Is Boolean Negation Explosive?
George Boole realised that truth and falsity could be analysed mathematically, and that negation is an operator that flips between the two. The negation of A can be computed with the formula: ~A = 1 - A. If you take 1 to be truth, and 0 to be false, and you assume that A is true, then ~A = 1 - A = 1 - 1 = 0, so ~A is false. And if A is false, then ~A = 1 - A = 1 - 0 = 1, so ~A is true. So we get that ~A = 0 iff A = 1 and ~A = 1 iff A = 0.
While Boole’s insight is quite simple from a modern point of view, it changed the way people research logic and made it a topic suitable for mathematical research. One feature that Boolean negation inherits from the mathematical treatment is the law of non-contradiction. If a proposition and its negation were both true, then we would get that 1 = 0, which is mathematically absurd. So A and its negation are mutually exhaustive.
My question is why Boolean negation is also explosive, in the sense that if a proposition A is true, and also its negation ~A is true, then everything is true. I’m not asking why it satisfies the law of explosion expressed in an object language as (A & ~A) -> B, which is easy to compute. I’m asking why the inference rule of explosion A, ~A |= B holds for Boolean negation.
The best answer I can think of is that the rule of explosion is necessary for mathematical reasoning because mathematics is maximally intolerant to contradiction and the rule of explosion is what reduces triviality to inconsistency. In mathematics, inconsistency is as bad as it gets and everything becomes true. There’s no room for coherence with inconsistent material. The second answer is that the rule is derivable, as per Alexander Neckham's old argument, which you can find in "Logic in the Wild."
Either way, that Boolean negation obeys the rule of explosion isn’t entailed by a mathematical treatment of negation like the law of non-contradiction follows, but rather obeys explosion because mathematical reasoning imposes it. That leaves wiggling room for a modern paraconsistent logician who has principled reasons for thinking that explosion isn’t a valid rule of inference. If it isn’t a valid rule, then Boolean negation doesn’t have to follow it, in which case the main characteristic of Boolean negation inherited from a mathematical treatment is that the law of non-contradiction holds. It leaves it open, however, for coherent reasoning in the presence of inconsistency.
