Non-classical Metatheory for Non-classical Logics

Journal of Philosophical Logic 42 (2):335-355 (2013)
Abstract
A number of authors have objected to the application of non-classical logic to problems in philosophy on the basis that these non-classical 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 non-classicality 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 non-classical logics are strong enough to formulate their own model theory in a corresponding non-classical 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 non-classical accounts of vagueness is drawn: there can be no axiomatic, and therefore precise, system which is determinately sound and complete
Keywords Non-classical logic  Metalogic  Vagueness  Higher order vagueness  Fuzzy logic
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    References found in this work BETA
    Kit Fine (1983). The Permutation Principle in Quantificational Logic. Journal of Philosophical Logic 12 (1):33 - 37.

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