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Reductio ad contradictionem: An Algebraic Perspective

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Abstract

We introduce a novel expansion of the four-valued Belnap–Dunn logic by a unary operator representing reductio ad contradictionem and study its algebraic semantics. This expansion thus contains both the direct, non-inferential negation of the Belnap–Dunn logic and an inferential negation akin to the negation of Johansson’s minimal logic. We formulate a sequent calculus for this logic and introduce the variety of reductio algebras as an algebraic semantics for this calculus. We then investigate some basic algebraic properties of this variety, in particular we show that it is locally finite and has EDPC. We identify the subdirectly irreducible algebras in this variety and describe the lattice of varieties of reductio algebras. In particular, we prove that this lattice contains an interval isomorphic to the lattice of classes of finite non-empty graphs with loops closed under surjective graph homomorphisms.

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References

  1. Beazer R.: Finitely subdirectly irreducible algebras with pseudocomplementation. Algebra Universalis 12(1), 376–386 (1981)

    Article  Google Scholar 

  2. Belnap, N. D., How a computer should think, in Contemporary Aspects of Philosophy, Oriel Press Ltd., Stocksfield, 1977, pp. 30–56.

  3. Belnap, N. D., A useful four-valued logic, in J. M. Dunn and G. Epstein (eds.), Modern Uses of Multiple-Valued Logic, vol. 2 of Episteme, Springer, Amsterdam, 1977, pp. 5–37.

  4. Belnap N.D., Spencer J.: Intensionally complemented distributive lattices. Portugaliae Mathematica 25, 99–104 (1966)

    Google Scholar 

  5. Blok, W. J., and D. Pigozzi, Abstract algebraic logic and the deduction theorem, Manuscript, 2001.

  6. Burris, S., and H. M. Sankappanavar, A Course in Universal Algebra, no. 78 in Graduate Texts in Mathematics, Springer, Berlin, 1981.

  7. Cornish W.H., Fowler P.R.: Coproducts of De Morgan algebras. Bulletin of the Australian Mathematical Society 16, 1–13 (1977)

    Article  Google Scholar 

  8. De M., Omori H.: Classical negation and expansions of Belnap-Dunn logic. Studia Logica 103(4), 825–851 (2015)

    Article  Google Scholar 

  9. Dunn, J. M., The algebra of intensional logics, Ph.D. Thesis, University of Pittsburgh, 1966.

  10. Dunn J.M.: Contradictory information: too much of a good thing. Journal of Philosophical Logic 39, 425–452 (2010)

    Article  Google Scholar 

  11. Font J.M.: Belnap’s four-valued logic and De Morgan lattices. Logic Journal of the IGPL 5, 1–29 (1997)

    Article  Google Scholar 

  12. Johansson I.: Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus. Compositio Mathematica 4, 119–136 (1937)

    Google Scholar 

  13. Kalman J.A.: Lattices with involution. Transactions of the American Mathematical Society 87(2), 485–491 (1958)

    Article  Google Scholar 

  14. Lakser H.: The structure of pseudocomplemented distributive lattices. I: Subdirect decomposition. Transactions of the American Mathematical Society 156, 335–342 (1971)

    Google Scholar 

  15. Lakser H.: Principal congruences of pseudocomplemented distributive lattices. Proceedings of the American Mathematical Society 37(1), 32–36 (1973)

    Article  Google Scholar 

  16. Lee K.B.: Equational classes of distributive pseudocomplemented lattices. Canadian Journal of Mathematics 22(4), 881–891 (1970)

    Article  Google Scholar 

  17. Odintsov, S. P., Constructive negations and paraconsistency, in Trends in Logic, Springer, Dordrecht, 2008.

  18. Priestley H.A.: The construction of spaces dual to pseudocomplemented distributive lattices. The Quarterly Journal of Mathematics 26(1), 215–228 (1975)

    Article  Google Scholar 

  19. Pynko A.P.: Implicational classes of De Morgan lattices. Discrete Mathematics 205(1), 171–181 (1999)

    Article  Google Scholar 

  20. Quackenbush R.W.: Completeness theorems for universal and implicational logics of algebras via congruences. Proceedings of the American Mathematical Society 103(4), 1015–1021 (1988)

    Article  Google Scholar 

  21. Rebagliato, J., and V. Verdú, Algebraizable Gentzen Systems and the Deduction Theorem for Gentzen Systems, no. 175 in Mathematic Preprint Series, University of Barcelona, 1995.

  22. Sankappanavar, H. P., A characterization of principal congruences of De Morgan algebras and its applications, in A. I. Arruda, R. Chuaqui, and N. C. A. da Costa (eds.), Mathematical Logic in Latin America: Proceedings of the IV Latin American Symposium on Mathematical Logic, vol. 99 of Studies in Logic and the Foundations of Mathematics, North Holland, 1980, pp. 341–349.

  23. Sankappanavar H.P.: Principal congruences of pseudocomplemented De Morgan algebras. Mathematical Logic Quarterly 33(1), 3–11 (1987)

    Article  Google Scholar 

  24. Vakarelov D.: Nelson’s negation on the base of weaker versions of intuitionistic negation. Studia Logica 80(2), 393–430 (2005)

    Article  Google Scholar 

  25. Wansing, H., Proofs, disproofs, and their duals, in C. Areces and R. Goldblatt (eds.), Advances in Modal Logic 7, 2010, pp. 483–505.

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  1. Institute of Computer Science, Pod Vodárenskou věží 271/2, Czech Academy of Sciences, Prague, Czech Republic

    Adam Přenosil

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  1. Adam Přenosil
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Přenosil, A. Reductio ad contradictionem: An Algebraic Perspective. Stud Logica 104, 389–415 (2016). https://doi.org/10.1007/s11225-015-9645-9

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