In classical logic, there is a principle known as the Principle of Explosion. But this principle allows absurd conclusions to follow from contradictions. A problem! It says that from a contradiction, anything can follow. For example, suppose we accept two statements: “It is raining” and “It is not raining.” In classical logic, accepting both as true allows us to logically conclude something completely unrelated, such as “The Moon is made of cheese.”
Paraconsistent logic is a family of logical systems that allow contradictions to be handled without making every statement provable. Paraconsistent logic simply says what we just showed, The Principle of Explosion is the problem. A problem of the mind, not reality.
For thinking well, paraconsistent logic supports the idea that reality takes precedent over our thoughts. It manages contradictions. The material world comes first, and ideas are tools for describing it. Simply put, when two claims contradict each other, that’s the problem. Fix it.
The Split helps us keep that distinction clear. Paraconsistent logic operates on the ideas side of the Split, allowing us to work with conflicting representations without assuming they are both accurate descriptions of nature.
A philosopher might object that TST gives classical logic priority without demonstrating that it applies universally. And they are right because Hume was right about induction, but it’s a solvable problem when the empirical precedes the rational.
The TST split makes a distinction between reality and our ideas about it. Handling a contradiction is not the same as establishing that a contradiction exists in nature. TST maintains the Law of Noncontradiction because induction supports it: across our observations, we have no reliable and reproducible demonstration of something both being and not being in the same respect, at the same time. Show me such a case in nature, and I’ll revise our understanding of how logic applies to reality. Reality is the standard. Logic must answer to it.