[FOM] question about relevance and variable-sharing

Arnon Avron aa at tau.ac.il
Sun Nov 24 02:12:24 EST 2013

I am not sure if the following result is the most important
(a subjective matter), or the deepest. However, it is a 
natural generalization of the variable-sharing property 
(see remark 2 below) which seems to be non-trivial:
the only proof I know uses Routley-Meyer semantics.

If A&B->C is a theorem of R while B->C and A share no variable,
Then B->C is a theorem of R. 

Remark 1:
This theorem implies the usual variable-sharing property, as well
as the following theorem: If A follows in R from the union
of T and S, and S has no variable in common with T or A,
then A follows in R from T alone (Here by "A follows in R from T" I mean
the most ordinary notion of "follows": that there is a finite sequence of
formulas ending with A, each element of which is either an axiom
of R, or an element of T, or can be derived from two previous
elements in the sequence by either MP or Adjunction).

Remark 2:
An equivalent formulation of the theorem above talks about
B->(C\/A) instead of A&B->C.

Remark 3: By R I mean here the system without propositional constants.
(I am not sure what "variable-sharing" means in case the use of
propositional constants is allowed.)

Remark 4:
The theorem applies not only to R, but to many other Relevance logics.

I hope that this helps.

Arnon Avron

On Thu, Nov 21, 2013 at 10:37:11AM -0800, Neil Tennant wrote:
> Could any member of this list please let me know what they judge to be the best/most important/deepest result about  variable-sharing that has ever been established for any (propositional) relevance logic of note?
> Thanks--
> Neil Tennant
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> FOM at cs.nyu.edu
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