Fibring Non-Truth-Functional Logics: Completeness Preservation

Fibring has been shown to be useful for combining logics endowed with truthfunctional semantics. However, the techniques used so far are unable to cope with fibring of logics endowed with non-truth-functional semantics as, for example, paraconsistent logics. The first main contribution of the paper is the development of a suitable abstract notion of logic, that may also encompass systems with non-truth-functional connectives, and where fibring can still be dealt with. Furthermore, it is shown that this extended notion of fibring preserves completeness under certain reasonable conditions. This completeness transfer result, the second main contribution of the paper, generalizes the one established in (Zanardo et al., 2001) but is obtained using new techniques that explore the properties of a suitable metalogic (conditional equational logic) where the (possibly) non-truth-functional valuations are specified. The modal paraconsistent logic of (da Costa and Carnielli, 1988) is studied in the context of this novel notion of fibring and its completeness is so established.