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METAVALUATIONS

Published online by Cambridge University Press:  04 December 2017

ROSS T. BRADY
ROSS T. BRADY*
Affiliation:
PHILOSOPHY LA TROBE UNIVERSITY MELBOURNE, VICTORIA3086, AUSTRALIAE-mail:[email protected]
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Abstract

This is a general account of metavaluations and their applications, which can be seen as an alternative to standard model-theoretic methodology. They work best for what are called metacomplete logics, which include the contraction-less relevant logics, with possible additions of Conjunctive Syllogism, (A→B) & (B→C) → .A→C, and the irrelevant, A→ .B→A, these including the logic MC of meaning containment which is arguably a good entailment logic. Indeed, metavaluations focus on the formula-inductive properties of theorems of entailment form A→B, splintering into two types, M1- and M2-, according to key properties of negated entailment theorems (see below). Metavaluations have an inductive presentation and thus have some of the advantages that model theory does, but they represent proof rather than truth and thus represent proof-theoretic properties, such as the priming property, if ├ A $\vee$ B then ├ A or ├ B, and the negated-entailment properties, not-├ ∼(A→B) (for M1-logics, with M1-metavaluations) and ├ ∼(A→B) iff ├ A and ├ ∼ B (for M2-logics, with M2-metavaluations). Topics to be covered are their impact on naive set theory and paradox solution, and also Peano arithmetic and Godel’s First and Second Theorems. Interesting to note here is that the familiar M1- and M2-metacomplete logics can be used to solve the set-theoretic paradoxes and, by inference, the Liar Paradox and key semantic paradoxes. For M1-logics, in particular, the final metavaluation that is used to prove the simple consistency is far simpler than its correspondent in the model-theoretic proof in that it consists of a limit point of a single transfinite sequence rather than that of a transfinite sequence of such limit points, as occurs in the model-theoretic approach. Additionally, it can be shown that Peano Arithmetic is simply consistent, using metavaluations that constitute finitary methods. Both of these results use specific metavaluational properties that have no correspondents in standard model theory and thus it would be highly unlikely that such model theory could prove these results in their final forms.

Keywords

03-0203B4703F52metavaluationmetacompleteproof theorypriming propertyexistential propertyconstructive logicmeaning containment
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 23 , Issue 3 , September 2017 , pp. 296 - 323
Copyright
Copyright © The Association for Symbolic Logic 2017 

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References

REFERENCES

Anderson, A. R. and Belnap, N. D. Jr., Entailment, The Logic of Relevance and Necessity, vol. 1, Princeton University Press, Princeton, NJ, 1975.Google Scholar
Brady, R. T., Completeness proofs for the systems RM3 and BN4 . Logique et Analyse, vol. 25 (1982), pp. 9–32.Google Scholar
Brady, R. T., The simple consistency of a set theory based on the logic CSQ . Notre Dame Journal of Formal Logic, vol. 12 (1983), pp. 447–453.Google Scholar
Brady, R. T., Relevant implication and the case for a weaker logic . Journal of Philosophical Logic, vol. 25 (1996), pp. 151–183.Google Scholar
Brady, R. T., Universal Logic, CSLI Publs, Stanford, 2006.Google Scholar
Brady, R. T., A rejection system for the first-degree formulae of some relevant logics . Australasian Journal of Logic, vol. 6 (2008), pp. 55–69.Google Scholar
Brady, R. T., Negation in metacomplete relevant logics . Logique et Analyse, vol. 51 (2008a), pp. 331–354.Google Scholar
Brady, R. T., Extending metacompleteness to systems with classical formulae . Australasian Journal of Logic, vol. 8 (2010), pp. 9–30.Google Scholar
Brady, R. T., Metavaluations, naive set theory and inconsistency , Logic without Frontiers, Festschrift for Walter Alexandre Carnielli on the Occasion of his 60th Birthday (Beziau, J.-Y. and Coniglio, M. E., editors), College Publications, London, 2011, pp. 339–360.Google Scholar
Brady, R. T., The consistency of arithmetic, based on a logic of meaning containment . Logique et Analyse, vol. 55 (2012), pp. 353–383.Google Scholar
Brady, R. T., The simple consistency of naive set theory using metavaluations . Journal of Philosophical Logic, vol. 43 (2014), pp. 261–281.Google Scholar
Brady, R. T., The Use of Definitions and their Logical Representation in Paradox Derivation. Presented to the World Congress in Universal Logic, University of Istanbul, Istanbul, June, 2015.Google Scholar
Brady, R. T. and Meinander, A., Distribution in the logic of meaning containment and in quantum mechanics , Paraconsistency: Logic and Applications (Tanaka, K., Berto, F., Mares, E., and Paoli, F., editors), Springer, Dordrecht, 2013, pp. 223–255.Google Scholar
Dunn, J. M., Relevance logic and entailment , Handbook of Philosophical Logic, vol. 3, first ed. (Gabbay, D. and Guenthner, F., editors), Reidel, Dordrecht, 1986, pp. 117–224.Google Scholar
Dunn, J. M. and Meyer, R. K., Gentzen’s cut and Ackermann’s γ, Directions in Relevant Logic (Norman, J. and Sylvan, R., editors), Kluwer, Dordrecht, 1989, pp. 229–240.Google Scholar
Dunn, J. M. and Restall, G., Relevance logic , Handbook of Philosophical Logic, vol. 6 (Gabbay, D. and Guenthner, F., editors), second ed., Kluwer, Dordrecht, 2002, pp. 1–128.Google Scholar
Fine, K., Modal Logic as Metalogic, manuscript, University of Edinburgh, 1971–3, pp. 1–5.Google Scholar
Harrop, R., On disjunctions and existential statements in intuitionistic systems of logic . Mathematische Annalen, vol. 132 (1956), pp. 347–361.Google Scholar
Harrop, R., Concerning formulas of types A → $B\, \vee \,C$ , A → (Ex)B(x) in intuitionistic formal systems . Journal of Symbolic Logic, vol. 25 (1960), pp. 27–32.Google Scholar
Kleene, S. C., Disjunction and existence under implication in elementary intuitionistic formalisms . Journal of Symbolic Logic, vol. 27 (1962), pp. 11–18.Google Scholar
Kleene, S. C., An addendum . Journal of Symbolic Logic, vol. 28 (1963), pp. 154–156.Google Scholar
Mares, E. D. and Meyer, R. K., The admissibility of γ in R4 . Notre Dame Journal of Formal Logic, vol. 33 (1992), pp. 197–206.Google Scholar
Meyer, R. K., Entailment and relevant implication . Logique et Analyse, vol. 11 (1968), pp. 472–479.Google Scholar
Meyer, R. K., On coherence in modal logics. Logique et Analyse, vol. 14 (1971), pp. 658–668.Google Scholar
Meyer, R. K., Metacompleteness . Notre Dame Journal of Formal Logic, vol. 17 (1976), pp. 501–516.Google Scholar
Meyer, R. K., Ackermann, Takeuti and Schmitt for higher-order relevant logic (abstract) . Bulletin of the Section of Logic, Polish Academy of Science, vol. 5 (1976a), pp. 138–144.Google Scholar
Meyer, R. K., Coherence Revisited, manuscript, A.N.U., 1972–1975.Google Scholar
Meyer, R. K. and Dunn, J. M., E, R and γ . Journal of Symbolic Logic, vol. 34 (1969), pp. 460–474.Google Scholar
Meyer, R. K. and Restall, G., ‘Strenge’ arithmetics . Logique et Analyse, vol. 42 (1999), pp. 205–220.Google Scholar
Routley, R., Meyer, R. K., Plumwood, V., and Brady, R. T., Relevant Logics and Their Rivals, vol. 1, Ridgeview, Atascadero, 1982.Google Scholar
Seki, T., The γ-admissibility of relevant modal logics II - the method using metavaluations . Studia Logica, vol. 97 (2011), pp. 351–383.Google Scholar
Slaney, J. K., A metacompleteness theorem for contraction-free relevant logics . Studia Logica, vol. 43 (1984), pp. 159–168.Google Scholar
Slaney, J. K., Reduced models for relevant logics without WI . Notre Dame Journal of Formal Logic, vol. 28 (1987), pp. 395–407.Google Scholar