Quid est veritas?
Paraconsistent mathematics is a type of mathematics in which contradictions may be true. In such a system it is perfectly possible for a statement A and its negation not A to both be true.
How can this be, and be coherent? (…)
This statement is false.
To be true, the statement has to be false, and vice versa. Many brilliant minds have been afflicted with many agonising headaches over this problem, and there isn’t a single solution that is accepted by all. But perhaps the best-known solution (at least, among philosophers) is Tarski’s hierarchy, a consequence of Tarski’s undefinability theorem.
In a nutshell, Tarski’s hierarchy assigns semantic concepts (such as truth and falsity) a level. To discuss whether a statement is true, one has to switch into a higher level of language. Instead of merely making a statement, one is making a statement about a statement. A language may only meaningfully talk about semantic concepts from a level lower than it. Thus a sentence such as the liar’s sentence simply isn’t meaningful. By talking about itself, the sentence attempts unsuccessfully to make a claim about the truth of a sentence of its own level. (…)
How does a paraconsistent perspective address these paradoxes? The paraconsistent response to the classical paradoxes and contradictions is to say that these are interesting facts to study, instead of problems to solve.
