I have always been puzzled by the fact that philosophers working with paraconsistent logic have always had a thing against disjunctive syllogism. Since someone dealing with paraconsistent logic wants a logic that does not allow contradiction explosion (if a contradiction is true, everything is true), the ostensible reason is that DS requires explosion:
(1) p & ~ p
(2) p
(3) p v q
(4) ~p
(5) q
(2) and (4) come from (1) by conjunction elimination; (3) from (2) by the disjunction rule called addition (also commonly called disjunction introduction); and (5) from (3) & (4) by DS. But this is not an adequate reason; you can also block the argument by rejecting addition, and there's a fairly obvious reason for it. Contradiction explosion occurs because any arbitrary proposition can be introduced; but DS does not introduce any propositions. Addition, however, does. Addition, in fact, is already explosive, in a mild sense: given addition, any proposition can be introduced into any argument. And we find that outside philosophy (e.g., computer science) it seems fairly common to explore paraconsistency by rejecting addition. So is there any other reason for picking on DS?
Graham Priest (An Introduction to Non-Classical Logic 1.10.4-5) gives the following sort of argument. The following principle seems reasonably plausible:
(*) 'If A then B' is true if there is some true statement C such that from C and A together we can deduce B.
But we can reason in the following way:
[Mat]
Assumed:
(1) ~A v B
Assumed:
(2) A
By Disjunctive Syllogism, from (1) and (2):
(3) B
Then by (*) we get 'If A then B'.
OK, so why is that supposed to be an issue? The above argument is apparently supposed to be an argument that 'If A then B' is a material conditional. But this is a problematic conclusion; and DS is certainly responsible if anything is.
I find the argument intriguing; I'm no fan of the material conditional interpretation, so I would consider an argument that DS requires that interpretation to be a strike against it. But this argument doesn't show that 'If A then B' is a material conditional; it simply shows that whenever you can assume both A and (~A v B) you can conclude 'If A then B' (if you assume DS). But this is not particularly interesting, because it is not distinctive to the material conditional interpretation. If the material conditional interpretation is true, you can conclude 'If A then B' from (~A v B), without having to combine (~A v B) with A; and the argument doesn't show that you can do this. All it does is show that on the material conditional interpretation DS and modus ponens are redundant: you can do with DS anything you can do with MP.
We can see the problem more clearly when we realize that there are other statements that can be plugged into (*)'s C. Consider this one:
Assumed:
(1) Necessarily (If A then B)
Assumed:
(2) A
From 1, by T axiom (i.e., in modal system M):
(3) If A then B
From (2) and (3):
(4) B
And by (*) we can conclude 'If A then B'. So by parallel reasoning, 'If A then B' must be the modal 'Necessarily (If A then B)'; and, being modal, this is definitely not a material conditional. But you can run the first argument in modal system M, too. So if the above argument were acceptable, it would imply that for M 'If A then B' is both a material conditional and not a material conditional; and this is a problem for the material conditional interpretation. This confirms the point above about the gist of the original argument.
So I'd say there's nothing problematic about [Mat], even for one who rejects the material conditional interpretation. This was so obvious to me on reading it that I really wonder if I'm missing something subtle and clever.
