At the heart of the development of fertilized eggs into fully formed organisms and the adaptation of cells to changed conditions are genetic regulatory networks (GRNs). In higher multicellular organisms, signal selection and multiplexing are performed at the cis-regulatory domains of genes, where combinations of transcription factors (TFs) regulate the rates at which the genes are transcribed into mRNA. To be able to act as activators or repressors of gene transcription, TFs must first bind to target sequences on the regulatory domains. Two TFs that act in concert may bind entirely independently of each other, but more often binding of the first one will alter the affinity of the other for its binding site. This article presents a systematic investigation into the effect of TF binding dependences on the predicted regulatory function of this bio-logic. Four extreme scenarios, commonly used to classify enzyme activation and inhibition patterns, for the binding of two TFs were explored: independent (the TFs bind without affecting each other's affinities), competitive (the TFs compete for the same binding site), ordered (the TFs bind in a compulsory order), and joint binding (the TFs either bind as a preformed complex, or binding of one is virtually impossible in the absence of the other). The conclusions are: (1) the laws of combinatorial logic hold only for systems with independently binding TFs; (2) systems formed according to the other scenarios can mimic the functions of their Boolean logical counterparts, but cannot be combined or decomposed in the same way; and (3) the continuously scaled output of systems consisting of competitively binding activators and repressors can be controlled more robustly than that of single TF or (quasi-)logical multi-TF systems.