Tuesday, June 23, 2015

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,…, PnQ, then the set {P1,…, Pn, ~Q} is unsatisfiable [i.e. a contradiction], and conversely.
For to say that {P1,…, Pn, ~Q} is unsatisfiable is just to assert that P1,…, Pn,  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,…, Pn are.
In particular, it follows that if {P1,…, Pn} is unsatisfiable, P1,…, PnQ, 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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