Description

• Title:
  Four-valued expansions of Dunn-Belnap's logic (I): Basic characterizations
• Creator:
  Pynko, Alexej P.
• Contributor:
  National Academy of Sciences of Ukraine V.M. Glushkov Institute of Cybernetics Department of Digital Automata Theory (100) Glushkov prosp. 40 Kiev, 03680, Ukraine;
  pynko@i.ua
• Description:
  Basic results of the paper are that any four-valued expansion L4 of Dunn-Belnap's logic DB4 is de_ned by a unique (up to isomorphism) conjunctive matrix ℳ4 with exactly two distinguished values over an expansion 4 of a De Morgan non-Boolean four-valued diamond, but by no matrix with either less than four values or a single [non-]distinguished value, and has no proper extension satisfying Variable Sharing Property (VSP). We then characterize L4's having a theorem / inconsistent formula, satisfying VSP and being [inferentially] maximal / subclassical / maximally paraconsistent, in particular, algebraically through ℳ4|4's (not) having certain submatrices|subalebras.Likewise, [providing 4 is regular / has no three-element subalgebra] L4 has a proper consistent axiomatic extension if[f] ℳ4 has a proper paraconsistent / two-valued submatrix [in which case the logic of this submatrix is the only proper consistent axiomatic extension of L4 and is relatively axiomatized by the Excluded Middle law axiom]. As a generic tool (applicable, in particular, to both classically-negative and implicative expansions of DB4), we also prove that the lattice of axiomatic extensions of the logic of an implicative matrix ℳ with equality determinant is dual to the distributive lattice of lower cones of the set of all submatrices of ℳ with non-distinguished values.
• Object availability:
  free access
• Rights: http://creativecommons.org/licenses/by-nc-nd/4.0
• Date:
  2021.05.11;
  2020.12.30
• Type:
  Other
• Language:
  English
• Format:
  application/pdf
• Publisher:
  Wydawnictwo Uniwersytetu Łódzkiego
• Relation:
  Bulletin of the Section of Logic;4;
  49
• Subject:
  propositional logic;
  logical matrix;
  Dunn-Belnap's logic;
  expansion;
  [bounded] distributive/De Morgan lattice;
  equality determinant
• Identifier:
  0138-0680;
  http://hdl.handle.net/11089/35465 ;
  2449-836X;
  10.18778/0138-0680.2020.19
• Data provider:
  Repozytorium Uniwersytetu Łódzkiego
• Can I use it?:
  check license
• Type:
  other