Abstract
The system R## of “true” relevant arithmetic is got by adding the ω-rule “Infer ∀xAx from A0, A1, A2, ....” to the system R# of “relevant Peano arithmetic”. The rule ⊃E (or “gamma”) is admissible for R##. This contrasts with the counterexample to ⊃E for R# (Friedman & Meyer, “Whither Relevant Arithmetic”). There is a Way Up part of the proof, which selects an arbitrary non-theorem C of R## and which builds by generalizing Henkin and Belnap arguments a prime theory T which still lacks C. (The key to the Way Up is a Witness Protection Program, using the ω-rule.) But T may be TOO BIG, whence there is a Way Down argument that produces a better theory TR, such that R## ⫅ TR ⫅ T. (The key to the Way Down is a Metavaluation, on which membership in T is combined with ordinary truth-functional conditions to determine TR.) The result is a theory that is Just Right, whence it never happens that A ⊃ C and A are theorems of R## but C is a non-theorem.
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REFERENCES
Meyer, R. K.: Relevant arithmetic, (abstract), Bulletin of the Section of Logic 5 (1976), 133–137.
Friedman, H. and Meyer, R. K.: Whither relevant arithmetic, The Journal of Symbolic Logic 57 (1992), 824–831.
Anderson, A. R., Belnap, N. D., Jr., and Dunn, J. M.: Entailment (vol. 2), Princeton, 1992.
Meyer, R. K.: Coherence revisited, typescript, ANU, 1976.
Ackermann, W.: Begründung einer strengen Implikation, The Journal of Symbolic Logic 21 (1956), 113–128.
Anderson, A. R.: Some open problems concerning the system E of entailment, Acta Philosophica Fennica 16 (1963), 7–18.
Meyer, R. K. and Dunn, J. M.: E, R, and γ, The Journal of Symbolic Logic 34(1969), 460–474.
Dunn, J. M.: Relevant predication 1: the formal theory, Journal of Philosophical Logic 16 (1987), 347–381.
Meyer, R. K.: Ackermann, Takeuti, und Schnitt: γ for higher-order relevant logics, (abstract), Bulletin of the Section of Logic 5 (1976), 138–144.
Routley, R. and Meyer, R. K.: The semantics of entailment (I), in: Truth, Syntax, Modality, H. Leblanc (ed), Amsterdam, 1973, 199–243.
Girard, J.-Y.: Linear logic, Theoretical Computer Science 50 (1987), 1–102.
Kalish, D. and Montague, R.: Logic: Techniques of Formal Reasoning, N. Y., 1964.
Curry, H. B.: Foundations of Mathematical Logic, McGraw-Hill, N. Y., 1963.
Meyer, R. K.: Arithmetic formulated relevantly, typescript, ANU, 1975.
Meyer, R. K. and Mortensen, C.: Inconsistent models for relevant arithmetics, The Journal of Symbolic Logic 49 (1984), 917–929.
Gabbay, D.: On second order intuitionistic propositional calculus with full comprehension, Archiv für mathematische Logik und Grundlagenforschung 16 (1974), 177–186.
Dunn, J. M. and Belnap, N. D., Jr.: The substitution interpretation of the quantifiers, Nous 2 (1968), 177–185.
Leblanc, H.: Truth-value Semantics, N. Holland, Amsterdam, 1976.
Heyting, A.: Die formalen Regeln der intuitionistischen Logik, in: Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physicalischmathematische Klasse (1930), 42–56.
Birkhoff, G. and Von Neumann, J.: On the logic of quantum mechanics, Ann. of Math. 37 (1936), 823–843.
Thistlewaite, P. B., McRobbie, M. A. and Meyer, R. K.: Automated Theorem-Proving in Non-Classical Logics, Pitman, London, 1988.
Routley, R. and Meyer, R. K.: The semantics of entailment III, Journal of Philosophical Logic 1 (1972), 192–208.
Slaney, J. K.: Reduced models for relevant logics without WI, Notre Dame Journal of Formal Logic 28 (1987), 395–407.
Mortensen, C.: Inconsistent mathematics, Kluwer, Dordrecht/Boston/London, 1994.
Mendelson, E: Introduction to Mathematical Logic, Van Nostrand, Princeton, 1964.
Schütte, K.: Beweistheoretische Erfassung der unendlichen Induktion in der Zahlentheorie, Math. Annalen 122 (1951), 369–389.
Dunn, J. M. and Meyer, R. K.: Gentzen's cut and Ackermann's gamma, in: J. Norman and R. Sylvan (eds), Directions in Relevant Logic, Kluwer, 1989, 229–240.
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Meyer, R.K. ⊃E is Admissible in “true” relevant arithmetic. Journal of Philosophical Logic 27, 327–351 (1998). https://doi.org/10.1023/A:1017990121294
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DOI: https://doi.org/10.1023/A:1017990121294