Generalized truth values, ordering relations defined on them, and the resulting lattice structures that give rise to various non-classical logics

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Logic, the theory of valid inferences, has often been described as the science of truth values. From the classical viewpoint, there are exactly two truth values, truth and falsity, and every statement is either true or false and not both. However, there are many philosophically important ways of generalizing the classical conception of truth values. In the literature one can find, for instance, conceptions of constructive versus non-constructive truth and falsity. The information human and artificial agents are confronted with is often incomplete and sometimes inconsistent, suggesting that statements may also be evaluated as neither true nor false or both true and false. Truth and falsity may receive other philosophically relevant qualifications: necessary truth, possible falsity, truth henceforth, commonly known truth, etc. Classical and non-classical truth values may be combined to obtain new, structured truth values. The idea of generalized, possibly structured truth values suggests ordering relations on these values. The values may be compared with respect to their degree of constructiveness, necessity, amount of information they give, etc. As a result, one obtains lattice structures of generalized truth values. Natural algebraic operations on these lattices correspond to logical operations, and the ordering relations give rise to entailment relations.