Abstract
Despite being fairly powerful, finite non-deterministic matrices are unable to characterize some logics of formal inconsistency, such as those found between \(\mathbf{mbCcl}\) and \(\mathbf{Cila}\). In order to overcome this limitation, we propose here restricted non-deterministic matrices (in short, RNmatrices), which are non-deterministic algebras together with a subset of the set of valuations. This allows us to characterize not only mbCcl and Cila (which is equivalent, up to language, to da Costa’s logic \(C_1\)) but the whole hierarchy of da Costa’s calculi \(C_n\). This produces a novel decision procedure for these logics. Moreover, we show that the RNmatrix semantics proposed here induces naturally a labelled tableau system for each \(C_n\), which constitutes another decision procedure for these logics. This new semantics allows us to conceive da Costa’s hierarchy of C-systems as a family of (non deterministically) \((n+2)\)-valued logics, where n is the number of “inconsistently true” truth-values and 2 is the number of “classical” or “consistent” truth-values, for every \(C_n\).
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Acknowledgements
We would like to thank the anonymous referees by their careful reading and useful suggestions which helped us to improve the overall quality of this paper. This paper is a corrected and expanded version of the preliminary draft [20]. The first author acknowledges support from the National Council for Scientific and Technological Development (CNPq), Brazil, under research Grant 306530/2019-8. The second author was supported by a doctoral scholarship from CAPES, Brazil.
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Dedicated to Newton C.A. da Costa, a permanent source of inspiration.
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Coniglio, M.E., Toledo, G.V. Two Decision Procedures for da Costa’s \(C_n\) Logics Based on Restricted Nmatrix Semantics. Stud Logica 110, 601–642 (2022). https://doi.org/10.1007/s11225-021-09972-z
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DOI: https://doi.org/10.1007/s11225-021-09972-z