Miscellaneous Extensions of Four-Valued Expansions of Belnap's Logic

EasyChair Preprint 4078

Abstract

As a generic tool,
we prove that the poset of (axiomatic) disjunctive [non-pseudo-axiomatic] extensions
of the logic of a finite set M of [(truth-non-empty)] finite disjunctive matrices
is dual to the distributive lattice of
relative universal (positive) Horn model subclasses of the set S
of [truth-non-empty] consistent submatrices of members of
M [(the duality preserving axiomatic relative axiomatizations)].
If M consists of a single matrix with equality
determinant, relative universal Horn model
subclasses of S
are proved constructively to be exactly
lower cones of S
that covers any four-valued expansion L4
of Belnap's four-valued logic B4.

Moreover, we find algebraic criteria of
the [inferential] paracompleteness of the extension
of L4  relatively axiomatized by the Resolution} rule.
We also find lattices of extensions of L4
satisfying certain rules (in particular, non-paracomplete
extensions)
under certain conditions
covering many interesting
four-valued expansions of B4 including both
itself and its bounded version
(as well as their purely implicative expansions).

Keyphrases: extension, logic, matrix, model

@booklet{EasyChair:4078,
  author    = {Alexej Pynko},
  title     = {Miscellaneous Extensions of Four-Valued Expansions of Belnap's Logic},
  howpublished = {EasyChair Preprint 4078},
  year      = {EasyChair, 2020}}