Counterexample Sufficiency in Modifications to Strict-Tolerant Logics

The three-valued strict-tolerant logic
$\mathsf{ST}$
that provides a substructural solution to semantic paradoxes appeals to the bounds consequence reading of validity in which provability of a sequent
$[\Gamma {>\!\!\!-} \Delta]$
(where
$\Gamma$
and
$\triangle$
are multisets of formulae) is understood as indicating that positions asserting
$\Gamma$
while denying
$\Delta$
violate some discursive norm. A modification
$\textsf{iST}$
(“intermediate ST”) following a broader notion of bounds consequence—including the representation of discursive norms rejecting positions on topical considerations—has been introduced. In
$\textsf{iST}$
the third truth value is understood as representing a topical defect (e.g. a sentence's including offensive subject-matter) but further variations on the setting have been proposed. This paper provides a characterization of those modifications that coincide with
$\textsf{ST}$
and
$\textsf{iST}$
, allowing characterizations of which modifications influence the consequence relations of these two systems.