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Larisa Maksimova’s Early Contributions to Relevance Logic

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Larisa Maksimova on Implication, Interpolation, and Definability

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 15))

Abstract

This paper presents an overview of the pioneering contributions of Larisa Maksimova to relevance logics. She is one of the first researchers who set out to methodically study systems of relevance logics, initially, focusing on Ackermann’s \(\varPi '\) of “Rigorous Implication,” and then extending her work to Anderson and Belnap’s systems E of Entailment and R of Relevant Implication, and other related logics. Not only did she develop an algebraic semantics for E, but also we find that a semantic definition of entailment via a ternary accessibility relation appears — the very first time — in an abstract by her.

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Notes

  1. 1.

    See Hollings (2014).

  2. 2.

    See, e.g., (Anderson and Belnap (1975), Sect. 21.1) and (Anderson et al. (1992), Sect. R2)

  3. 3.

    See Anderson and Belnap (1975, Sect. 26.2 and 29.2).

  4. 4.

    This portion of Dunn (1966) was published as Sect. 28.2.2 in the 1st volume of Entailment, Anderson and Belnap (1975).

  5. 5.

    We should point out that Maksimova (1967a) is missing from the references in Anderson and Belnap (1975) and the content of the mentioned section is attributed to Maksimova (1967b), which does not contain those results or their proofs — as should become obvious from our account of the content of the latter paper.

  6. 6.

    She seems to use the term “rigorous implication” not only to denote Ackermann’s \(\varPi '\) calculus, but other closely related calculi too, each of which avoids the paradoxes of material implication. The “S” in “SE” may stand for “strenge” or for “(sub)system.”

  7. 7.

    Maksimova is careful to distinguish between the components of the language of the logic SE and the particles of a model — as she did earlier too. We continue to re-use the same symbols for the connectives in order to shorten our presentation.

  8. 8.

    Some earlier work of Maksimova (e.g., her 1967c) concerned connections between quasi-ordered sets and closure operators.

  9. 9.

    See Urquhart (1984), Anderson et al. (1992, Sect. 65) and Urquhart (2007).

  10. 10.

    See Dunn (1986, Sect. 4.3–4.4) for more on De Morgan lattices and their representation including the fact that Białynicki-Birula and Rasiowa (1957) first originated the use of an involution in the representation of De Morgan lattices. The latter are the algebraic counterparts of first-degree entailments. (Cf. Lemma 3.2.10.).

  11. 11.

    These were introduced in Dunn’s (1966) dissertation, where he independently investigated the algebra of R. See also Anderson and Belnap (1975, Sect. 28.2).

  12. 12.

    For a detailed algebraic explanation of this point, see Anderson and Belnap (1975, Sect. 28.2).

  13. 13.

    A detailed proof would be even longer than this two-page outline, and we do not include that here. Somewhat similar claims and proofs may be found in Bimbó and Dunn (2008, Chap. 9), as well as in Urquhart (1996), Bimbó (2007a) and Bimbó (2009).

  14. 14.

    See Anderson and Belnap (1975, pp. 351–352).

  15. 15.

    For later discussions of the same point in the Routley–Meyer semantics, see Dunn (1993) and Dunn (2001).

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Acknowledgements

We would like to thank Larisa Maksimova for providing us with e-copies of some of her publications that were otherwise inaccessible to us. We are thankful to Sergeǐ Odintsov for his help in this matter, and to the anonymous referees for their comments.

Parts of this paper belong to our larger project concerning the relational semantics of intensional logics, which is supported by an Insight Grant (IG #435–2014–0127), awarded by the Social Sciences and Humanities Research Council of Canada.

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Authors and Affiliations

  1. Department of Philosophy, University of Alberta, 2–40 Assiniboia Hall, Edmonton, AB, T6G 2E7, Canada

    Katalin Bimbó

  2. Department of Philosophy and School of Informatics, Computing, and Engineering, Indiana University, 901 East 10th Street, Bloomington, IN, 47408–3912, USA

    J. Michael Dunn

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  1. Katalin Bimbó
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Correspondence to J. Michael Dunn .

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Editors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Sergei Odintsov

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Bimbó, K., Dunn, J.M. (2018). Larisa Maksimova’s Early Contributions to Relevance Logic. In: Odintsov, S. (eds) Larisa Maksimova on Implication, Interpolation, and Definability. Outstanding Contributions to Logic, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-69917-2_3

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