History and Philosophy of Logic 39 (4):365-395 (2018)

Authors
Curtis Franks
University of Notre Dame
Abstract
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, whet...
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DOI 10.1080/01445340.2018.1439257
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Philosophy of Logic.W. V. O. Quine - 1970 - Harvard University Press.
Philosophy of Logic.Michael Jubien & W. V. Quine - 1988 - Journal of Symbolic Logic 53 (1):303.

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