The λμ-calculus is an extension of the λ-calculus that has been introduced by M Parigot to give an algorithmic content to classical proofs. We show that Böhm's theorem fails in this calculus.

A completeness theorem is established for logics with congruence endowed with general semantics (in the style of general frames). As a corollary, completeness is shown to be preserved by fibring logics with congruence provided that congruence is retained in the resulting logic. The class of logics with equivalence is shown to be closed under fibring and to be included in the class of logics with congruence. Thus, completeness is shown to be preserved by fibring logics with equivalence and general semantics. An example is provided showing that completeness is not always preserved by fibring ligics endowed with standard (non general) semantics. A categorial characterization of fibring is provided using coproducts and cocartesian liftings.