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

逻辑，即有效推理的理论，通常被描述为真理价值的科学。从经典的观点来看，真假恰好有两个真值，每一种说法都是真或假，而不是两者兼而有之。然而，有许多哲学上重要的方法来概括经典的真理价值概念。例如，人们可以在文献中找到建设性与非建设性真假的概念。人类和人工代理人面临的信息往往是不完整的，有时甚至不一致，这表明陈述也可能被评估为既不真实也不虚假，或者既真实又虚假。真理和谬误可以获得其他哲学上相关的资格：必然真理、可能的谬误、从今以后的真理、众所周知的真理等等。经典真值和非经典真值可以结合在一起，以获得新的结构化真值。广义的、可能是结构化的真值的想法暗示了对这些值的排序关系。这些值可以根据它们的建设性程度、必要性、它们提供的信息量等进行比较。结果，人们获得了广义真值的格子结构。这些格上的自然代数运算对应于逻辑运算，而序关系产生蕴涵关系。

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.