For the reduced matrix semantics of a propositional logic, truth is almost parametrically equationally definable exactly when the Leibniz operator is almost completely order-reflecting, and Leibniz-injectivity transfers from theories to arbitrary filters exactly for countable languages.
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A study of truth predicates in matrix semantics
For the reduced matrix semantics of a propositional logic, truth is almost parametrically equationally definable exactly when the Leibniz operator is almost completely order-reflecting, and Leibniz-injectivity transfers from theories to arbitrary filters exactly for countable languages.