Marcelo Coniglio

Departamento de Filosofia, Instituto de Filosofia e Ciências Humanas, Universidade Estadual de Campinas (Brazil)

conigliocle.unicamp.br

Non-Classical Logics/ Philosophical Logic

Non-deterministc semantics for non-classical logics

The relationship between algebra and logic begins with George Boole seminal works from 1850, which were improved and enhanced by Ernst Schröder and others by the end of the XIX century. A fundamental step was given by Adolf Lindenbaum and Alfred Tarski through the construction of the so-called Lindenbaum-Tarski algebras, namely ordered algebras naturally associated to the theories of a given logic. This techique was generalized by Wim Blok and Don Pigozzi, giving origin to a wide theory of algebraization of logics now called Blok-Pigozzi algebraization. This originates a subarea of algebraic logic known as Abstract algebraic logic, AAL (see [11]).

Despite the wide scope of AAL, not every logic can be analyzed under this perspec- tive. For instance, the well-known hierachy Cn (n ≥ 1) paraconsistent logics introduced by Newton da Costa cannot be semantically characterized by a single finite matrix. Moreover, it lies outside the scope of AAL techniques. The same holds for several log- ics in the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (in short LFIs, see [5, 4]), which generalizes da Costa’s logics, as well as some non- normal modal systems (see [12]). Several alternative semantical tools based on some form of non-determinsim were introduced in the literature in order to deal with such systems: (non-truth-functional) bivaluations, Fidel structures, possible-translations semantics, non-deterministic matrices (or Nmatrices) and swap structures, among others, obtaining so decision procedures for these logics.

In this tutorial, some non-deterministic semantical approaches to non-classical logics will be presented and compared, emphasizing on swap structures and Fidel structures. In order to illustrate the scope and limitations of each method, several LFIs and non- normal modal logics will be considered, as well as Nelson’s paraconsistent logic N4. In particular, the uncharacterizability of da Costa’s logic C1 by finite Nmatrices will be analyzed. By contrast, a semantical characterization of C1 by means of finite Fidel structures semantics, on the one hand, and by finite possible-translations semantics, on the other, will be show, which constitute decision procedures for this logic.

Finally, an ouline of Fidel structures and swap tructures from the point of view of category thery and model theory will also presented.

• [1] A. Avron, Non-deterministic matrices and modular semantics of rules, Logica Universalis, (Jean-Y. Béziau, editor), Birkhäuser Verlag Basel, Switzerland, 2005, pp. 149–167.
• [2] W. J. Blok and D. Pigozzi, Algebraizable Logics, Memoirs of the American Mathematical Society, vol. 77 (1989).
• [3] W. A. Carnielli, Possible-Translations Semantics for Paraconsistent Logics, Frontiers of Paraconsistent Logic: Proceedings of the I World Congress on Paraconsistency (Ghent, Belgium), (D. Batens, C. Mortensen, G. Priest, and J.P. Van Bendegem, editors), Logic and Computation Series, Hertfordshire: Research Studies Press, 1997, pp. 149–163.
• [4] W. A. Carnielli and M. E. Coniglio, Paraconsistent Logic: Consistency, Contradiction and Negation, Logic, Epistemology, and the Unity of Science, vol. 40, Springer, (2016).
• [5] W. A. Carnielli, M. E. Coniglio, and J. Marcos, Logics of Formal Incon- sistency, Handbook of Philosophical Logic (2nd. edition) (D. M. Gabbay and F. Guenthner, editors), volume 14, Springer, 2007, pp. 1–93.
• [6] M. E. Coniglio and A. Figallo-Orellano, A model-theoretic analysis of Fidel-structures for mbC. Submitted, 2016.
• [7] M. E. Coniglio, A. Figallo-Orellano and A. C. Golzio, Towards an hy-peralgebraic theory of non-algebraizable logics, CLE e-Prints, vol. 16 (2016), no. 4.
• [8] M. E. Coniglio and A. C. Golzio, Swap structures for some non-normal modal logics. Submitted, 2017.
• [9] N. C. A da Costa, Sistemas Formais Inconsistentes (in Portuguese), Habilitation thesis, Universidade Federal do Paraná, Curitiba, Brazil, 1963. Republished by Editora UFPR, Curitiba, Brazil,1993.
• [10] M. M. Fidel, The decidability of the calculi Cn, Reports on Mathematical Logic, vol. 8 (1977), pp. 31–40.
• [11] J. M. Font, Abstract Algebraic Logic. An Introductory Textbook, College Publications, 2016.
• [12] Y. V. Ivlev, A semantics for modal calculi, Bulletin of the Section of Logic, vol. 17 (1988), no. 3/4, pp. 114–121.
• [13] F. Marty, Sur une g ́en ́eralisation de la notion de groupe, Proceedings of 8th Congress Math. Scandinaves, 1934, pp. 45–49.

CV

Marcelo E. Coniglio obtained his PhD in Mathematics from the University of São Paulo (USP), Brazil, in 1997, under the supervision of Francisco Miraglia. He joined in 1998 the Philosophy Department of the Institute of Philosophy and the Humanities (IFCH) of the University of Campinas (UNICAMP), Brazil. In 2004 he got his Habilitation(Livre-Docência) in Logic from UNICAMP. Since 2013 he is Full Professor at the Philosophy Department of IFCH/UNICAMP.

Since 2016 he is Director of the Centre for Logic, Epistemology and the History of Science (CLE) of UNICAMP. Since 2014 he is President of the Brazilian Logic Society (SBL). Since 2007 he is collaborator of the Security and Quantum Information Group (SQIG) of the Instituto de Telecomunicações (IT) associated to the Instituto Superior Técnico (IST), Lisbon, Portugal.

He is a fellow researcher of the National Council for Scientific and Technological Development (CNPq), Brazil, since 2006. Together with Jean-Yves Beziau he is Editor-in-Chief of the South-American Journal of Logic (SAJL).

His research interests include non-classical logics and combination of logics.