Non-classical Metatheory for Non-classical Logics
- Andrew Bacon
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
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.
- Field, H (2008) Saving truth from paradox. Oxford University Press, Oxford CrossRef
- Field, H (2000) Indeterminacy, degree of belief, and excluded middle. Nous 34: pp. 1-30 CrossRef
- Field, H. (2007). Solving the paradoxes, escaping revenge. In: Revenge of the liar: New essays on the paradox (p. 78).
- Fine, K (1983) The permutation principle in quantificational logic. Journal of Philosophical Logic 12: pp. 33-37 CrossRef
- Grišin, VN (1982) Predicate and set-theoretic calculi based on logic without contractions. Mathematics of the USSR-Izvestiya 18: pp. 41 CrossRef
- Hájek P. (2001). Metamathematics of fuzzy logic. Kluwer Academic Publishers.
- Hardegree, GM (1974) The conditional in quantum logic. Synthese 29: pp. 63-80 CrossRef
- Kreisel, G (1962) On weak completeness of intuitionistic predicate logic. Journal of Symbolic logic 27: pp. 139-158 CrossRef
- Kreisel, G (1967) Informal rigour and completeness proofs. Studies in Logic and the Foundations of Mathematics 47: pp. 138-186 CrossRef
- Leitgeb, H. (2007). On the Metatheory of field’s ‘Solving the Paradoxes, Escaping Revenge’. In: Revenge of the liar: New essays on the paradox (p. 159).
- McCarty, DC (1991) Incompleteness in intuitionistic metamathematics. Notre Dame journal of formal logic 32: pp. 323-358 CrossRef
- McCarty, DC (1994) On theorems of Gödel and Kreisel: Completeness and Markov’s principle. Notre Dame Journal of Formal Logic 35: pp. 99-107 CrossRef
- McGee, V (1997) How we learn mathematical language. The Philosophical Review 106: pp. 35-68 CrossRef
- McGee, V (2001) Review: John etchemendy, the concept of logical consequence. Bulletin of Symbolic logic 7: pp. 379-380 CrossRef
- Montague, R (1963) Syntactical treatments of modality, with corollaries on reflexion principles and finite axiomatizability. Acta philosophica fennica 16: pp. 153-167
- Rayo, A, Uzquiano, G (1999) Toward a theory of second-order consequence. Notre Dame Journal of Formal Logic 40: pp. 315-325 CrossRef
- Rayo, A., & Williamson, T. (2004). A completeness theorem for unrestricted first-order languages. In: Liars and heaps: New essays on paradox (pp. 331–356).
- Tarski, A. (2002). On the concept of logical consequence. In D. Jacquette (Ed.), Philosophy of logic: An anthology (pp. 210–215). Blackwell..
- Tye, M (1994) Sorites paradoxes and the semantics of vagueness. Philosophical Perspectives 8: pp. 189-206 CrossRef
- Welch, P (2008) Ultimate truth vis-a-vis stable truth. The Review of Symbolic Logic 1: pp. 126-142 CrossRef
- Whittle, B (2004) Dialetheism, logical consequence and hierarchy. Analysis 64: pp. 318-326 CrossRef
- Williamson, T. (1994). Vagueness. Routledge.
- Non-classical Metatheory for Non-classical Logics
Journal of Philosophical Logic
Volume 42, Issue 2 , pp 335-355
- Cover Date
- Print ISSN
- Online ISSN
- Springer Netherlands
- Additional Links
- Non-classical logic
- Higher order vagueness
- Fuzzy logic
- Andrew Bacon (1)
- Author Affiliations
- 1. Magdalen College, Oxford University, Oxford, UK