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

逻辑学是关于有效推论的理论，常被描述为关于真值的科学。从古典观点来看，真理值只有真值和假值两种，每个陈述不是真就是假，而不是两者都是。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，在文献中，我们可以找到建设性与非建设性真理和谬误的概念。人类和人工智能体所面对的信息往往是不完整的，有时是不一致的，这表明陈述也可能被评估为非真非假或既真又假。真理和谬误可以获得其他哲学上相关的限定：必要真理、可能谬误、从此以后的真理、众所周知的真理等等。经典和非经典真值可以结合起来得到新的、结构化的真值。广义的，可能是结构化的真值的概念建议在这些值上排序关系。这些价值可以根据它们的建设性程度、必要性、所提供的信息量等进行比较。结果，得到了广义真值的格结构。这些格上的自然代数运算对应于逻辑运算，排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.