 In order to exemplify the concept of PTS as a splitting methodology, consider the paraconsistent logic bC, introduced in Carnielli and Marcos 2002. This logic is, in particular, a logic of formal inconsistency (LFI), in the sense that there exists a unary connective \(\circ\) expressing the consistency of a formula. Thus, from \(\varphi\) and \(\neg\varphi\) does not follow, in general, an arbitrary formula \(\psi\). However, \(\{\varphi, \neg\varphi, \circ\varphi\}\) entails any formula \(\psi\). The signature \(C\) of bC consists of a paraconsistent negation \(\neg\), a consistency operator \(\circ\), and classical connectives \(\wedge,\vee,\to\). It was proved in Carnielli et al.  2007 that bC, and many other logics of formal inconsistency extending it, cannot be characterized by finite matrices. Nonetheless, bC is decomposed into several copies of a three-valued logic by means of possible-translations as follows: consider the signature \(C_1=\{\neg_1,\neg_2,{\circ}_1,{\circ}_2, {\circ}_3, \wedge,\vee,\to\}\) consisting of two negations, three consistency operators, a conjunction, a disjunction and an implication. Let \(M\) be the matrix over \(C_1\) with domain \(\{T,t,F\}\) displayed below, where \(\{T,t\}\) is the set of designated values.