Paraconsistent logic

About: Paraconsistent logic is a research topic. Over the lifetime, 1610 publications have been published within this topic receiving 28842 citations.

On a Problem of da Costa

Abstract: 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 Antonio Mario 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.

When data dependencies over SQL tables meet the logics of paradox and S-3

Abstract: We study functional and multivalued dependencies over SQL tables with NOT NULL constraints. Under a no-information interpretation of null values we develop tools for reasoning. We further show that in the absence of NOT NULL constraints the associated implication problem is equivalent to that in propositional fragments of Priest's paraconsistent Logic of Paradox. Subsequently, we extend the equivalence to Boolean dependencies and to the presence of NOT NULL constraints using Schaerf and Cadoli's S-3 logics where S corresponds to the set of attributes declared NOT NULL. The findings also apply to Codd's interpretation "value at present unknown" utilizing a weak possible world semantics. Our results establish NOT NULL constraints as an effective mechanism to balance the expressiveness and tractability of consequence relations, and to control the degree by which the existing classical theory of data dependencies can be soundly approximated in practice.

Paraconsistent vagueness: Why not?

Abstract: The idea that the phenomenon of vagueness might be modelled by a paraconsistent logic has been little discussed in contemporary work on vagueness, just as the idea that paraconsistent logics might be fruitfully applied to the phenomenon of vagueness has been little discussed in contemporary work on paraconsistency. This is prima facie surprising given that the earliest formalisations of paraconsistent logics presented in Jaskowski and Hallden were presented as logics of vagueness. One possible explanation for this is that, despite initial advocacy by pioneers of paraconsistency, the prospects for a paraconsistent account of vagueness are so poor as to warrant little further consideration. In this paper we look at the reasons that might be offered in defence of this negative claim. As we shall show, they are far from compelling. Paraconsistent accounts of vagueness deserve further attention.