The rules of inference in classical logic seem to imply the principle of explosion (
ex contradictione quodlibet ECQ). There are some very simple proofs. So the question becomes, how can the rules of classical logic best be modified so as to eliminate volatility. But introducing changes creates the danger of implications rippling through the logical system and creating inconsistencies elsewhere.
This is the province of so-called 'para-consistent logic' of which there are several different varieties. See the little history in section 1.2 in the article below (by Graham Priest, a very big name):
https://plato.stanford.edu/entries/logic-paraconsistent/
"In antiquity, however, no one seems to have endorsed the validity of ECQ. Aristotle presented what is sometimes called the connexive principle: "it is impossible that the same thing should be necessitated by the being and by the not-being of the same thing" (Prior Analytic II 4 57b3)...
...The principle was taken up by Boethius (480-524) and Abelard (1079-1142), who considered two accounts of consequences. The first one is a familiar one: it is impossible for the premises to be true but conclusion false. The first account is thus similar to the contemporary notion of truth-preservation. The second one is less accepted recently: the sense of the premises contains that of the conclusion. This account, as in relevant logics, does not permit an inference whose conclusion is arbitrary. Abelard held that the first account fails to meet the connexive principle and that the second account (the account of containment) captured Aristotle's principle.
Abelard's position was shown to face a difficulty by Alberic of Paris in the 1130's. Most medieval logicians didn't, however, abandon the account of validity based on containment or something similar. But one way to handle the difficulty is to reject the connexive principle. This approach, which has become most influential, was accepted by the followers of Adam Balsham or Parvipontanus... The Parvipontanians embraced the truth-preservation account of consequences and the "paradoxes" that are associated with it. In fact, it was a member of the Parvipontanians, William of Soissons, who discovered in the twelfth century what we now call the C.I. Lewis argument for ECQ...
The containment account, however, did not disappear. John Duns Scotus (1266-1308) and his followers accepted the containment account. The Cologne School of the late fifteenth century argued against ECQ by rejecting disjunctive syllogism..."
This little proof of ECQ is from the
'Principle of Explosion' article in Wikipedia
1. P & ~P ...our premise, some arbitrary contradiction
2. P ...from 1 by conjunction elimination
3. ~P ...from 1 by conjunction elimination
4. P or Q ...from 2 by disjunction introduction, where Q is any arbitrary proposition
5. Q ...our conclusion, from 3 and 4 by disjunctive syllogism