Expressing logical disagreement from within

Abstract Against the backdrop of frequent comparison theories truth in literature on semantic paradoxes with regard to which inferences and metainferences are deemed valid, this paper develops a novel approach defining binary predicate for representing valid theory within itself under assumption that is defined classical meta-theory. The aim obtain tool facilitates between its competitors itself, thereby expressing disagreement theories. After discussing what we can should require an object-linguistic representation purpose, proposes restrict locally metainferences, requirement turns out be $$\omega $$ ω -consistent conservative over first-order arithmetic. This then applied four definable strong Kleene models using labelled nested sequent calculus.