on whether a contradiction and bivalence conjointly imply explosion.

An explosionist is one who believes that any statement follows from a contradiction. Here’s an argument for explosionism:

1   A • ~A  ∴ B

2   A                1, conjunction elimination (•E)

3   A v B          2, disjunction introduction (vI)

4   ~A              1, •E

5   B                3, 4, disjunctive syllogism (DS)

Thus, if one wishes to be an anti-explosionist, one must say that either •E, vI, or DS is invalid.¹

According to John Bell and company, explosionism follows from bivalence:

Proposition 1.1 If P₁,…, P_n╞ Q, then the set {P₁,…, P_n, ~Q} is unsatisfiable [i.e. a contradiction], and conversely.

For to say that {P₁,…, P_n, ~Q} is unsatisfiable is just to assert that P₁,…, P_n,  and ~Q are never simultaneously true, which, given the principle of bivalence, amounts to asserting that ~Q is false, i.e. Q is true, whenever all of P₁,…, P_n are.

In particular, it follows that if {P₁,…, P_n} is unsatisfiable, P₁,…, P_n╞ Q, for any statement Q. That is, inconsistent premises yield any conclusion whatsoever.²

How so, exactly? How exactly does explosionism follow form bivalence?

1   A • ~A  ∴ B

2   A v ~A       (bivalence)

3   B v ~B        (bivalence)

4   ???? 

¹ If I were an anti-explosionist, I would deny vI. Many anti-explosionists deny DS rather than vI. I am utterly baffled by such anti-explosionists.

² John Bell, David DeVidi, and Graham Solomon, Logical Options (Toronto: Broadview Press, 2001), 14. Emphasis is Bell’s et al.