Three-Valued Paraconsistent Logics with Subclassical Negation and Their Extensions

EasyChair Preprint 5564

Abstract

We first prove that
any [conjunctive/dis\-jun\-c\-ti\-ve/implicative]
3-valued pa\-ra\-con\-sis\-tent logic
with subclassical negation (3VPLSN) is defined by a
unique \{modulo isomorphism\} [conjunctive/dis\-jun\-c\-ti\-ve/implicative]
3-va\-lu\-ed matrix
and provide
effective
algebraic criteria of any  3VPLSN's
being {\em subclassical\/}$|$being {\em maximally\/}
pa\-ra\-con\-sis\-tent$|$having no (inferentially) consistent non-subclassical extension
implying that any [co\-n\-j\-un\-c\-ti\-ve/disjunctive]$|$co\-n\-j\-u\-n\-c\-ti\-ve/``both disjunctive
and \{non-\}subclassical''/``refuting {\em Double Negation
Law\/}''$|$``co\-n\-j\-un\-c\-ti\-ve/disjunctive subclassical''
3VPLSN ``is subclassical if[f]
its defining 3-valued matrix has a 2-valued sub\-mat\-rix''$|$``is
\{pre-\}ma\-ximally paraconsistent''$|$``has a theorem but no consistent non-sub\-clas\-si\-cal extension''.
Next, any disjunctive/implicative 3VPLSN has no proper consistent
non-clas\-si\-cal disjunctive/ax\-i\-o\-ma\-tic extension, any classical extension being
disjunctive/axiomatic and relatively axiomatized by the ``{\em Resolution\/}
rule''/``{\em Ex Contradictione Qu\-o\-d\-li\-bet\/} axiom''.
Further, we provide an effective
algebraic criterion of a [subclassical] ``3VPLSN with lattice conjunction and
di\-s\-j\-u\-n\-c\-t\-i\-on'''s having no proper [consistent non-classical] extension
but that [non-]inconsistent one which is relatively axiomatized by the
{\em Ex Contradictione Qu\-o\-d\-li\-bet\/} rule
[and defined by the product of any defining 3-valued matrix
and its 2-valued submatrix].
Finally, any disjunctive and conjunctive 3VPLSN
with classically-va\-lu\-ed connectives
has an infinite increasing chain of finitary
extensions.

Keyphrases: extension, logic, matrix

@booklet{EasyChair:5564,
  author    = {Alexej Pynko},
  title     = {Three-Valued Paraconsistent Logics with Subclassical Negation and Their Extensions},
  howpublished = {EasyChair Preprint 5564},
  year      = {EasyChair, 2021}}