In his book Formalization of Logic (1943), Carnap pointed out that there exist what he called non-normal interpretations of classical propositional logic (PL), i.e. interpretations of some of the usual propositional connectives that differ from the standard truth tables, but are consistent with all classical laws of PL. Carnap thought this was a major problem for formalizations of PL and suggested two kinds of remedy. His observations are not very well known among logicians or philosophers today, but recently they have been the subject of some debate, e.g. in Analysis: T. Smiley (1996), P. Raatikainen (2008), J. Murzi & O. Hjortland (2009), L. Incurvati & P. Smith (2010). These authors all take Carnap’s observations for granted but differ about their import. We take a new look at Carnap’s question from the perspective of modern (formal) semantics (a perspective Carnap could not have taken in 1943). We show that under very reasonable assumptions about such a semantics—the main one being that meaning assignment is *compositional*—there are no non-normal interpretations of PL. In fact, most of the relevant facts about PL can be found already in Carnap’s book, but he didn’t assign any special role to compositionality. We also suggest that some of the modern logical study of PL might profit from taking compositionality seriously. Finally, we raise Carnap’s question for first-order logic (FOL): Are there non-normal interpretations of the quantifiers consistent with all classical FOL laws? The answer (which is less easy to find than for PL) is Yes, but only to the extent that quantification is interpreted as in *free logic*. And with an extra, also natural, assumption (topic neutrality, or permutation invariance), this ’non-normal’ interpretation is eliminated too.