction in the classical
implicative calculus, the completeness of the latter, as well as the
completeness of the classical calculus of material implication and conjunction,
will be shown to follow from the completeness of the former in quite simple and
direct ways. It is worth noting that all these completeness proofs will be
constructive, that is to say, we can easily extract from each of them an
effective method for producing formal deductions of all theses of the axiomatic
system concerned.
