On a Problem of da Costa

The two main founders of paraconsistent logic, Stanis law Jaśkowski and Newton da Costa, built their systems on distinct grounds. Starting from different projects, they used different tools and ultimately designed quite different calculi to attend their needs. How successful were their enterprises? Here we discuss the problem of defining paraconsistent logics following the original instructions laid down by da Costa. We present a new approach to P, the first full solution —proposed by Antônio Mário Sette— to the problem of da Costa, and argue in favor of yet another solution we shall study here: the logic P. Both P and P constitute maximal 3-valued paraconsistent fragments of classical logic. Constructive completeness proofs are here presented for both logics. 1 Requisites to paraconsistent calculi When proposing the first paraconsistent propositional system, in 1948, Jaśkowski expected it to enjoy the following properties (see [17]): Jas1 when applied to inconsistent systems it should not always entail their trivialization; Jas2 it should be rich enough to enable practical inferences; Jas3 it should have an intuitive justification. A few years later, in 1963, da Costa would independently tackle a similar problem, this time proposing a whole hierarchy of paraconsistent propositional calculi, known as Cn, for 0 < n < ω. His requisites to these calculi were the following (see [12]): NdC1 in these calculi the principle of non-contradiction, in the form ¬(A∧¬A), should not be a valid schema; NdC2 from two contradictory formulae, A and ¬A, it would not in general be possible to deduce an arbitrary formula B; NdC3 it should be simple to extend these calculi to corresponding predicate calculi (with or without equality); NdC4 they should contain the most part of the schemata and rules of the classical propositional calculus which do not interfere with the first conditions.