This chapter discusses paraconsistent logics (PL) and paraconsistency. PL are the logics of inconsistent but nontrivial theories. A deductive theory is paraconsistent if its underlying logic is paraconsistent. A theory is inconsistent if there is a formula (a grammatically well-formed expression of its language) such that the formula and its negation are both theorems of the theory; otherwise, the theory is called consistent. A theory is trivial if all formulas of its language are theorems. In a trivial theory "everything" (expressed in its language) can be proved. If the underlying logic of a theory is classical logic, or even any of the standard logical systems such as intuitionistic logic, inconsistency entails triviality, and conversely. This chapter discusses da Costa's C-logics. This chapter elaborates on paraconsistent set theories, and shows, in particular, how they accommodate inconsistent objects, such as the Russell set. Jáskowski's discussive logic is examined, and it is showed how it can be used in the formulation of the concept of partial truth. The chapter also examines annotated logic, and some of its applications.
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