It was argued in a series of experimental studies that Japanese tits (Parus minor) have an ABC call that has an alert function, a D call that has a recruitment function, and an ABC-D call that is compositionally derived from ABC and D, and has a mobbing function. A key conclusion was that ABC-D differs from the combination of separate utterances of ABC and of D (e.g. as played by distinct but close loudspeakers). While the logic of the argument is arguably sound, no explicit rule has been proposed to derive the meaning of ABC-D from that of its parts. We compare two analyses. One posits a limited instance of semantic compositionality ('Minimal Compositionality'); the other does without compositionality, but with a more sophisticated pragmatics ('Bird Implicatures'). Minimal Compositionality takes the composition of ABC and D to deviate only minimally from what would be found with two independent utterances: ABC means that 'there is something that licenses an alert', D means that 'there is something that licenses recruitment', and ABC-D means that 'there is something that licenses both an alert and recruitment'. By contrast, ABC and D as independent utterances yield something weaker, namely: 'there is something that licenses an alert, and there is something that licenses recruitment', without any 'binding' across the two utterances. The second theory, Bird Implicatures, only requires that ABC-D should be more informative than ABC, and/or than D. It builds on the idea, proposed for several monkey species, that a less informative call competes with a more informative one ('Informativity Principle'): when produced alone, ABC and D trigger an inference that ABC-D is false. We explain how both Minimal Compositionality and Bird Implicatures could have evolved, and we compare the predictions of the two theories. Finally, we extend the discussion to some chimpanzee and meerkat sequences that might raise related theoretical problems.