## Abstract

The main aim of this paper is to introduce first-order versions of logics of evidence and truth, together with corresponding sound and complete Kripke semantics with variable and constant domains. According to the intuitive interpretation proposed here, these logics intend to represent possibly inconsistent and incomplete information bases over time. The paper also discusses the connections between Belnap-Dunn’s and da Costa’s approaches to paraconsistency, and argues that the logics of evidence and truth combine them in a very natural way.

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## Notes

For more precise characterizations of \(LFI\)s, \(LFU\)s, and

*LFIU*s, see [18], Definitions 4.3, 4.5, and 4.7.For a more detailed account of the notion of evidence in \(LET\)s, see [46, Sects. 1 and 2].

The relational semantics for \(FDE\) proposed by Dunn [26] in terms of subsets of \(\{ 0 ,1 \}\) is essentially the same idea, though in different clothing.

This point is clearer with the valuation semantics for \(FDE\) presented in [47, Sect. 2.1], which makes the non-deterministic character of the negation of \(FDE\) explicit.

This terminology has been borrowed from Fitting [33].

A definition of evidence based on the notion of information as meaningful data supplemented by possibly non-factive justifications is proposed and discussed in [46, Sect. 2.3], where these ideas are worked out in more detail.

This is illustrated in Remark 4.8 below. Note that this idea applies to any context of reasoning that receives new information as time passes, not only to databases in the strict sense of an information base accessed by a computer system.

A formula is an

*alphabetic variant*of another if they only differ in (some of) their bound variables. See [31, pp. 126-7] for details.The reader may wonder at this point what would it take to review a certain piece of information (e.g., when it leads to an inconsistency), that is, how to ‘delete’ something that has already been inserted into the database. The suitable tools for dealing with this problem are provided by

*belief revision*, which can take one or another \(LET\) as the underlying logic. Belief revision based on \(LET\)s will be investigated elsewhere.The other quantifier rules of \(Q_{v}LET_{F}^-\) would also have to be modified if we opted for an inclusive free version of \(Q_{v}LET_{F}^-\).

It is worth mentioning that the failure to recognize this point has been the source of the error in Thomason’s proof of the soundness of first-order intuitionistic logic in [48]. According to Posy [44], when this omission was pointed out to him, Thomason claimed that a free logic is indeed more appropriate for intuitionism than the logic he originally proposed in [48]. The strategy adopted here to avoid the need for free quantifiers, viz., that of requiring every individual constant to denote at every stage, is due to van Dalen [24, Sect. 5.3].

Item (2) above is an immediate consequence of rules \(\lnot \! \rightarrow I\) and \(\lnot \! \rightarrow E\) and the properties of regular Henkin sets.

Note that there is no need for Kripke models for a constant domain first-order version of \(LET_{F}^-\), since in this case all semantic clauses are local.

On the distinction between pure and applied logics, see e.g. [7, pp. 157ff.].

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The research of the first author is supported by the grants 311911/2018-8, Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Brazil), and APQ-02093-21, Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG, Brazil). We would like to thank Alexandre Barros, Carlos Barth, Walter Carnielli, and Marcelo Coniglio for valuable discussions that helped to shape some of the ideas presented in this text.

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Rodrigues, A., Antunes, H. First-order Logics of Evidence and Truth with Constant and Variable Domains.
*Log. Univers.* (2022). https://doi.org/10.1007/s11787-022-00306-8

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DOI: https://doi.org/10.1007/s11787-022-00306-8

### Keywords

- Logics of evidence and truth
- Paraconsistency
- Kripke models
- Variable domains

### Mathematics Subject Classification

- Primary 03B53
- Secondary 03A05