## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 58, Number 2 (2017), 221-239.

### Dunn–Priest Quotients of Many-Valued Structures

#### Abstract

J. Michael Dunn’s *Theorem in 3-Valued Model Theory* and Graham Priest’s *Collapsing Lemma* provide the means of constructing first-order, three-valued structures from classical models while preserving some control over the theories of the ensuing models. The present article introduces a general construction that we call a *Dunn–Priest quotient*, providing a more general means of constructing models for arbitrary many-valued, first-order logical systems from models of any second system. This technique not only counts Dunn’s and Priest’s techniques as special cases, but also provides a generalized Collapsing Lemma for Priest’s more recent plurivalent semantics in general. We examine when and how much control may be exerted over the resulting theories in particular cases. Finally, we expand the utility of the construction by showing that taking Dunn–Priest quotients of a family of structures commutes with taking an ultraproduct of that family, increasing the versatility of the tool.

#### Article information

**Source**

Notre Dame J. Formal Logic, Volume 58, Number 2 (2017), 221-239.

**Dates**

Received: 25 April 2014

Accepted: 30 July 2014

First available in Project Euclid: 21 February 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1487646411

**Digital Object Identifier**

doi:10.1215/00294527-3838853

**Mathematical Reviews number (MathSciNet)**

MR3634978

**Zentralblatt MATH identifier**

06751300

**Subjects**

Primary: 03C90: Nonclassical models (Boolean-valued, sheaf, etc.)

Secondary: 03C20: Ultraproducts and related constructions

**Keywords**

many-valued logic model theory Dunn–Priest techniques

#### Citation

Ferguson, Thomas Macaulay. Dunn–Priest Quotients of Many-Valued Structures. Notre Dame J. Formal Logic 58 (2017), no. 2, 221--239. doi:10.1215/00294527-3838853. https://projecteuclid.org/euclid.ndjfl/1487646411