I've recently learned about paraconsistent and intuitionistic logic, and dialetheism.
According to the Stanford Encyclopedia of Philosophy's page on Dialetheism, it states:
Dialetheism is the view that some contradictions are true.
Likewise, it also states:
Dialetheic paraconsistency has it that some inconsistent but non-trivial theories are true.
As per the following answer on Math StackExchange, it is stated that there is a system of logic which essentially rejects the standard principles found in Classical Logic:
Lukasiewicz 3-valued logic rejects both the law of noncontradition and the law of the excluded middle as general laws applicable to all propositions, although they do still apply in special cases. As a propositional logic, it does not make specific use of the law of identity.
However, even then, contradiction is not entirely rejected, but merely restricted so that there are different appropriate contexts as far as I understand:
Proof by contradiction becomes much more difficult
Encountering systems like this in logic made me wonder if there is any system where all contradictions are considered acceptable and true, rather than in dialetheism where only some are considered true.
If it's possible, please tell me which system would allow for this, and how it works.
If it's not possible, please give a brief explanation of why or a link to a page which explains/proves why it's not possible if that is easier.
Please understand that I am a beginner in logic and mathematics, and thus, if I say something incorrect, please let me know.