The logic of quasi-truth
The need for paraconsistent fuzzy logic has become more pressing in recent years by practical applications. For instance, information systems have become contradictory because of their size and diversity. If inconsistency is not addressed in a coherent way, query answering in such systems would be impossible. In paraconsistent fuzzy logic, a proposition is not restricted to being true or false. Its truth value can also be partial, something between true and false. The Polish logician and philosopher Jan Łukasiewicz dedicated much of his attention to such many-valued logics and developed his own many-valued propositional calculus. The EU-funded project PARFUZGENQ (Paraconsistent fuzzy logic with generalized quantifiers) was focused on perfect many-valued algebras. In these algebraic structures, below absolute truth, there are infinite many quasi-truths and above absolute false, there are infinite many quasi-falsehoods. Scientists studied a generalised Łukasiewicz logic with a particular feature: quasi-true propositions never lead to conclusions that would be false. By associating certain values with propositions and the conclusion, they proved that the validity of intermediate syllogisms is determined by a simple multi-valued algebra equation. Specifically, in traditional syllogisms, like 'if all M are P and all M are S, then some S are not P', there are only two quantifiers involved: the universal 'all' and the particular 'some'. But additional quantifiers exist, including 'few', 'many' and 'most' and intermediate syllogisms related to these quantifiers. The PARFUZGENQ scientists applied many-valued algebra semantics to intermediate propositions and showed that the validity of intermediate syllogisms can be calculated using these semantics. In other words, they showed that empirical facts on intermediate syllogisms form a multi-valued algebra. Paraconsistent logics are applicable to solving complex decision-making problems, as demonstrated in two project publications in renowned peer-reviewed journals. The project has resulted in a sound mathematical basis that will allow fuzzy logics to be adapted to more real-world applications.
Keywords
Paraconsistent fuzzy logic, fuzzy logic, Łukasiewicz, many-valued algebras, intermediate syllogisms
