Relations between paraconsistent logic and many-valued logic
Newton C. A. da Costa, Elias Humberto Alves · 1981
Let us consider informally a theory T (deductive system). We say that T is trivial (or overcomplete) if all formulas of T are theorems of T ; otherwise, we say that T is non-trivial (or not overcomplete). T is inconsistent if it has a negation symbol and there are, at least, two theorems of T such that one is the negation of the other; if this is not the case, T is consistent. If the underlying logic of T is the classical logic, T is inconsistent if and only if it is trivial. The same occurs with a great part of well-known systems of logic. So, if we intend to study inconsistent but non-trivial theories, we must construct new types of logic as a foundation to those theories. The logical systems constructed with this intention, have been called paraconsistent systems (see [1]). In general, the systems of paraconsistent logic must satisfy the following conditions: