  We consider a certain class of infinitary rules of inference, called here
restriction rules, using of which allows us to deduce complete theories of
given models. The first instance of such rules was the $\omega$-rule introduced
by Hilbert, and generalizations of the $\omega$-rule were first considered by
Henkin. Later on Barwise showed that within countable languages, for any
countable model $\mathfrak M$, first-order logic expanded by the corresponding
$\mathfrak M$-rule deduces from the diagram of $\mathfrak M$ all formulas that
are true in all models that include $\mathfrak M$, in particular, it deduces
the (relativized) complete theory of $\mathfrak M$. We show that, if the aim is
only deducing the complete theory of a given model, these countability
assumptions can be omitted. Moreover, similar facts hold for infinitary and
higher-order logics, even if these logics are highly incomplete. Finally, we
show that Barwise's theorem in its stronger form, for vocabularies of arbitrary
cardinality, holds for infinitary logics $\mathscr L_{\kappa,\lambda}$ whenever
$\kappa$ is supercompact. The note also contains brief historical remarks.
