On Universally Free First-Order Extensions of Belnap-Dunn's Four-Valued Logic and Nelson's Paraconsistent Logic N4

TL;DR

This paper introduces the logics FFDE and FN4, which are free first-order extensions of Belnap-Dunn's four-valued logic and Nelson's N4, capable of modeling dynamic, inconsistent, and incomplete information in databases with Kripke semantics.

Contribution

It proposes universally free versions of FDE and N4 with Kripke semantics, better capturing the development of information states over time, including empty domains and names.

Findings

FFDE and FN4 support empty domains and names.

They provide a better representation of information development over time.

Both systems include an identity predicate interpreted with positive and negative rules.

Abstract

The aim of this paper is to introduce the logics FFDE and FN4, which are universally free versions of Belnap-Dunn's four-valued logic, also known as the logic of first-degree entailment (FDE), and Nelson's paraconsistent logic QN4 (N-). Both FDE and QN4 are suitable to be interpreted as information-based logics, that is, logics that are capable of representing the deductive behavior of possibly inconsistent and incomplete information in a database. Like QN4 and some non-free first-order extensions of FDE, FFDE and FN4 are endowed with Kripke-style variable domain semantics, which allows representing the dynamic aspect of information processing, that is, how a database receives new information over time, including information about new individuals. We argue, however, that FFDE and FN4 can better represent the development of inconsistent and incomplete information states (i.e., configurations of a database) over time than their non-free versions. First, because they allow for empty domains, which corresponds to the idea that a database may acknowledge no individual at all at an early stage of its development. Second, because they allow for empty names, which get interpreted as information about new individuals is inserted into the database. Also, both systems include an identity predicate that is interpreted along the same lines of the other logical operators, viz., in terms of independent positive and negative rules.