  The problems of logical translation of axiomatizations and the choice of
primitive operators have surfaced several times over the years. An early issue
was raised by H. Hi{\. z} in the 1950s on the incompleteness of translated
calculi. Further pertinent work, some of it touched on here, was done in the
1970s by W. Frank and S. Shapiro, as well as by others in subsequent decades.
As we shall see, overlooking such possibilities has led to incorrect claims of
completeness being made (e.g. by J. L. Bell and A. B. Slomson as well as J. N.
Crossley) for axiomatizations of classical predicate logic obtained by
translation from axiomatizations suited to differently chosen logical
primitives. In this note we begin by discussing some problematic aspects of an
early article by W. Frank on the difficulties of obtaining completeness
theorems for translated calculi. Shapiro had established the incompleteness of
Crossley's axiomatization by exhibiting a propositional tautology that was not
provable. In contrast, to deal with Bell and Slomson's system which is complete
for propositional tautologies, we go on to show that taking a formal system for
classical predicate calculus with the primitive $ \exists$, setting $\forall x
\phi(x) \stackrel{\text{def}}{=}\neg \exists x \neg \phi(x)$, and writing down
a set of axioms and rules complete for the calculus with $\forall $ instead of
$ \exists$ as primitive, does not guarantee completeness of the resulting
system. In particular, instances of the valid schema $\exists x \phi (x)
\rightarrow \exists x \neg \neg\phi (x)$ are not provable, which is analogous
to what occurs in modal logic with $\Box$ and $\Diamond$.
