On Sublogics in Vasiliev Fragment of the Logic Definable with A. Arruda’s Calculus V1

  • Vladimir M. PopovEmail author
  • Vasily O. Shangin
Part of the Synthese Library book series (SYLI, volume 387)


In Arruda (On the imaginary logic of N.A.Vasiliev. In: Proceedings of fourth Latin-American symposium on mathematical logic, pp 1–41, 1979) presented calculi in order to formalize some ideas of N.A. Vasiliev. Calculus V1 is one of the calculi in question. Vasiliev fragment of the logic definable with A. Arruda’s calculus V1 (that is, a set of all V1-provable formulas with the following property: each propositional variable occurring in a formula is a Vasiliev propositional variable) is of independent interest. It should be noted that the fragment in question is equal to the logic definable with calculus P 1 in Sette (Math Jpn 18(3):173–180, 1973). In our paper, (a) we define logics I 1,1, I 1,2, I 1,3, … I 1,ω (see Popov (Two sequences of simple paraconsistent logics. In: Logical investigations, vol 14-M., pp 257–261, 2007a (in Russian))), which form (in the order indicated above) a strictly decreasing (in terms of the set-theoretic inclusion) sequence of sublogics in Vasiliev fragment of the logic definable with A. Arruda’s calculus V1, (b) for any j in {1, 2, 3,…ω}, we present a sequent-style calculus GI 1,j (see Popov (Sequential axiomatization of simple paralogics. In: Logical investigations, vol 15, pp 205–220, 2010a. IPHRAN. M.-SPb.: ZGI (in Russian))) and a natural deduction calculus NI 1,j (offered by Shangin) which axiomatizes logic I 1,j , (c) for any j in {1, 2, 3,…ω}, we present an I 1,j -semantics (built by Popov) for logic I 1,j , (d) for any j in {1, 2, 3,…}, we present a cortege semantics for logic I 1,j (see Popov (Semantical characterization of paraconsistent logics I1,1, I1,2, I1,3,…. In: Proceedings of XI conference “modern logic: theory and applications”, Saint-Petersburg, 24–26 June. SPb, pp 366–368, 2010b (in Russian))). Below there are some results obtained with the presented semantics and calculi.


Vasiliev logic Imaginary logic Valuation semantics Cortege semantics Sequent calculus Natural deduction 


  1. Arruda, A. I. (1979). On the imaginary logic of N.A.Vasiliev. In Proceedings of Fourth Latin-American Symposium on Mathematical Logic (pp. 1–41).Google Scholar
  2. Bolotov, A. E., Shangin, V. (2012). Natural deduction system in paraconsistent setting: Proof search for pcont. Journal of Intelligent Systems, 21(1), 1–24. ISSN (Online) 2191-026X, ISSN (Print) 0334-1860. doi:
  3. Bolotov, A., Grigoryev, O., Shangin, V. (2007). Automated natural deduction for propositional linear-time temporal logic. In Proceedings of the 14th International Symposium on Temporal Representation and Reasoning (Time2007), Alicante, 28–30 June (pp. 47–58). ISBN:0-7695-2836-8Google Scholar
  4. Gentzen, G. (1969). Investigations into logical deduction. The collected papers of Gerhard Gentzen (Ed. M.E. Szabo). North-Holland Pub. Co.Google Scholar
  5. Kleene, S. C. (1952). Introduction to Metamathematics. North-Holland Publishing Co., Amsterdam, and P. Noordhoff, Groningen, 1952; D. van Nostrand Company, New York and Toronto 1952; X + 550 pp.Google Scholar
  6. Popov, V. M. (2007a). Two sequences of simple paraconsistent logics. In Logical investigations (Vol. 14-M., pp. 257–261) (in Russian).Google Scholar
  7. Popov, V. M. (2007b). Intervals of simple paralogics. In Proceedings of the V Conference “Smirnov Readings in Logic”, 20–22 June 2007 (pp. 35–37). M. (in Russian).Google Scholar
  8. Popov, V. M. (2010a). Sequential axiomatization of simple paralogics. In Logical investigations (Vol. 15, pp. 205–220). IPHRAN. M.-SPb.: ZGI (in Russian).Google Scholar
  9. Popov, V. M. (2010b). Semantical characterization of paraconsistent logics I1,1, I1,2, I1,3,…. In Proceedings of XI Conference “Modern Logic: Theory and Applications”, Saint-Petersburg, 24–26 June (pp. 366–368). SPb (in Russian).Google Scholar
  10. Sette, A. M. (1973). On the propositional calculus P1. Mathemetica Japonicae, 18(3), 173–180.Google Scholar
  11. Vasiliev, N. A. (1989). Imaginary (non-Aristotelian) logic. Imaginary logic (pp. 53–94). Selected works. M. Nauka Publishers (in Russian).Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityGSP-1MoscowRussian Federation

Personalised recommendations