Minimally Many-Valued Maximally Paraconsistent Minimal Unary Subclassical Expansions of LP

EasyChair Preprint no. 5358, version 1

Versions: 12history
37 pagesDate: April 20, 2021

Abstract

Here, for any $n>2$,
we propose a {\em minimally\/} $n$-valued
(i.e., $m$-valued, for no $0<m<n$)
{\em maximally\/} paraconsistent
(i.e., having no proper paraconsistent extension)
subclassical (i.e., having a classical extension)
expansion $C_n$ of the
{\em logic of paradox\/} $LP$ by solely unary connectives,
none of which can be eliminated with retaining both
minimal $n$-valuedness and maximal paraconsistency,
$C_3$ being exactly $LP$.
And what is more, we prove that, in case $n=[>]4$,like for $LP$ [resp., $HZ/LA$],
there are just two proper consistent extensions of $C_n$ ---
the classical one, defined by the two-valued submatrix
$\mc{A}_{n:2}$ of the $n$-valued matrix $\mc{A}_n$
defining $C_n$ and relatively axiomatized by the
{\em Resolution/Modus Ponens''\/} rule
/for {\em material\/} implication''
[or (\{un\}like $HZ/LA$ \{resp., $LP$\}) by a single axiom],
and its proper sublogic,
defined by the direct product of $\mc{A}_n$ and
$\mc{A}_{n:2}$ (in which case having the same theorems as $C_n$ has,
and so not being an axiomatic extension of $C_n$)
and relatively axiomatized by
the {\em Ex Contradictione Quodlibet\/} rule.
Finally, we find both a sequent axiomatization
of $C_n$ with Cut Elimination Property that is
algebraizable iff $n\neq4$, $C_n$ as such being algebraizable
iff $n>4$, in which case it is equivalent to its
sequent axiomatization, and a finite Hilbert-style one
as well as, in case $n>4$, finite equational axiomatizations
of the discriminator variety equivalent to both $C_n$ and its
sequent axiomatization.

Keyphrases: Calculus, extension, logic, matrix