From Paraconsistent Logic to Dialetheic Logic

Abstract

The only condition for a logic to be paraconsistent is to invalidate the so-called explosion. However, the understanding of the only connective involved in the explosion, namely negation, is not shared among paraconsistentists. By returning to the modern origin of paraconsistent logic, this paper proposes an account of negation, and explores some of its implications. These will be followed by a consideration on underlying logics for dialetheic theories, especially those following the suggestion of Laura Goodship. More specifically, I will introduce a special kind of paraconsistent logic, called dialetheic logic, and present a new system of paraconsistent logic, which is dialetheic, by expanding the Logic of Paradox of Graham Priest. The new logic is obtained by combining connectives from different traditions of paraconsistency, and has some distinctive features such as its propositional fragment being Post complete. The logic is presented in a Hilbert-style calculus, and the soundness and completeness results are established.

Date: March 16, 2016.

The author was a Postdoctoral Fellow for Research Abroad of the Japan Society for the Promotion of Science (JSPS) at the time of submission, and now a Postdoctoral Research Fellow of JSPS. I would like to thank Holger Andreas and Peter Verdée for their encouragement and patience. I would also like to thank Diderik Batens, Filippo Casati, Petr Cintula, Michael De, Graham Priest, Dilip Raghavan, Greg Restall, Daniel Skurt, Heinrich Wansing and Zach Weber for their valuable suggestions, comments and discussions. Earlier versions of the paper were presented to conference Paraconsistent Reasoning in Science and Mathematics in Munich, Workshop on Non-Classical [Meta]Mathematics in Otago and seminars in Ghent, Munich, Singapore and Melbourne, and many thanks go to organizers and audiences. Finally, I would like to thank the referees for their helpful comments and suggestions which substantially improved the paper.

Appendix

Details of Remark 10 Consider the algebra $\langle \{ \mathbf {t}, \mathbf {b}, \mathbf {f} \} , \{ \bot , {\sim }, \wedge , \vee , \rightarrow \} \rangle$ where the operations are defined as follows:

The aim here is to show that $\circ$ is not definable in this algebra. To this end, we prove the following lemma.

Lemma 4

Let $\varphi (p)$ be any formula in the language $\mathcal {L}_\bot$ whose only propositional variable is p. Then, there are seven cases for the value of $\varphi (p)$ depending on the value assigned to p, namely:

Proof

We proceed by induction on the complexity of $\varphi (p)$. For the base case, if $\varphi (p)$ is p or $\bot$, then it satisfies the condition (2) or (7) respectively. For the induction step, we cover only three of the four cases, as the others are similar.

Case 1: let $\varphi (p)$ be of the form ${\sim }\psi (p)$. Then, by induction hypothesis, $\psi (p)$ satisfies one of the seven cases. And with the truth table for ${\sim }$ in mind, $\varphi (p)$ satisfies the condition $(8-i)$ when $\psi (p)$ satisfies (i) ($i\in \{ 1, 2, \dots , 7 \}$) respectively.

Case 2: let $\varphi (p)$ be of the form $\psi ( p ) \wedge \xi (p)$. Then, by induction hypothesis, $\psi (p)$ and $\xi (p)$ both satisfy one of the eight conditions. And with the truth table for $\wedge$ in mind, $\varphi (p)$ behaves as follows:

The case for disjunction is similar to the case for conjunction.

Case 3: let $\varphi (p)$ be of the form $\psi ( p ) \rightarrow \xi (p)$. Then, by induction hypothesis, $\psi (p)$ and $\xi (p)$ both satisfy one of the eight conditions. And with the truth table for $\wedge$ in mind, $\varphi (p)$ behaves as follows:

This completes the proof.    $\square$

This implies that $\circ$ is not definable. Indeed, if $\circ$ is definable, then we will have the case in which $\circ \mathbf {t}=\mathbf {t}$ and $\circ \mathbf {b}=\mathbf {f}$, but this is not the case in view of the above lemma.

Details of Remark 20 In view of Theorem 2, and that $\rightarrow$ is essentially binary, it suffices to show that all unary functions are definable in the algebra $\langle \{ \mathbf {t}, \mathbf {b}, \mathbf {f} \} , \{ {\sim }, \circ , \rightarrow \} \rangle$. In other words, we need to show $27(=3^3$) functions are definable. This can be done as follows.

This completes the proof.    $\square$