In 1986, Mikenberg et al. introduced the semantic notion of quasi-truth defined by means of partial structures. In such structures, the predicates are seen as triples of pairwise disjoint sets: the set of tuples which satisfies, does not satisfy and can satisfy or not the predicate, respectively. The syntactical counterpart of the logic of partial truth is a rather complicated first-order modal logic. In the present article, the notion of predicates as triples is recursively extended, in a natural way, to any complex formula of the first-order object language. From this, a new definition of quasi-truth is obtained. The proof-theoretic counterpart of the new semantics is a first-order paraconsistent logic whose propositional base is a 3-valued logic belonging to hierarchy of paraconsistent logics known as Logics of Formal Inconsistency, which was proposed by Carnielli and Marcos in 2002.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)State Univ Campinas UNICAMP, Ctr Log Epistemol & Hist Sci CLE, BR-13083859 Campinas, SP, BrazilState Univ Campinas UNICAMP, Dept Philosophy, BR-13083859 Campinas, SP, BrazilSao Paulo State Univ UNESP, Dept Math, BR-17033360 Bauru, SP, BrazilSao Paulo State Univ UNESP, Dept Math, BR-17033360 Bauru, SP, BrazilFAPESP: 10/51038-0