Paraconsistent logic is a method of reasoning about inconsistencies without falling into absurdity.

"Sometimes it is the people no one can imagine anything of who do the things no one can imagine." - Alan Turing

The belief that reasoning with inconsistent information in a managed and discriminating fashion should be feasible drives paraconsistent logic. Explosion theory rules it out. Thus that option is out. One inconsistent theory in non-paraconsistent logic is the trivial theory that treats every phrase as a theorem. Paraconsistent logic allows us to identify and reason about incoherent theories.

Dialetheism

The philosophical school of dialetheism was founded as a result of research into paraconsistent logic, and it maintains that authentic contradictions occur in reality, such as communities with contrasting moral perspectives. The alternative to paraconsistent logic for a dialetheist is to embrace trivialism, the view that all contradictions (and, by extension, all truths) are true. However, a dialetheist perspective is not required to study paraconsistent logic. For instance, Bas van Fraassen proposes that one can choose empirical adequacy over genuine theories or true contradictions.

Paraconsistent logic vs classical logic

Paraconsistent logic is propositionally weaker than classical logic; they accept fewer propositional conclusions. Paraconsistent logic can never be a propositional extension of classical logic, meaning it can never validate all it accomplishes. Paraconsistent logic is thus more conservative or cautious than classical logic. Because of this conservatism, paraconsistent languages can be more expressive than their classical counterparts, including the metalanguage hierarchy proposed by Alfred Tarski et al. Solomon Feferman states, "Natural language abounds with directly or indirectly self-referential yet harmless expressions—all of which are excluded from the Tarskian framework." In paraconsistent logic, this expressive barrier can be solved.

Paraconsistent Logic in Mathematics

The mathematics that is or may be inconsistent but not nonsensical can be developed thanks to the concept of paraconsistency. According to conventional logic, contradictions are interchangeable, making any inconsistent theory equally valid (or invalid).

Employing paraconsistent logic

The idea of employing paraconsistent logic to evaluate neural network outcomes arose from the review of an industry-level model's results. The outputs had an unusual shape: the model could identify one class but turned random with another. As a result, the precision value was disguised. Consider a balanced dataset as an example. Allow the model to correctly identify 50% of the test entries and randomly classify the remaining 50%. The outcome should be more than 75% accuracy, with each iteration incorrectly recognising different entries. In such a model, the real accuracy should be considered 50% rather than 75%.

Objective

The notion of paraconsistency states that not all contradictions produce irrational absurdities. Once we get past that initial claim, there is a wide range of perspectives and mechanisms for paraconsistent logic, from somewhat weak to relatively strong.

• At the very least, paraconsistent logic protects against human fallibility. As humans, we are fallible; as such, paraconsistent logic is necessary to keep us from stumbling into incoherence as we modify our theories, hold incorrect beliefs, and make mistakes. Claims of this modesty and conservatism offer little about the nature of truth. Weak paraconsistency is consistent with the idea that if a contradiction were true, then everything would be true since contradictions cannot be true regardless of one's beliefs or theories.
• In the end, the argument that some contradictions are true is supported by paraconsistent logic. According to this view, called dialetheism, the best theory (in mathematics, metaphysics, or even factual reality) is often incompatible with other theories. Although certain contradictions are true, strong paraconsistency insists they are all false. So, at this extreme, dialetheism is a contradiction in and of itself.

Image source: Unsplash