A structure \(\) is a bilattice if i) \(\) and \(\) are both lattices; and ii) the meet and join operations of each lattice are monotone with respect to the order relation of the other lattice. Bilattices were introduced in computer science by \textit{M. L. Ginsberg} [Multi-valued logics, Proc. AAAI-86, Morgan Kaufmann, 243-247 (1986)]. This paper shows that bilattices are natural generalisations of structures to be found in the semantics of First Degree Entailment (due to Dunn and Belnap), in the semantics of languages with a partial truth predicate (due to Woodruff, Kripke et al.) and elsewhere. This allows many known features of these semantics to be seen as special cases of more general theorems, and also delivers some novel results, e.g., concerning the relationships between various fixed points.