The principle of implosion (verum ex quodlibet) states that a valid formula follows from anything. It is expressed:
B ⊨ A ∨ ¬A
Consider some paracomplete or intuitionistic logic in which the law of excluded middle is rejected, such that there are some statements of the language of the logic which are false and who's negations are also false (or statements which are simply undetermined and without a truth-value).
How does the principle of implosion "react" to such gaps? We know that the principle of explosion leads to triviality given a contradiction or glut, but would implosion + gaps do the same? Or perhaps this would lead to a complete implosion of the logic, where no statements are true?
This may be a simple question for many of you, but I am still fairly new to mathematical logic. The way I am approaching this is by taking the contrapositive,
¬(A ∨ ¬A) ⊨ ¬B
which suggests to me that a negated proposition follows from a truth-value gap, but I'm wondering whether ¬B is equivalent to B, given that any B is arbitrary? Or whether this is stating that an arbitrary B does not hold given a gap.
I hope I've made myself clear enough, and would appreciate some guidance.