Four-Valued Logics with Infinitely Many Extensions

Here, we we prove that there is a strictly increasing countable chain of finitary relatively finitely-axiomatizable extensions of ({the} truth-singular {version/extension of})[{the} bounded {expansion of}] first-degree entailments - (TS)[B]FDE, for short - /``relatively axiomatized by the Modus Ponens rule for material implication'', in which case the chain does not contain its join,and so this, being a finitary extension of (TS)[B]FDE, is not {relatively} finitely-axiomatizable. ([As a consequence, applying one of our previous works, we immediately get a strictly decreasing chain of finitely-axiomatizable quasi-varieties of bounded De Morgan lattices including the variety of bounded Kleene lattices with non-finitely-axiomatizable intersection.])