Before I get to the question in the title of the post, let me give a quick rehearsal of some uncontroversial material, for readers innocent of this particular topic. In classical logic, anything follows from a contradiction:
A & ~A, ∴ B
is a valid argument. This argument form is known as ex falso quodlibet (EFQ); Graham Priest calls it 'explosion.' This clearly runs counter to our intuitions about what follows from what: it just doesn't seem like '2+2=200' follows from 'Grass is green and grass is not green.' Yet it does in classical logic.
Because this seems counterintuitive, people have devised logics in which EFQ is not a valid argument form. The most prominent is the family of relevance logics. So relevance logics score a point because they fit our intuitions about EFQ.
How do relevance logics avoid EFQ? Semantically/ Model-theoretically, they allow 'truth-value gluts', that is, a sentence can be both true and false. Now we can see why (A & ~A), ∴ B is invalid in logics allowing gluts: assign A both true and false, and assign B false. Then all the premises are true and the conclusion is not true.
That was all set-up. Now the question: suppose it turns out that there are no truth-value gluts, i.e., no sentence (or proposition or whatever) is both true and false. Would (some) defenders of relevance logics then accept EFQ? Well, perhaps all the glut people need is that gluts are possible, and not that actually some sentence is both true and false. Then my question would be: would EFQ-deniers accept EFQ if gluts were impossible? From my limited exposure to the literature (Priest, C. Mortensen NDJFL 1983), it seemed like the answer might be yes, because they say things like 'Disjunctive Syllogism (which is valid classically but not relevantly) is valid in all consistent reasoning-contexts.' I would've hoped that we could discard EFQ without taking on such a contentious idea as truth-value gluts...
And a further question just out of ignorance: does anyone characterize logical validity in such a way that it (i) avoids EFQ and (ii) does not require truth-preservation? I don't see any other way besides gluts to declare EFQ invalid, if we stick to the standard characterization of validity.