The principle of explosion is this:
$$ X \land \lnot X \rightarrow Y $$
And similarly, the rule of weakening/monotonicity of entailment is this:
$$ (X \rightarrow Y) \rightarrow (X \land A \rightarrow Y) $$
If in a given logic, the rule of weakening holds, you could substitute $X = P$, $Y = Q$, and $A = \lnot P$ (or $P = \lnot P$ and $A = P$) in the rule of weakening to get something that gets close assuming the law of the excluded middle holds (see line 4) - it matches the principle of explosion at least when $Q = X = \top$:
$$ (P \rightarrow Q) \rightarrow (P \land \lnot P \rightarrow Q) \tag{1} $$ $$ (\lnot P \rightarrow Q) \rightarrow (P \land \lnot P \rightarrow Q) \tag{2} $$ $$ ((P \lor \lnot P) \rightarrow Q) \rightarrow ((P \land \lnot P) \rightarrow Q) \tag{3} $$ $$ Q \rightarrow (P \land \lnot P \rightarrow Q) \tag{4} $$
Similarly, most logics I've found that reject the law of the excluded middle also reject the principle of explosion, either implicitly like minimal logic or explicitly like relevant logic. This pervasive pattern led me to question why that pattern exists in the first place - if nobody is making their logic work with it when they drop the rule of weakening, is it even possible at all (despite the broad opinionatedness)?
My question is two-fold:
Does the principle of explosion depend on the rule of weakening in any case? Or in more formal terms, does proving the rule of weakening to hold also prove the principle of explosion to hold or vice versa? If not generally, then under what constraints? (The above proof sets two constraints, but I'm not fully convinced either one of them are absolute.)
Are there any known non-monotonic logics where the principle of explosion does hold? (This would serve as a counterexample to my hypothesis.)
