The concept of logical consequence is one of the most central concepts of logic. It naturally involves de idea of truth preservation. Nonetheless, another adequacy criterion could be devised and its development would still remain in the central area of logical matters. We are thinking in the concept of falsity preservation. In our exposition, we want to go through a few logical systems in which falsity preservation is the adequacy criterion adopted. As we pretend to show quickly, there is a huge amount of systems to which this criterion can be applied such that some traditional remarkable logical metatheorems still hold. We will use examples of natural deduction rules to present our ideas in an intuitive fashion. These rules are easily related to axiomatic systems and also sequent calculi.