## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 55, Number 1 (2014), 129-155.

### Revisiting $\mathbb{Z}$

Mauricio Osorio, José Luis Carballido, and Claudia Zepeda

#### Abstract

Béziau developed the paraconsistent logic $\mathbb{Z}$, which is definitionally equivalent to the modal logic $\mathbb{S}5$, and gave an axiomatization of the logic $\mathbb{Z}$: the system HZ. Omori and Waragai proved that some axioms of HZ are not independent and then proposed another axiomatization for $\mathbb{Z}$ that includes two inference rules and helps to understand the relation between $\mathbb{S}5$ and classical propositional logic. In the present paper, we analyze logic $\mathbb{Z}$ in detail; in the process we also construct a family of paraconsistent logics that are characterized by different properties that are relevant in the study of logics.

#### Article information

**Source**

Notre Dame J. Formal Logic, Volume 55, Number 1 (2014), 129-155.

**Dates**

First available in Project Euclid: 20 January 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1390246443

**Digital Object Identifier**

doi:10.1215/00294527-2377905

**Mathematical Reviews number (MathSciNet)**

MR3161417

**Zentralblatt MATH identifier**

1326.03033

**Subjects**

Primary: X001 Y002

Secondary: Z003

**Keywords**

paraconsistent logic logic $\mathbb{Z}$ modal logic nonmonotonic reasoning

#### Citation

Osorio, Mauricio; Carballido, José Luis; Zepeda, Claudia. Revisiting $\mathbb{Z}$. Notre Dame J. Formal Logic 55 (2014), no. 1, 129--155. doi:10.1215/00294527-2377905. https://projecteuclid.org/euclid.ndjfl/1390246443