Project

Dialetheism and Paraconsistent Logic

Philosophy aims at making sense of things in a very general level. And the large majority of philosophers take the laws of excluded middle and non-contradiction as unquestionable. Moreover, logic is concerned with reasoning in one way or another. And the large majority of logicians accepts the so-called classical logic that reflects the above philosophical principles. This is of course not a surprise when we consider the success of philosophical theories as well as logical theories.

But, it is far from obvious that there is only one way to make sense of things. For example, Luitzen Egbertus Jan Brouwer, an outstanding mathematician known for Brouwer's fixpoint theorem, suggested to make sense of mathematics in a different way from the more standard one. In brief, Brouwer developed the idea that mathematics is a creation of the mind. This idea is at the core of what is now known as intuitionism which gave rise to a different way of reasoning, and thus a nonclassical logic called intuitionistic logic.

Yet another path for sense making goes through dialetheism, the view that some sentences (or propositions) are both true and false. That is, dialetheists of the likes of Graham Priest and Richard Routley, challenge the long tradition that runs since Aristotle who claimed that no sentence (or proposition) is both true and false. Moreover, dialetheism motivates a different kind of logic known as paraconsistent logic. In brief, paraconsistent logics block the inference, called ex contradictione quodlibet (also known as explosion), from a sentence (or a proposition) and its negation to arbitrary sentence. More formally,

A, ~ A ⊢ B

where ~ is intended to be negation, and ⊢ is a logical consequence relation.

However, there are some major problems in the developments of dialetheism and paraconsistent logics. First, for dialetheism, there are a number of interesting attempts at formulating dialetheic theories of truth and set, but still far from complete. Moreover, there are intensive discussions on the possibility of applying dialetheism to some philosophical problems, as well as history of philosophy, but not receiving the attention it deserves in the literature. Second, for paraconsistent logics, even though there are infinitely many systems of paraconsistent logics, there is no agreement on what the tilde, namely `~', is at all. It is intended to be a negation, but there are no common ground among paraconsistentists on what a negation is.

The aims of this project are twofold. The first aim is to offer a systematic and deep understanding of paraconsistent logics including those that are less known. The second aim is to develop and advance dialetheism through dialetheic theories with the aid of formal logic, and examine the dialetheic positions in various philosophical problems, both in the Western and the Eastern traditions.

About the Project

The project is funded by the Alexander von Humboldt Foundation in the context of the Sofja Kovalevskaja Award granted to Hitoshi Omori. The project started in December, 2018, and will run for 5 years with the amount of 1,592,000 euros, financed by the German Federal Ministry of Education and Research.