True is a set: Many-valued logics with matrix semantics

Originating in the framework of philosophical preoccupations with paradoxes and determinism, the many-valued logics are today essential, finding abundant applications in STEM fields from electronics to knowledge bases. Essentially characterized by many-valuedness, or the property of having more than the two classical truth values, the many-valued logical systems are often presented only axiomatically and/or truth-functionally, but this presentation largely obliterates the intended meanings of many-truthfulness, the property that in these logics more than a single truth value can be distinguished as true. Matrix semantics, a type of algebraic semantics, handles many-truthfulness in a set-theoretic way that typically allows for a meaning account. In this review article, the core aspects of many-valued logics with matrix semantics are elaborated on from the viewpoint of logical consequence.