Abstract
The purpose of this paper is to present a paraconsistent formal system and a corresponding intended interpretation according to which true contradictions are not tolerated. Contradictions are, instead, epistemically understood as conflicting evidence, where evidence for a proposition A is understood as reasons for believing that A is true. The paper defines a paraconsistent and paracomplete natural deduction system, called the Basic Logic of Evidence (BLE), and extends it to the Logic of Evidence and Truth (\(\textit{LET}_{J}\)). The latter is a logic of formal inconsistency and undeterminedness that is able to express not only preservation of evidence but also preservation of truth. \(\textit{LET}_{J}\) is anti-dialetheist in the sense that, according to the intuitive interpretation proposed here, its consequence relation is trivial in the presence of any true contradiction. Adequate semantics and a decision method are presented for both BLE and \(\textit{LET}_{J}\), as well as some technical results that fit the intended interpretation.
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Notes
Although the dialetheist view has some antecedents in the history of philosophy, especially in Hegel and, according to some interpreters, also Heraclitus, the claim that there are ‘ontological contradictions’ is rather contentious, both inside and outside philosophy. It is not our aim here, however, to discuss the legitimacy of, nor argue against, dialetheism.
We defend this view in Carnielli and Rodrigues (2016). It is not unlikely that a view that accepts a non-constructive proof of the truth of a given proposition, but distinguishes such a proof from a (perhaps) more informative constructive proof, is the predominant approach to intuitionism nowadays (see Dubucs 2008).
BLE, although differently motivated, turns out to be equivalent to the well-known Nelson’s logic N4. (See also Sect. 5.3, and footnotes 11 and 21). All the technical results presented here with respect to BLE, including a valuation semantics and a decision method, also hold for N4. In a Fregian spirit, we may say that N4 and BLE are two different names, with different senses, that happen to have the same reference.
In van Benthem and Pacuit (2011) and van Benthem et al. (2015) we find a proposal of ‘evidence logics’, designed to give an account of “epistemic agents faced with possibly contradictory evidence from different sources”. Their approach is mainly semantical, in terms of ‘neighborhood semantics’ and differs from ours, motivated by proof-theoretical insights. The resulting logics proposed by them are quite different from both BLE and \(\textit{LET}_{J}\) presented here.
The logic of evidence proposed by us does not intend to represent cognitive relations between agents and propositions. It is not an account of any kind of propositional attitude. The idea is that there is some objective criterion that, when satisfied, indicates the existence of evidence for a proposition A. So, again, there can be evidence for A and an agent may be aware of such evidence and still does not believe in A. Evidence, thus, is not a name for the epistemic attitude of an agent w.r.t. a proposition. We say that our approach is epistemic because the property of propositions that is being preserved is an epistemic notion.
Melvin Fitting, in a forthcoming paper (Fitting 2017), proposes a formalization of the notion of evidence that includes an embedding of the logic BLE into the modal logic KX4, that is S4 minus \(\square A \rightarrow A\) plus \(\square \square A \rightarrow \square A\). In the well-known embedding of intuitionistic logic in S4, \(\square \) represents provability; dually, in the embedding of BLE into KX4, \(\square \) represents evidence which, in contrast to proof, permits contradictions.
The logic of evidence proposed by us is adjunctive because we want to express a notion of preservation of evidence for which \(\wedge \)-introduction holds in general. Suppose a document is (non-conclusive) evidence for the truth of A, i.e. such a document is evidence \(\kappa \) for A. Now, suppose that from another source comes another document that is (also non-conclusive) evidence for the falsity of A. The latter would be an evidence \(\kappa '\) for \(\lnot A\). We can, of course, put these two documents together, say, in a folder, or even merge them in an electronic file. These two pieces of evidence \(\kappa \) and \(\kappa '\) together are evidence that the conjunction \(A \wedge \lnot A\) is true. So, it is not the case that evidence for the latter fails when there is conflicting evidence for A. Such a situation, evidence for \(A \wedge \lnot A\), is nothing but an indication that further investigation is necessary. This is a central point of the non-dialetheist approach to paraconsistency proposed by us: it may be that evidence for a contradiction is available, but only as an indication that something is wrong and should be fixed.
A paraconsistent system in which explosion does not hold but \(A, \lnot A \vdash \lnot B\) holds is called partially explosive. An example of a partially explosive formal system is Kolmogorov’s ‘logic of judgment’ (see Kolmogorov 1925).
Notice that the negation rules exhibit a symmetry with respect to the corresponding assertion rules for the dual operators.
The rules for negation are the same as the rules for constructive falsity presented by Prawitz (1965, pp. 96–97). Although the guiding idea that led us to BLE is different, BLE can be easily proved equivalent to Nelson’s logic N4 (see Fact 10). Besides, BLE is also equivalent to the propositional fragment of refutability calculus presented by López-Escobar (1972).
In the so-called proof-theoretic semantics, the notion of proof is semantical in the sense that it gives the ‘meanings’ involved in the inferences of the formal system. Our approach has a similar spirit.
As far as we know, non-functional valuation semantics (and the respective quasi-matrices) for paraconsistent logics were presented for the first time by Costa and Alves (1977) where we find a sound and complete semantics for da Costa’s \(C_{1}\). The notion of semivaluation was introduced in Loparic (1986), where we find semantic clauses for an intuitionistic implication. In Loparic (1986) and Loparic (2010) we find adequate valuation semantics and decision procedures respectively for \(C_{\omega }\) and Heyting intuitionistic logic.
The conditions for implication are ‘global’ in the sense that one has to look at all semivaluations in order to establish whether a given semivaluation is a valuation.
Notice that v is a semivaluation that satisfies condition Val, given \(Val'\), and is thus a valuation.
The idea of expressing a metalogical notion within the object language is not new in the literature. It is found, e.g. in the \(C_{n}\) hierarchy introduced by da Costa (1963), through the idea of ‘well-behavedness’ of a formula. In da Costa’s hierarchy, however, this is done by means of a definition: in \(C_{1}\), for instance, it is expressed by \(A{}^{\circ }\), an abbreviation of \(\lnot (A\wedge \lnot A)\), which makes the ‘well-behavedness’ of A equivalent to saying that A is non-contradictory. On the other hand, in the LFIs, \(\circ A\) is introduced in such a way that allows \(\circ A\) and \(\lnot (A\wedge \lnot A)\) to be logically independent (non-equivalent). The family of LFIs incorporate a wide class of paraconsistent logics, as shown in Carnielli et al. (2007) and Carnielli and Coniglio (2016).
The notion of LFUs is introduced in Marcos (2005).
The account of logical pluralism given by Beall and Restall (2006) is based on preservation of truth, but it considers different types of cases: Tarskian models, constructions and situations that are, respectively, cases for classical, intuitionistic and relevant (i.e. paraconsistent) logics.
In Nelson (1949, p. 17), we read: “This notion of [constructive] truth will be made precise by defining a syntactical predicate ‘The natural number a P-realizes the formula A.’ At the same time a correlative concept of constructible falsity will be expressed by a predicate ‘The natural number a N-realizes the formula A’.”
The line of reasoning that lead to the logic BLE started in a modification in the logic mbC, an LFI presented in Carnielli et al. (2007). mbC is an extension of classical positive propositional logic, and excluded middle holds in mbC. In order to make mbC suitable for expressing contradictions as conflicting evidence, we presented in Carnielli and Rodrigues (2015) the logic mbCD, in which \(A \vee \lnot A\) has been replaced by the axiom \(\circ A \rightarrow (A \vee \lnot A)\), that corresponds to the rule \(PEM^\circ \). Then, in the search for a logic that could express preservation of evidence for both truth and falsity, we adopted the natural deduction system for PIL as the starting point, thus rejecting \(A \vee (A \rightarrow B)\), that holds in mbC, and we found out that, w.r.t. negation, natural deduction rules equivalent to De Morgan’s laws and double negation should hold, as well as the equivalence between \(\lnot (A \rightarrow B)\) and \(A \wedge \lnot B\). The logic so obtained is \(\textit{LET}_{J}\). BLE is \(\textit{LET}_{J}\) without the rules \(EXP^\circ \) and \(PEM^\circ \).
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Acknowledgements
We would like to thank Henrique Almeida, Antonio Coelho, Décio Krause and Wagner Sanz for some valuable comments on a previous version of this text. The first author acknowledges support from FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo, thematic project LogCons) and from a CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) research grant. The second author acknowledges support from FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais, research project 21308).
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Carnielli, W., Rodrigues, A. An epistemic approach to paraconsistency: a logic of evidence and truth. Synthese 196, 3789–3813 (2019). https://doi.org/10.1007/s11229-017-1621-7
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DOI: https://doi.org/10.1007/s11229-017-1621-7