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

### EasyChair Preprint 5358, version 1

37 pages•Date: 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