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Paraconsistent Orbits of Logics

Abstract

Some strategies to turn any logic into a paraconsistent system are examined. In the environment of universal logic, we show how to paraconsistentize logics at the abstract level using a transformation in the class of all abstract logics called paraconsistentization by consistent sets . Moreover, by means of the notions of paradeduction and paraconsequence we go on applying the process of changing a logic converting it into a paraconsistent system. We also examine how this transformation can be performed using multideductive abstract logics. To conclude, the conceptual notion paraconsistent orbit of a logic is proposed.

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Notes

  1. 1.

    When we take these logics as those systems eliminating, avoiding or denying some manifestation of the law of explosion, we face a problem proposed by J-Y. Beziau in [9]. He argues that paraconsistent logics should also respect some positive property, and not only reject ex falso, because non-explosive unary operators can be defined in many systems which are not traditionally conceived as paraconsistent. Moreover, other criticism to the idea and definition of paraconsistent logic is suggested by H. Slater in [35]. He states that negation in a paraconsistent logic does not generate a contradictory relation, but rather a relation of subcontrariety and, for this reason, explosion fails. We can understand his view as a defence of only one possible type of negation: truth-functional classical negation.

  2. 2.

    For an extension of this work, a new approach to well-behavior by means of a consistency operator has been proposed in [12].

  3. 3.

    For more on these varieties of concrete methods of paraconsistentization, see [14].

  4. 4.

    Although defined in a very abstract perspective, these methods are applicable to particular logical systems, indeed.

  5. 5.

    There are in the literature many proposed attempts to deal with consistent subsets of sets containing some types of contradictory of inconsistent information. In this tradition we can mention [1, 2, 4, 5, 11, 34, 36] and [39]. In our case, though consistent subsets play a substantial role, our target is to paraconsistentize, to alter the nature of a given logic which is not paraconsistent (for comparison, see [22]).

  6. 6.

    More on consequence relations without any special law or a priori conditions can be found at [10].

  7. 7.

    See A. Loparic and N. da Costa in [28] for more on valuation theory.

  8. 8.

    The concept of multideductive logic has been introduced by E.G. de Souza in [18] and [20]: it is argued, in particular, that multideductive logics can be applied to any attempt to unify (physical) theories which are incompatible. In some sense, all logics can be seen as a multideductive logic. The idea of a multideductive logic is somewhat related with procedures in combination of logics, especially fusions of logics (cf. D. Gabbay in [25]). Although paraconsistentization usually decreases the inferential power of a logic by adding to it a paraconsistent property it is not a genuine method for combining logics, though it is obvioulsy possible to develop ways of paraconsistentizing through combining logics.

  9. 9.

    It is interesting to note that in this case we could have paraconsistency even with classical negation and, in this sense, those criticisms mentioned in footnote 1 do not follow.

  10. 10.

    Needless to say that the way we choose to slice the original logic depends, of course, of what we intend to accomplish.

  11. 11.

    Elements on paraclassical logic have been studied by [17] and [19]. We have shown in [21] that paraclassical logic can be obtained from classical logic using a specific paraconsistentization by consistent sets and, as such, it is a paraconsistentization of classical propositional logic.

  12. 12.

    Paraconsistent logics does not commit us with the limits of consistency. For example, there were fragments of formal sciences which were used despite being inconsistent (for more on inconsistent mathematics, cf. C. Mortensen in [30]). In addition, some philosophers inflates inconsistencies and contradictions and even claim a strong metaphysical (and ontological) thesis: there exist in reality true contradictions (i.e dialetheias) (cf. G. Priest in [33]). An extensive and detailed discussion on the philosophical implications of paraconsistency can be found in [16]. For philosophical issues concerning paraconsistentization, see also D.H.B. Dias in [24]).

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de Souza, E.G., Costa-Leite, A. & Dias, D.H.B. Paraconsistent Orbits of Logics. Log. Univers. 15, 271–289 (2021). https://doi.org/10.1007/s11787-021-00284-3

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Keywords

  • Paraconsistent logic
  • Abstract logic
  • Paraconsistentization
  • Universal logic
  • Multideduction
  • Orbits of logics

Mathematics Subject Classification

  • Primary 03B53
  • Secondary 03B22