⊃E is admissible in “true” relevant arithmetic

Journal of Philosophical Logic 27 (4):327 - 351 (1998)
The system R## of "true" relevant arithmetic is got by adding the ω-rule "Infer VxAx from AO, 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
Keywords Philosophy
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Reprint years 2004
DOI 10.1023/A:1017990121294
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The Semantics of Entailment — III.Richard Routley & Robert K. Meyer - 1972 - Journal of Philosophical Logic 1 (2):192 - 208.
Relevant Predication 1: The Formal Theory. [REVIEW]J. Michael Dunn - 1987 - Journal of Philosophical Logic 16 (4):347 - 381.

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Relevant Logic and the Philosophy of Mathematics.Edwin Mares - 2012 - Philosophy Compass 7 (7):481-494.

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