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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.

Tankut Beygu
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setszu
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  • 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 $\bot$. – PW_246 Dec 03 '23 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): https://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. – setszu Dec 03 '23 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. – PW_246 Dec 03 '23 at 18:26
  • It seems quite useless. You can start with a single axiom schema that allows you to assert every formula. This means that $A$ is an axiom, as well as $\lnot A$ and $A \land \lnot A$ and so on. The only application that I can imagine is to politics. – Mauro ALLEGRANZA Feb 02 '25 at 15:46

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For such an attitude, the generic name is trivialism. In logic, triviality is a technical notion; in his A Dictionary of Philosophical Logic, Cook defines it as

A theory is trivial if it has all statements as theorems. In the presence of the rule of inference ex falso quodlibet triviality is equivalent to inconsistency. A theory that is not trivial is a non-trivial theory.

Trivialism, like solipsism, is deemed as an absurd position. It should not be hard to see that it collapses all the distinctions reasoning demands, and affirming each and every statement, it is subversive to rationality. We may illustrate this with a miniature model: Suppose we have adopted fatalism for which a plain expression of its principle is $\Box\neg A\vee\Box A$, where $\Box$ is the necessity operator and want to query the results in modal logic system T. So, we have

  1. $\;\Box A\rightarrow A\tag{by the system $\mathbf{T}$}$
  2. $\;A\rightarrow\neg\Box\neg A\tag{propositional equivalent of 1}$
  3. $\;\Box\neg A\vee\Box A\tag{the assumption of fatalism}$
  4. $\;\neg\Box\neg A\rightarrow\Box A\tag{propositional equivalent of 3}$
  5. $\;\neg\Box\neg A\rightarrow A\tag{hypothetical syllogism 2, 4}$
  6. $\;A\rightarrow\Box A\tag{propositional equivalent of 5}$
  7. $\;A\leftrightarrow\Box A\tag{material equivalence 1, 6}$

Hence, the principle of fatalism has trivialised modality and any significant difference between contingent truths and necessary truths are removed from our reasoning.

Thereby, trivialism is used in reductio ad absurdum arguments to undermine a challenged contention by demonstrating that it has untenable consequences. Along the same vein, it has a negative relation to paraconsistency and dialetheism.

Paraconsistency concerns the nature of logical consequence, rejecting the principle of ex falso (to put a nuance, contradictione might be better) quodlibet (i.e., $A, \neg A\vDash B$ whatever statement $B$ is), whilst dialetheism concerns the nature of logical truth, asserting there are dialetheia. Dialetheia is a statement $A$ such that $A$ and $\neg A$ are true. Hence, dialetheism holds that some contradictions are true.

Notice that paraconsistency and dialetheism claim to characterise different aspects of logic and neither of them necessarily implies the other one. They share trivialism as their limiting conditions; the systems built on them are formulated so as to avoid trivialism.

Whether personal or institutionalised, some belief systems may embrace the incoherence of trivialism, as some schools of Buddhism do, however, they are off our topic.

Tankut Beygu
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