Nonclassical Metatheory for Nonclassical Logics
 Andrew Bacon
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A number of authors have objected to the application of nonclassical logic to problems in philosophy on the basis that these nonclassical logics are usually characterised by a classical metatheory. In many cases the problem amounts to more than just a discrepancy; the very phenomena responsible for nonclassicality occur in the field of semantics as much as they do elsewhere. The phenomena of higher order vagueness and the revenge liar are just two such examples. The aim of this paper is to show that a large class of nonclassical logics are strong enough to formulate their own model theory in a corresponding nonclassical set theory. Specifically I show that adequate definitions of validity can be given for the propositional calculus in such a way that the metatheory proves, in the specified logic, that every theorem of the propositional fragment of that logic is validated. It is shown that in some cases it may fail to be a classical matter whether a given sentence is valid or not. One surprising conclusion for nonclassical accounts of vagueness is drawn: there can be no axiomatic, and therefore precise, system which is determinately sound and complete.
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 Title
 Nonclassical Metatheory for Nonclassical Logics
 Journal

Journal of Philosophical Logic
Volume 42, Issue 2 , pp 335355
 Cover Date
 20130401
 DOI
 10.1007/s1099201292239
 Print ISSN
 00223611
 Online ISSN
 15730433
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Nonclassical logic
 Metalogic
 Vagueness
 Higher order vagueness
 Fuzzy logic
 Authors

 Andrew Bacon ^{(1)}
 Author Affiliations

 1. Magdalen College, Oxford University, Oxford, UK