# Is there any logic system which ENTIRELY rejects non-contradiction of any kind for any sentence (i.e. all contradictions are true)? Is this possible?

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.

• Depending on your inference rules/axioms, this is just the inconsistent logic/language. You may also just take $A \land \neg A$ as an axiom schema with no inference rules or other axioms, but I don’t see how that’s much different. Finally, there are versions of Free FOL and Modal Logic in which you can have something of an inconsistency. Namely you may have formulas of the form $\forall x \bot$ and $\Box \bot$ respectively without also having $\bot$. Dec 3, 2023 at 16:35
• @PW_246 I'm not sure if I understand this properly since I'm only a beginner, but isn't what I'm looking for (based on my research thus far) closer to trivialism but without the first part (that all propositions must be true. I don't know if such a system can exist though): en.wikipedia.org/wiki/Trivialism. I don't know if what you told me would entail that all contradictions are true in a proper sense. Dec 3, 2023 at 16:38
• @setszu there aren’t a lot of options when it comes to having the second part without the first, especially semantically. Syntactically, all you have to do is have a proof system with $A \land \neg A$ as an axiom schema and with no inference rules that would allow you to freely infer a conjunct from a conjunction. Again I’m not sure how that would work semantically. Dec 3, 2023 at 18:26