How to justify banning $A\to (B\to (A\land B))$ in relevance logic? (Follow-up to Which line(s) of my proof of $A\to (\neg A \to B)$ are not allowed in relevance logic? )
Theorem: $A\to (B\to (A\land B))$
Proof:


*

*$A\space\space$ (assume)

*$B\space\space$ (assume)

*$A\land B\space\space$ (intro $\land$ 1,2)

*$B \to (A\land B)\space\space$ (intro $\to$ 2, 3)

*$A\to (B\to (A\land B))\space\space$ (intro $\to$ 1, 4)


Apparently the inference on line 3 is banned by relevance logic. Commenting on the same issue (as I understand it) in a similar proof, Greg Restall writes:

We must do something to the rule for conjunction introduction to ban
  this proof. The required emendation is to only allow conjunction
  introduction when the two subproofs have exactly the same open
  assumption.

-- Relevant and Substructural Logics, page 20
My question: Why must we do anything to the rule for conjunction? Does the usual form of the rule result in any logical inconsistencies? Given the stunning successes of classical logic in just about every field of human endeavour, it seems to me that only logical inconsistencies could justify tinkering with the basic rules of logic in this way. Also, has relevance logic been used by any mathematicians to any significant degree?
 A: 
Why must we do anything to the rule for conjunction?

In Relevance Logic, the conditional symbol, $\to$, has a weaker meaning than in classical logic.   The claim $A\to B$ is only considered true when $B$ relevantly follows from $A$.   (Recollect the perannial questions of "Why is $A\to B$ true then $A$ is false, since surely $B$ cannot follow from $A$ when $A$ isn't even true?" Relevance Logic tries to capture that intuition that there is more to conditionality than material implication.)
As such, conjunction inherits this weaker meaning.   In Relevance Logic, the claim $A\land B$ is true exactly when both $A$ and $B$ are true and they have the same relavance (follow from the same assumptions).

Does the usual form of the rule result in any logical inconsistencies?

Internally.   If you don't restrict conjunction then you can prove $A\to (B\to A)$ when $B$ is not relevant to $A$, which should not be valid in Relevance Logic.   The system should not do what we want it to not do.

Given the stunning successes of classical logic in just about every field of human endeavour, it seems to me that only logical inconsistencies could justify tinkering with the basic rules of logic in this way. 

The universe does not always subscribe to classical logic (re: Quantum Logic).   Exploring alternate proof systems at least provides the tools to think flexibly about such things.
