explosion sans disjunctive syllogism. partie trois.

Consider the following inference rule:

False Antecedent (FA):

       φ

∴  ¬φ ⊃ ψ

FA is underwritten by the truth conditions for the material conditional: a statement of the form φ ⊃ ψ is not true iff φ is true and ψ is not true, and true otherwise. Thus, given φ, the conditional ¬φ ⊃ ψ is true because ¬φ is not true.

Now suppose that some contradiction holds:

1.   A • ¬A      ∴ B

2.   A               1, •E

3.   ¬A             1, •E

4.   ¬A ⊃ B     2, FA

5.   B               3, 4, MP

Thus, if one wishes to say that there is at least one true contradiction and deny that anything follows from this fact, one must give up either •E, MP, or FA. Suppose FA is the culprit. Then one must affirm that there it is not true that a statement of the form φ ⊃ ψ is not true iff φ is true and ψ is not true.