Yes, I realized that you're on a paraconsistent logic bend a while ago, Baruch. Thing is, paraconsistency can't save you from the implications of the principle of explosion. It's just a matter of truth tables. A truth table analysis reveals that if p and p⊃q are both true, then the only truth value of q that makes sense is "true." Similarly, if p is false, then a truth table analysis reveals that p⊃q is true no matter what q is, therefore p⊃q is true for any q in the case that p is false. The trouble comes in when you consider p to be both true and false. It doesn't matter if paraconsistent logic says that there's no problem and the explosion doesn't occur, the problem is staring you straight in the face. There is a reason why the rules of valid logical argument have the form they do in classical logic.

Yes, the motivation behind paraconsistency is to resolve inconsistencies in a controlled way. I don't think that it does adequately because it still kind of leaves you in the lurch. Not only does it not really resolve the principle of explosion in a satisfying way, if you have to base an action on whether a particular implication is true or false, and those two choices are exclusive (if you try to, for instance, go to Panama and Canada at the same time, you're going to fail at least one of them), then you're sunk — you consider the critical statement both true and false, which leaves you no closer to resolving which action to take. It's much better to instead try to solve the inconsistency (resolve which of the cases actually matches reality), or work within a more powerful system that captures the uncertainty or vagueness presented in the inconsistencies, than to go to paraconsistent logics.

Oh, and another thing, the statements used in Godel's incompleteness theorem are NOT the same sort used in the liar paradox — they're very similar, but not the same. All such statements are of the form, "This statement is not provable in system S." It has a clear meaning independent of its truth, is therefore well-grounded (unlike the liar paradox, where the meaning of the liar statement, "This statement is false," is dependent on its truth value). That is the power of the theorem, that these statements, though self-referential, are nonetheless well-grounded in meaning — you know exactly what scenario is needed for the statements to be true or false, unlike with the liar paradox. Godel's theorems are not grounds for dialetheism, nor have I encountered any statement that would give it grounds. Every paradoxical statement used in support of dialetheism turns out to be a mere problem in semantics. This makes sense, because ultimately, what we operate on in logic is *statements,* not actual states of affairs.