[FOM] Tennant on relevant logic

[A.P. Hazen]

    I have an awful feeling that two different things are being 
discussed without  distinguishing between them.  Could Professor 
Tennant say something about the  relation between  his "Classical 
Relevant Logic" (and his real favorite, "Intuitionistic  Relevant 
Logic") and the systems studied in the tradition starting with 
Ackermann, Anderson and Belnap?
    Tennant's systems amount to classical or intuitionistic logic with 
certain restrictions on the sorts of proofs allowed.  In particular, 
they have the same logical vocabulary as Classical (resp. 
Intuitionistic) logic.  What the other tradition calls relevant (or 
relevance) logics extend classical logic by the addition of 
non-classical connectives (the implication connective, and sometimes 
"fusion" and "fission", which are analogous to Girard's 
multiplicatives).  They also do NOT involve a restriction to cut-free 
proofs.
     Since the stronger relevant logics contain classical logic, I 
suspect Tennant's theorem (classical consequences of consistent 
axioms are relevant consequences of them) may well apply to them as 
well.  I am less confident (but basically ignorant) of whether it 
would apply to the weaker relevance logics  of interest to 
(dialetheic and other) fans of non-trivial inconsistent "naive" set 
theory.  (Cut-free formulations are known for many relevance logics, 
but the metatheory of these systems is very difficult!)

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Allen Hazen
Philosophy Department
University of  Melbourne