⊃E is admissible in “true” relevant arithmetic
Journal of Philosophical Logic 27 (4):327 - 351 (1998)
| Abstract | The system R## of true relevant arithmetic is got by adding the -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 -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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