There is an ambiguity in the concept of deductive validity that went unnoticed until the middle of the twentieth century. Sometimes an inference rule is called valid because its conclusion is a theorem whenever its premises are. But often something different is meant: The rule's conclusion follows from its premises even in the presence of other assumptions. In many logical environments, these two definitions pick out the same rules. But other environments are context-sensitive, and in these environments the second notion is stronger. Sorting out this ambiguity has led to profound mathematical investigations with applications in complexity theory and computer science. The origins of this ambiguity and the history of its resolution deserve philosophical attention, because our understanding of logic stands to benefit from their details. I am eager to examine together with you, Crito, whether this argument will appear in any way different to me in my present circumstances, or whether it remains the same, whether we are to abandon it or believe in it.—Plato Crito, 46d

中文翻译：

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推理的上下文

直到 20 世纪中叶才注意到演绎有效性的概念存在歧义。有时，推理规则之所以被称为有效，是因为只要其前提是，其结论就是定理。但通常有一些不同的意思：即使存在其他假设，规则的结论也是从其前提得出的。在许多逻辑环境中，这两个定义选择相同的规则。但是其他环境是上下文敏感的，在这些环境中，第二个概念更强。解决这种歧义导致了对复杂性理论和计算机科学应用的深刻数学研究。这种歧义的起源及其解决的历史值得哲学关注，因为我们对逻辑的理解将受益于它们的细节。