Abstract

The aim of this paper is to introduce the logics $$\textit{FFDE}$$ and $$\textit{FN}{4}$$, which are universally free versions of Belnap-Dunn’s four-valued logic, also known as the logic of first-degree entailment ( $$\textit{FDE}$$ ), and Nelson’s paraconsistent logic $$N^{-}$$ (a.k.a. $$Q\!N {4}$$ ). Both $$\textit{FDE}$$ and $$Q\!N {4}$$ 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 $$Q\!N {4}$$ and some non-free first-order extensions of $$\textit{FDE}$$, $$\textit{FFDE}$$ and $$\textit{FN}{4}$$ 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 $$\textit{FFDE}$$ and $$\textit{FN}{4}$$ 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.