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Contradiction and (in)Consistency

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Paraconsistent Logic: Consistency, Contradiction and Negation

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 40))

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Abstract

This chapter intends to clarify the whole project behind LFIs, explaining why and how contradiction and triviality cease to coincide, and why and how contradiction ceases to coincide with inconsistency. It also intends to explain that there is no opposition to the classical stance, besides the awareness that ‘classical’ logic involves some hidden assumptions that are made clear in this chapter.

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Notes

  1. 1.

    This chapter corresponds in part with the tutorial on Logics of Formal Inconsistency presented in the \(\textit{5}{th}\) World Congress on Paraconsistency (Kolkata, India, February 2014), see [1]. Parts of that material have already appeared in [2].

  2. 2.

    Example 14 (p. 15) of [5] provides an example of a logic that respects the principle of Ex Falso Sequitur Quodlibet, but not the ECSQ, showing that those principles do not need to be identified, contrary to what is commonly held in the literature.

  3. 3.

    To the best of our knowledge, the exact expressions Ex Contradictione Sequitur Quodlibet and Ex Contradictione Quodlibet have been independently coined by, respectively Priest and Bobenrieth-Miserda, see [6].

  4. 4.

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

  5. 5.

    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 clear in each case whether the negation is being used in a paracomplete or a paraconsistent sense.

  6. 6.

    This idea has some consequences for Harman’s arguments [9] against non-classical logics, a point that we intend to develop elsewhere.

  7. 7.

    For a more detailed account on the duality between paracompleteness and paraconsistency, see e.g. [11].

  8. 8.

    Actually, da Costa defined a hierarchy of systems, starting with the system \(C_{1}\). A full hierarchy of calculi \(C_{n}\), for n natural, is defined and studied in [12]. Each \(C_{n}\) has its own definition of well-behavedness.

  9. 9.

    The reader is warned that, along this book, the expression ‘strong negation’ is reserved for a negation with a Boolean character.

  10. 10.

    See ‘Carta de F.M. Quesada a N.C.A. da Costa, 29.IX.1975’ in [13], p. 609.

  11. 11.

    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 the Introduction to [24].

  12. 12.

    See [27], p. 13: ‘they are boundary stones set in an eternal foundation, which our thought can overflow, but never displace’.

  13. 13.

    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, in some way, actual reasoning.

  14. 14.

    Brouwer [29]: “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 [30].

  15. 15.

    See, for example, [31]: “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”.

  16. 16.

    In [30], p. 1, we read “You ought to consider what Brouwer’s program was [...]. It consisted in the investigation of mental mathematical construction as such, without reference to questions regarding the nature of the constructed objects, such as whether these objects exist independently of our knowledge of them”.

  17. 17.

    This tripartite approach is also found in [34], where these three versions are called, respectively, ontological, doxastic and semantic.

  18. 18.

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

  19. 19.

    The idea that there are opposing elements existing simultaneously in a whole is found in many places in Hegel’s writings. The following passage from [38], Sect. 20, illustrates this interpretation and the weaker sense of contradiction: “The True is the whole. But the whole is nothing other than the essence consummating itself through its development. Of the Absolute it must be said that it is essentially a result, that only in the end is it what it truly is; and that precisely in this consists its nature, viz. to be actual, subject, the spontaneous becoming of itself. Though it may seem contradictory that the Absolute should be conceived essentially as a result, it needs little pondering to set this show of contradiction in its true light” .

  20. 20.

    In the logic mbC, to be studied in detail in Chap. 2 of this Book, the consistency of \(\alpha \), represented by \(\circ \alpha \), is not equivalent to non-contradictoriness of \(\alpha \): \(\circ \alpha \) implies \(\lnot (\alpha \wedge \lnot \alpha )\), but the converse does not hold in general.

  21. 21.

    With the help of this definition, \(\alpha \vee {\sim } \alpha \) and \(\alpha \rightarrow ({\sim } \alpha \rightarrow \beta )\) is also provable in mbC. Since mbC is, by the very definition, an extension of classical propositional positive logic, classical logic may be restored within mbC. Notice the difference between, on the one hand, restoring classical consequence by means of a definition of a classical negation and, on the other, by means of a DAT which, roughly speaking, states that each derivation within classical logic can be recovered in mbC, by adding as additional hypothesis the consistency of certain formulas (see Sect. 2.4 and Theorem 2.4.7). In the latter case, the point is the information that has to be available in order to restore classical logic. The former shows that, in a certain sense, although the idea is to restrict inferences valid classically, mbC is an extension of classical logic.

  22. 22.

    See [50], pp. 147 and 237: “[A]t the macroscopic level, the experience seems to indicate that there are no contradictions; however, at a microscopic level, there is nothing to prevent real contradictions. [...] [R]eal contradictions are not impossible, although there is nothing so far proving that they exist”.

  23. 23.

    Aristotle had already been clear in saying that an object having different properties at different moments of time, or from different perspectives, would not be a counterexample for the principle of non-contradiction (see Metaphysics, 1009b1 and 1010b10). Actually, saying this one more time is almost a platitude.

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Authors and Affiliations

  1. Department of Philosophy and Centre for Logic, Epistemology and the History of Science (CLE), University of Campinas (UNICAMP), Campinas, São Paulo, Brazil

    Walter Carnielli & Marcelo Esteban Coniglio

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Correspondence to Walter Carnielli .

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Carnielli, W., Coniglio, M.E. (2016). Contradiction and (in)Consistency. In: Paraconsistent Logic: Consistency, Contradiction and Negation. Logic, Epistemology, and the Unity of Science, vol 40. Springer, Cham. https://doi.org/10.1007/978-3-319-33205-5_1

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