Abstract
This paper is devoted to a consequence relation combining the negation of Classical Logic (\(\mathbf {CL}\)) and a paraconsistent negation based on Graham Priest’s Logic of Paradox (\(\mathbf {LP}\)). We give a number of natural desiderata for a logic \(\mathbf {L}\) that combines both negations. They are motivated by a particular property-theoretic perspective on paraconsistency and are all about warranting that the combining logic has the same characteristics as the combined logics, without giving up on the radically paraconsistent nature of the paraconsistent negation. We devise the logic \(\mathbf {CLP}\) by means of an axiomatization and three equivalent semantical characterizations (a non-deterministic semantics, an infinite-valued set-theoretic semantics and an infinite-valued semantics with integer numbers as values). By showing that this logic is maximally paraconsistent, we prove that \(\mathbf {CLP}\) is the only logic satisfying all postulated desiderata. Finally we show how the logic’s infinite-valued semantics permits defining different types of entailment relations.
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Notes
\(\mathbf {FDE}\) is the four-valued logic named ‘First Degree Entailment’ first proposed by Belnap and Dunn in Belnap (1977), also called Belnap–Dunn logic.
A set of formulas is trivial iff it contains all well-formed formulas of the language.
A formula or set of formulas is meaningful iff there is a non-trivial interpretation for it.
For now this is vague language which will be made more precise in Req1 on p. 14.
The Equivalence Rules says that a formula is equivalent to the formula resulting from substituting a subformula of the original formula by one that is equivalent to it.
This holds for all formulas, no specific clause for sentential letters is needed.
To see that \(\mathrm {\sim }A\nvDash _\mathbf {CLP} \lnot A\), observe that nothing prohibits the existence of a model M for which \(M\models \mathrm {\sim }p\), \(M\models p\) and \(M\not \models \lnot p\), as it is in accordance with IS5 and IS6.
In the equation we use the subtraction symbol “−” both for the subtraction of numbers and as for the subtraction of sets.
This can be proven by showing that \(\varGamma \vdash _\mathbf {CLP'} B\) iff \(\varGamma \cup \{\lnot (\Box ^i A \wedge \mathrm {\sim }\Box ^i A)\mid A\in \mathcal {S}, i\in \mathbb {N}\} \vdash _\mathbf {CLP} B\), using Theorem 1.
Of course, one can easily devise many other paraconsistent logics where inconsistencies come in degrees (e.g. paraconsistent fuzzy logics), but here the degrees are the result of doing nothing but adding classical negation to a standard paraconsistent logic in such a way that the resulting logic preserves as many features as possible of the combined logics.
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Acknowledgements
I would like to thank the two reviewers of this paper and the two reviewers of an unpublished submission of an early version of this paper to another journal for their invaluable remarks, corrections, criticisms and suggestions.
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Verdée, P. Obtaining infinitely many degrees of inconsistency by adding a strictly paraconsistent negation to classical logic. Synthese 198 (Suppl 22), 5415–5449 (2021). https://doi.org/10.1007/s11229-020-02638-8
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DOI: https://doi.org/10.1007/s11229-020-02638-8