### 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**P*_{1},…,*P*╞_{n}*Q,**then the set*{*P*_{1},…,*P*, ~_{n}*Q*}*is unsatisfiable*[i.e. a contradiction]*, and conversely*.
For to say that {

*P*_{1},…,*P*, ~_{n}*Q*} is unsatisfiable is just to assert that*P*_{1},…,*P*, and ~_{n}*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*P*_{1},…,*P*are._{n}^{}
In particular, it follows that if
{

*P*_{1},…,*P*} is unsatisfiable,_{n}*P*_{1},…,*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.*

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