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On the Philosophy and Mathematics of the Logics of Formal Inconsistency

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 152)


The aim of this text is to present the philosophical motivations for the Logics of Formal Inconsistency (LFIs), along with some relevant technical results. The text is divided into two main parts (besides a short introduction). In Sect. 3.2, we present and discuss philosophical issues related to paraconsistency in general, and especially to logics of formal inconsistency. We argue that there are two basic and philosophically legitimate approaches to paraconsistency that depend on whether the contradictions are understood ontologically or epistemologically. LFIs are suitable to both options, but we emphasize the epistemological interpretation of contradictions. The main argument depends on the duality between paraconsistency and paracompleteness. In a few words, the idea is as follows: just as excluded middle may be rejected by intuitionistic logic due to epistemological reasons, explosion may also be rejected by paraconsistent logics due to epistemological reasons. In Sect. 3.3, some formal systems and a few basic technical results about them are presented.


  • Logics of Formal Inconsistency
  • Contradictions
  • Philosophy of paraconsistency

Mathematics Subject Classication (2000)

  • Primary 03B53
  • Secondary 03A05
  • 03-01

The first author acknowledges support from FAPESP (São Paulo Research Council) and CNPq, Brazil (The National Council for Scientific and Technological Development). The second author acknowledges support from the Universidade Federal de Minas Gerais (project call 12/2011) and FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais, research project 21308). We would like to thank Henrique Almeida, Marcos Silva and Peter Verdée for some valuable comments on a previous version of this text.

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  1. 1.

    This paper corresponds, with some additions, to the tutorial on Logics of Formal Inconsistency presented in the 5th World Congress on Paraconsistency that took place in Kolkata, India, in February 2014. Parts of this material have already appeared in other texts by the authors, and other parts are already in print elsewhere [12, 13]. A much more detailed mathematical treatment can be found in Carnielli and Coniglio [10] and Carnielli et al. [15].

  2. 2.

    We do not use the term ‘information’ here in a strictly technical sense. We might say, in an attempt not to define but rather to elucidate, that ‘information’ means any ‘amount of data’ that can be expressed by a sentence (or proposition) in natural language. Accordingly, there may be contradictory or conflicting information (in a sense to be clarified below), vague information, or lack of information.

  3. 3.

    The symbol \({\sim }\) will always denote the classical negation, while \(\lnot \) usually denotes a paraconsistent negation but sometimes a paracomplete (e.g. intuitionistic) negation. The context will make it clear in each case whether the negation is used in a paracomplete or paraconsistent sense.

  4. 4.

    This idea has some consequences for Harman’s arguments (see [31]) against non-classical logics, a point that we intend to develop elsewhere.

  5. 5.

    Or sentences, if one prefers—here, we do not go into the distinction between sentences and propositions.

  6. 6.

    For a more detailed explanation of the duality between paracompleteness and paraconsistency, see Marcos [36].

  7. 7.

    Actually, da Costa has a hierarchy of systems, starting with the system \(C_{1}\), where \(A^{\circ }\) is an abbreviation of \(\lnot (A\wedge \lnot A)\). A full hierarchy of calculi \(C_{n}\), for n natural, is defined and studied in da Costa [20].

  8. 8.

    See ‘Carta de Francisco Miro Quesada a Newton da Costa, 29.IX.1975’ in Gomes [28, p. 609].

  9. 9.

    A rejection of the linguistic conception of logic, and a defense of logic as a theory with ontological and epistemological aspects, can be found in Chateaubriand [17, Introduction].

  10. 10.

    Cf. Frege [25, p. 13]: ‘they are boundary stones set in an eternal foundation, which our thought can overflow, but never displace.’

  11. 11.

    There is a sense in which for Frege laws of logic are descriptive: they describe reality, as well as laws of physics and mathematics. But we say here that a logic is descriptive when it describes reasoning.

  12. 12.

    Brouwer [5, pp. 51, 73–74]: “Mathematics can deal with no other matter than that which it has itself constructed. In the preceding pages it has been shown for the fundamental parts of mathematics how they can be built up from units of perception. (...) The words of your mathematical demonstration merely accompany a mathematical construction that is effected without words (...) While thus mathematics is independent of logic, logic does depend upon mathematics.” A more acessible presentation of the motivations for intuitionistic logic is to be found in Heyting [32, Disputation].

  13. 13.

    See, for example, van Dalen [46, p. 225]: “two [logics] stand out as having a solid philosophical-mathematical justification. On the one hand, classical logic with its ontological basis and on the other hand intuitionistic logic with its epistemic motivation.”

  14. 14.

    It is worth noting that Brouwer’s and Heyting’s attempts to identify truth with a notion of proof have failed, as Raatikainen [44] shows, because the result is a concept of truth that goes against some basic intuitions about truth.

  15. 15.

    All passages from Aristotle referred to here are from [2].

  16. 16.

    This tripartite approach is also found in Gottlieb [29], where these three versions are called, respectively, ontological, doxastic, and semantic.

  17. 17.

    For example, the issue of particulars/universals, the Fregean distinction between object and function, and even Quine’s attacks to the notion of property.

  18. 18.

    A more comprehensive and detailed presentation of mbC can be found in Carnielli et al. [15].

  19. 19.

    An equivalent system is obtained by substituting Axiom 9 by Peirce’s Law, \(((A \rightarrow B) \rightarrow A) \rightarrow A\). Indeed, it is well known that Axioms 1 and 2 plus Peirce’s Law define positive implicative classical logic.

  20. 20.

    LFIs may also be obtained as extensions of positive intuitionistic propositional logic, IPL+, the system given by Axioms 1–8 plus modus ponens. Different formal systems may be obtained depending on the positive logic one starts with and the desired behavior of negation and the operator \(\circ \). The inconsistency of a formula A may also be primitive, represented by \(\bullet A\), that may or may not be equivalent to \(\lnot \circ A\) (see details in Carnielli and Coniglio [10]). We have chosen here to take CPL+ as a basis and mbC as the first LFI, because this allows for a simpler and more didactic presentation, more suitable to the aims of this text.

  21. 21.

    A precise characterization of the principle of gentle explosion is to be found in Carnielli et al. [15, pp. 19–20].

  22. 22.

    As far as we know, DATs were proposed for the first time by Diderik Batens, one of the main researchers in the field of paraconsistency. His inconsistency-adaptive logics are a kind of paraconsistent logic that restricts the validity of the principle of explosion according to the information available in some context. As we see the proposal of inconsistency-adaptive logics, they share with the Logics of Formal Inconsistency the possibility of interpreting contradictions epistemologically. However, an important difference is that in adaptive logics everything is supposed to be consistent unless proven otherwise. LFIs, in contrast, do not presuppose consistency.

  23. 23.

    In fact, not only mbC but all logics of the da Costa hierarchy \(C_{n}\), and most LFIs, are not characterizable by finite matrices (see Carnielli et al. [15, p. 74, Theorem 121]).


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Carnielli, W., Rodrigues, A. (2015). On the Philosophy and Mathematics of the Logics of Formal Inconsistency. In: Beziau, JY., Chakraborty, M., Dutta, S. (eds) New Directions in Paraconsistent Logic. Springer Proceedings in Mathematics & Statistics, vol 152. Springer, New Delhi.

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