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." /> On sublogics in Vasiliev fragment of the logic definable with A. Arruda's calculus v1 - Popov Vladimir M.; Shangin Vasily O. | sdvig press