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The problem of Byzantine consensus has been key to designing secure distributed systems. However, it is particularly difficult, mainly due to the presence of Byzantine processes that act arbitrarily and the unknown message delays in general networks. Although it is well known that both safety and liveness are at risk as soon as $n/3$ Byzantine processes fail, very few works attempted to characterize precisely the faults that produce safety violations from the faults that produce termination violations. In this paper, we present a new lower bound on the solvability of the consensus problem by distinguishing deceitful faults violating safety and benign faults violating termination from the more general Byzantine faults, in what we call the Byzantine-deceitful-benign fault model. We show that one cannot solve consensus if $n\leq 3t+d+2q$ with $t$ Byzantine processes, $d$ deceitful processes, and $q$ benign processes. In addition, we show that this bound is tight by presenting the Basilic class of consensus protocols that solve consensus when $n > 3t+d+2q$. These protocols differ in the number of processes from which they wait to receive messages before progressing. Each of these protocols is thus better suited for some applications depending on the predominance of benign or deceitful faults. Finally, we study the fault tolerance of the Basilic class of consensus protocols in the context of blockchains that need to solve the weaker problem of eventual consensus. We demonstrate that Basilic solves this problem with only $n > 2t+d+q$, hence demonstrating how it can strengthen blockchain security.