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On Split Negation, Strong Negation, Information, Falsification, and Verification

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J. Michael Dunn on Information Based Logics

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 8))

Abstract

This paper deals with some criticism that has been put forward against strong, constructive negation in comparison to a certain example of Galois connected negations. The general background to this discussion is the informational interpretation of substructural logics, and the key issue is whether there exists an asymmetry or not between positive and negative information and between verification and falsification. The present paper confirms the view that a symmetrical conception is adequate for both direct and indirect variants of verification and falsification.

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Notes

  1. 1.

    The association with Fregean thoughts may prompt further discussion. In his famous paper on sense and reference (Frege 1892), Frege considers an example of a fictional sentence that has no truth value but nevertheless expresses a thought. In the later paper “Compound Thoughts” (Frege 1923), however, he characterizes a thought as “something which must be either true or false, tertium non datur.”

  2. 2.

    Sequoiah-Grayson uses ‘\(\lnot \)’ instead of ‘\(\lnot ^r\)’ and ‘\(\sim \)’ instead of ‘\(\lnot ^l\)’. The superscripts have the advantage of reminding one of the directionality of the implication that is involved.

  3. 3.

    We may assume that there is a typing error in the versions of (3) and (4) in Sequoiah-Grayson (2009). With these versions:  

    (3)\(_{SG}\) :

    \(x\Vdash A\rightarrow B\text { iff for all }y,z\in S,\text { if }x \bullet y\sqsubseteq z\text { and }y\Vdash A,\text { then }z\Vdash B\)

    (4)\(_{SG}\) :

    \(x\Vdash A\leftarrow B\text { iff for all }y,z\in S, \text { if }y\bullet x\sqsubseteq z\text { and }y\Vdash A,\text { then }z\Vdash B\)

        \(x \Vdash (A\rightarrow B)\otimes A\) implies \(x\Vdash B\) and \(x\Vdash A\otimes (A \leftarrow B)\) implies \(x\Vdash B\), and, as a result, we do not obtain (6) and (7).

  4. 4.

    He writes (Sequoiah-Grayson 2009, p. 236):

    One might wish to understand ‘supports’ as ‘makes true’ if one holds to a dialethic paraconsistentism whereby at least some contradictions are taken to be true. However, we will sidestep this particular debate and stay with the interpretation of ‘supports’ that takes it to be the subtler relative of ‘makes true’ in a manner aligned with Mares’ informational interpretation.

    .

  5. 5.

    Note that in this paper I do not pay close attention to the mention/use distinction when there is no risk of confusion.

  6. 6.

    An anonymous reviewer raised the question whether there are other ways of defining a semantics in order to obtain a formal reconstruction of the presentation in (Sequoiah-Grayson 2009). I have nothing else to offer than the preservation of support-of-information-that, which captures derivability in NL\(\mathbf{0 }\).

  7. 7.

    Assuming sequent rules that capture Sequoiah-Grayson’s (3)\(_{SG}\) and (4)\(_{SG}\) instead of (3) and (4) would not help.

  8. 8.

    Analogous remarks apply to the inference pattern (21) and the “prohibited procedure” (22). Also the endorsed inference patterns

    1. (23)

      \(A\rightarrow \lnot B\vdash {\sim }A\leftarrow B\), and

    2. (24)

      \(A\rightarrow {\sim }B\vdash \lnot A\leftarrow B\)

    are not provable, so that they are not underpinned by the formulas numbered (25) and (26) in (Sequoiah-Grayson 2009).

  9. 9.

    It is complicated in the constructive setting of intuitionistic logic and N3 already, cf. (Wansing 2006).

  10. 10.

    Note that the strong negation \({\sim }\) is different from Sequoiah-Grayson’s \({\sim }\), i.e., \(A\rightarrow \mathbf{0 }\).

  11. 11.

    Since in ordinary natural deduction a formula A is a proof of A from \(\{\,A\,\}\), the cancellation of formulas amounts to the cancellation of proofs.

  12. 12.

    In these tables, Ep stands for “elimination from proofs,” Ip for “introduction into proofs,” Edp for “elimination from dual proofs,” and Idp for “introduction into dual proofs.”

  13. 13.

    Another translation presented in (Wansing 2013) gives one a faithful embedding of 2Int into dual intuitionistic logic with respect to dual entailment.

  14. 14.

    Cf. (Wansing 2010), where I consider extensions of Heyting–Brouwer logic (Rauszer 1980) by strong negation and refer to reductions to non-truth as dual proofs.

  15. 15.

    In (Kapsner 2014), a detailed analysis is presented of Michael Dummett’s views on the interaction between verification and falsification in an inferentialist theory of meaning. In that context Kapsner argues for “the superiority of the Nelson account over the intuitionistic one,” (Kapsner 2014, p. 198).

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Acknowledgments

I would like to thank Katalin Bimbó, Yaroslav Shramko, Stanislav Speranski and two anonymous referees for their very helpful comments.

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  1. Department of Philosophy II, Ruhr-University Bochum, Bochum, Germany

    Heinrich Wansing

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  1. Department of Philosophy, University of Alberta, Edmonton, Alberta, Canada

    Katalin Bimbó

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Wansing, H. (2016). On Split Negation, Strong Negation, Information, Falsification, and Verification. In: Bimbó, K. (eds) J. Michael Dunn on Information Based Logics. Outstanding Contributions to Logic, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-29300-4_10

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