Towards a Non-classical Meta-theory for Substructural Approaches to Paradox

Journal of Philosophical Logic 50 (5):1007-1055 (2021)


In the literature on self-referential paradoxes one of the hardest and most challenging problems is that of revenge. This problem can take many shapes, but, typically, it besets non-classical accounts of some semantic notion, such as truth, that depend on a set of classically defined meta-theoretic concepts, like validity, consistency, and so on. A particularly troubling form of revenge that has received a lot of attention lately involves the concept of validity. The difficulty lies in that the non-classical logician cannot accept her own definition of validity because it is given in a classical meta-theory. It is often suggested that this mismatch between the consequence relation of the account being espoused and the consequence relation of the meta-theory is a serious embarrassment. The main goal of the paper is to explore whether certain substructural accounts of the paradoxes can avoid this sort of embarrassment. Typically, these accounts are expressively incomplete, since they cannot assert in the object language that certain invalid arguments are in fact invalid. To overcome this difficulty I develop a novel type of hybrid proof-procedure, one that takes invalidities to be just as fundamental as validities. I prove that this proof-procedure enjoys a number of interesting properties and I analyze the prospects of applying it to languages capable of expressing self-referential statements.

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Author's Profile

Lucas Rosenblatt
University of Buenos Aires

References found in this work

Saving Truth From Paradox.Hartry Field - 2008 - Oxford, England: Oxford University Press.
Spandrels of Truth.J. C. Beall - 2009 - Oxford, England: Oxford University Press.
Paradoxes and Failures of Cut.David Ripley - 2013 - Australasian Journal of Philosophy 91 (1):139 - 164.
The Liar: An Essay on Truth and Circularity.Jon Barwise & John Etchemendy - 1987 - Oxford, England and New York, NY, USA: Oxford University Press USA.

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