Proof theory and mathematical meaning of paraconsistent C-systems

A proof theoretic analysis and new arithmetical semantics are proposed for some paraconsistent C-systems, which are a relevant subclass of Logics of Formal Inconsistency (LFIs) introduced by W. A. Carnielli et al. [8,9]. The sequent versions BC, CI,CIL of the systems bC, Ci, Cil presented in [8,9] are introduced and examined. BC, CI,CIL admits the cut elimination property and, in general, a weakened subformula property. Moreover, a formal notion of constructive paraconsistent system is given, and the constructivity of CI is proven. Further possible developments of proof-theory and provability logic of CI-based arithmetical systems are sketched, and a possible weakened Hilbert’s program is discussed. As to the semantical aspects, arithmetical semantics interprets C-system formulas into Provability Logic sentences of classical Arithmetic PA [2,19,15,22]: thus, it links the notion of truth to the notion of provability inside a classical environment. It makes true infinitely many contradictions B ∧ ¬B and falsifies many arbitrarily complex instances of non-contradiction principle ¬(A∧¬A). Moreover, arithmetical models falsify both classical logic LK and intuitionistic logic LJ, so that a kind of metalogical completeness property of LFI-paraconsistent logic w.r.t. arithmetical semantics is proven. As a work in progress, the possibility to interpret CI-based paraconsistent Arithmetic PACI into Provability Logic of classical Arithmetic PA is discussed, showing the role that PACI arithmetical models could have in establishing new meta-mathematical properties, e.g. in breaking classical equivalences between consistency statements and reflection principles.