Cut-free sequent calculi for C-systems with generalized finite-valued semantics

Abstract

In [5], a general method was developed for generating cut-free ordinary sequent calculi for logics that can be characterized by finite-valued semantics based on non-deterministic matrices (Nmatrices). In this paper, a substantial step towards automation of paraconsistent reasoning is made by applying that method to a certain crucial family of thousands of paraconsistent logics, all belonging to the class of C-systems. For that family, the method produces in a modular way uniform Gentzen-type rules corresponding to a variety of axioms considered in the literature.

Taming Paraconsistent (and Other) Logics

“…It should be noted that the method in Avron et al [2012Avron et al [ , 2013 is quite different from ours: it works by first constructing the corresponding Nmatrix and then extracting an analytic sequent calculus from that. Note also that although this last step is algorithmic, the construction of Nmatrices out of the Hilbert calculi is done manually, and it requires some ingenuity.…”

“…Paraconsistent logics are logics that are not trivialized in the presence of inconsistency, that is, for which there are some formulas ψ, ϕ, such that: ψ, ¬ψ ϕ. The family of C-systems [Da Costa 1974;Carnielli and Marcos 2002;Carnielli et al 2007;Avron 2007;Avron et al 2012Avron et al , 2013 is a well-known family of paraconsistent logics in which the notion of consistency is internalized into the object language by employing a unary consistency operator •, the intuitive meaning of •ψ being "ψ is consistent." C-systems are defined by adding to HCL + the axioms (b) and (n 1 ), as well as different subsets of the axioms (r ), (k), (i), (o 1 ), (o 2 ), (a ¬ ), (e k ), (a ), and (c k ) in Figure 1.…”

“…A second option for (ii) is the sequent calculus in Avron et al [2012Avron et al [ , 2013 for the basic Logic of Formal Inconsistency BK, which is LK + extended with the rule (⇒ ¬) {{ f : p 1 }}/{t : ¬ p 1 } for ¬ and the two (invertible) rules for the connective •:…”

“…When selecting BK as base calculus in step (ii), Paralyzer also exploits the invertibility of the rules (• ⇒) and (⇒ •) for defining the calculus rules. Notice that for the C-systems with finite-valued semantics, its output produces the same sequent calculi and associated semantics as those introduced in a semiautomated 6 way in Avron et al [2012Avron et al [ , 2013.…”

“…While analytic sequent calculi and/or adequate semantics for some of these logics were already available, the novelty of our approach is that their introduction is fully automated (as opposed, e.g., to Avron et al [2012Avron et al [ , 2013, where the construction of semantics is done manually). Moreover, it applies to infinitely many logics (and is not tailored to the C-systems as in Avron et al [2012Avron et al [ , 2013 or to specific logics as in Kamide [2013]), many of which have had so far no adequate calculi or available semantics.…”