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Intuitionistic mereology

  • S.I.: Mereology and Identity
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Abstract

Two mereological theories are presented based on a primitive apartness relation along with binary relations of mereological excess and weak excess, respectively. It is shown that both theories are acceptable from the standpoint of constructive reasoning while remaining faithful to the spirit of classical mereology. The two theories are then compared and assessed with regard to their extensional import.

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Notes

  1. Actually there is some ambiguity also in what counts as classical mereology tout court. Here we are thinking of the familiar theory based on classical first-order logic (Varzi 2016). Historically, however, this theory came to us in different guises. Leśniewski’s ‘Mereologia’ (1916, 1927–1931) was based on Ontology and Protothetic; Leonard and Goodman’s ‘Calculus of Individuals’ (1940) made use of quantification over classes. Such systems are not elementarily axiomatizable (Pietruszczak 2015) and are, therefore, strictly stronger than their first-order approximation, which is due to Goodman (1951).

  2. There is a literature on so-called ‘Heyting mereologies’, as in Forrest (2002), Mormann (2013), and Russell (2016). Despite the name, however, such theories are based on classical logic and differ from classical mereology with regard to their proper axioms (they lack Weak Supplementation), so they are non-classical in the first sense introduced above. Essentially, they deliver a parthood relation whose structure is not a Boolean but a Heyting algebra (about which see Johnstone 1982).

  3. For a review of the arguments, see Lando (2017, pt. ii). For an explicit defense of extensionality on behalf of classical mereology, see Varzi (2008).

  4. In Dutch: ‘verwijderd’, ‘plaatselijk verschillend’ (Brouwer 1923, §2); in German: ‘entfernt’ (Brouwer 1925, p. 254), ‘örtlich verschieden’ (Brouwer 1919, p. 3).

  5. Similar axioms for constructive ordering relations may be found in Bridges (1999); cf. also Scott (1968, §1).

  6. Classically, overlap is defined as sharing of a common part, and here we shall go along with that definition. We leave it to future work to investigate the possibility of adopting a notion of overlap with greater constructive appeal, such as the intuitionistic overlap relation developed by Ciraulo (2013) and Ciraulo et al. (2013) in the context of Sambin’s ‘overlap algebra’ (Sambin Forthcoming).

  7. We assume familiarity with Kripke models for intuitionistic logic, referring to Kripke’s original article (Kripke 1965) and to Troelstra and van Dalen (1988) for a systematic presentation. Specific applications to intuitionistic theories may be found in Smorynski (1973) and, with special reference to order theories, in Greenleaf (1978).

  8. See the original formulation in Simons (1987); the formulation with \(\leqslant \) is from Casati and Varzi (1999). See also Hovda (2009) and Varzi (2016, §3.1).

  9. Thanks to a referee for pressing us on this point.

  10. This result also follows from Theorem 8 below.

  11. See Varzi (2016, §3.2). A notable exception is Leonard and Goodman’s ‘Calculus of Individuals’ (1940), which is based on a primitive of mereological disjointness and includes EO (or, rather, its equivalent formulation in terms of disjointness) as an axiom. See also Niebergall (2011) for a systematic overview of classical mereology based on overlap, following Goodman (1951).

  12. Some authors actually identify Strong Supplementation with OP; see e.g. Hovda (2016, p. 187).

  13. Here, again, we are indebted to a referee for bringing this point to our attention.

  14. Classical mereology is not only extensional; it is also closed under unrestricted mereological fusion. In this paper we have not discussed any notion of fusion, which we leave for future work. Some indications on how to accommodate fusions constructively may be found in von Plato’s work and in Baroni’s work on constructive suprema (Baroni 2005).

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Authors and Affiliations

  1. Department of Philosophy, University of Barcelona, Barcelona, Spain

    Paolo Maffezioli

  2. Department of Philosophy, Columbia University, New York, USA

    Achille C. Varzi

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  1. Paolo Maffezioli
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  2. Achille C. Varzi
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Correspondence to Achille C. Varzi.

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Maffezioli, P., Varzi, A.C. Intuitionistic mereology. Synthese 198 (Suppl 18), 4277–4302 (2021). https://doi.org/10.1007/s11229-018-02035-2

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