Three-valued paraconsistent logics and their extensions: Three-valued paraconsistent logics

We first prove that

any [conjunctive/disjunctive/implicative]

3-valued paraconsi-tent logic

with subclassical negation (3VPLSN) is defined by a

unique {modulo isomorphism} [conjunctive/disjunctive/implicative]

3-valued matrix and provide

effective

algebraic criteria of any 3VPLSN's

"being subclassical"|$"being maximally

paraconsistent"|"having no (inferentially) consistent non-subclassical extension"

implying that any [conjunctive/disjunctive]|conjunctive/"both disjunctive

and {non\}subclassical''/"refuting Double Negation

Law"'|"conjunctive/disjunctive subclassical''

3VPLSN "is subclassical if[f]

its defining 3-valued matrix has a 2-valued submatrix"|"is

{pre-}maximally paraconsistent"|"has a theorem but no consistent non-subclassical extension''.

Next, any disjunctive/implicative 3VPLSN has no proper consistent

non-classical disjunctive/axiomatic extension, any classical extension being

disjunctive/axiomatic and relatively axiomatized by the "Resolution

rule"'/"Ex Contradictione Quodlibet axiom''.

Further, we provide an effective

algebraic criterion of a [subclassical] "3VPLSN with lattice conjunction and

disjunction"''s having no proper [consistent non-classical] extension

but that [non-]inconsistent one which is relatively axiomatized by the

Ex Contradictione Quodlibet 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-valued connectives

has an infinite increasing chain of finitary

extensions.