Abstract
Proof-theoretic semantics provides meanings to the connectives of intuitionistic logic without the need for a semantics in the standard sense of an attribution of semantic values to formulas. Meanings are given by the inference rules that, in this case, do not express preservation of truth but rather preservation of availability of a constructive proof. Elsewhere we presented two paraconsistent systems of natural deduction: the Basic Logic of Evidence (BLE) and the Logic of Evidence and Truth (\(LET_{J}\)). The rules of BLE have been conceived to preserve a notion weaker than truth, namely, evidence, understood as reasons for believing in or accepting a given proposition. \(LET_{J}\), on the other hand, is a logic of formal inconsistency and undeterminedness that extends \(BLE\) by adding resources to recover classical logic for formulas taken as true, or false. We extend the idea of proof-theoretic semantics to these logics and argue that the meanings of the connectives in BLE are given by the fact that its rules are concerned with preservation of the availability of evidence. An analogous idea also applies to \(LET_{J}\).
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Notes
- 1.
- 2.
An example is the Goldstein theorem whose proof is independent of Peano Arithmetic (formal number theory), and so cannot be proved by constructive means.
- 3.
We defend this view in Carnielli and Rodrigues (2016, Sect. 3).
- 4.
The same idea of a “peaceful coexistence” between intuitionism and classicism is found in Dubucs (2008).
- 5.
For a more detailed account of paraconsistency as preservation of evidence and the duality between paraconsistent and paracomplete logics understood as concerned, respectively, with a notion weaker and stronger than truth, we refer the reader to Carnielli and Rodrigues (2016, Sect. 4) and (2017, Sects. 1 and 2).
- 6.
- 7.
We quote here from the French edition of Heyting’s text originally publish in German in 1934.
- 8.
Another remarkable feature of \(BLE\) is the symmetry between the introduction and elimination positive rules for \(\wedge \) and \(\vee \) on the one hand, and the introduction and elimination negative rules for the dual operators \(\vee \) and \(\wedge \) on the other (there is no dual of \(\rightarrow \) in \(BLE\)).
- 9.
The logic \(BLE\), besides being equivalent to Nelson’s N4, is also equivalent to the propositional fragment of the refutability calculus proposed by López-Escobar (1972), where an explanation of refutation as a primitive notion in number theory that is similar to the clauses [E4]–[E7] above can be found (López-Escobar 1972, pp. 363ff). This is not a surprise, though, since the logics are equivalent, but his motivations are completely different from ours. Regarding the motivations and the “discovery” of \(BLE\) and \(LET_{J}\), see Sect. 5.3 and footnote 21 of Carnielli and Rodrigues (2017).
- 10.
- 11.
These two conditions exclude contradictions due to properties instantiated in different moments of time and contradictions that depend on two different perspectives. Heraclitean oppositions, thus, are not dialetheias, and contradictions in history, like the ones asserted by Hegelians, do not qualify as dialetheias either.
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Acknowledgements
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, grants PEP 157-16 and 701-16). We would like to thank Eduardo Barrio, the audience of the Workshop CLE-BsAs Logic Group (Buenos Aires, April 2016), and two anonymous referees for some valuable comments on a previous version of this text.
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Carnielli, W., Rodrigues, A. (2019). Inferential Semantics, Paraconsistency, and Preservation of Evidence. In: Başkent, C., Ferguson, T. (eds) Graham Priest on Dialetheism and Paraconsistency. Outstanding Contributions to Logic, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-25365-3_9
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