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The Ontology of Mereological Systems: A Logical Approach

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Theory and Applications of Ontology: Philosophical Perspectives

Abstract

Mereology is the theory of parthood relations. These relations pertain to part to whole, and part-to-part within a whole. This area of research is today some of the core topics of ontology and of conceptual modelling in computer science and artificial intelligence. The present paper addresses a number of relevant topics of this research field. First, the paper presents an overview on the main abstract mereological systems, axiomatized in first Order Logic (FOL). Second, basic relations between merelogy and set theory are discussed. This section is based mainly on the results of D. Lewis. Third, the paper is devoted to a systematic classification of merelogical systems. We present a partial classification of the consistent complete extensions of two theories, of the general extension mereology (GEM) including the second order variant, and of the classical merelogy CM. Then, we present some new systems which are extensions of the ground mereology M by introducing the notion of the tree-skeleton of a partial ordering. A complete and general description of the notion of whole and part which works for every situation seems to be impossible. Hence, we purpose a logical framework which allows to formally capture the main aspects of parts and wholes.

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Notes

  1. 1.

    This notion of being dense in a set derives from a well-known topological interpretation used in model theory where the types are points in a topological space which is called the Stone Space of the theory (Chang and Keisler 1973, Hodges 1993).

  2. 2.

    The stronger a theory the smaller the class of models, i.e. the following principle is satisfied for arbitrary theories T, S in first-order logic: Cn(T) ⊆ Cn(S) if and only if Mod(S) ⊆ Mod (T). Cn(T) is the deductive closure of the theory T.

  3. 3.

    It is stated on p. 53 (Brentano 1968). ‘Unter den Wesen welche Teile zeigen, finden sich einige, deren Ganzes sich nicht aus einer Mehrheit von von Teilen zusammensetzt; es erscheint vielmehr als eine Bereicherung eines Teiles, aber nicht durch Hinzukommen eines zweiten Teils.’

  4. 4.

    This is not the case for every definable binary relation. In fact, every binary relational system can be defined within a suitable partial ordering.

  5. 5.

    ∃!!(n)x has the meaning “there exists exactly n many x ”, whereas ∃!(n)x has the meaning “there exist at most n many x ”. The index (n)is omitted if n =1.

  6. 6.

    Universals are sometimes considered as categories being independent from any subject; they are associated to invariants of reality. Such universals cannot be immediately communicated, concepts must be related to them which, in turn, may be communicated by using symbols and tokens denoting them.

  7. 7.

    There is a relation between singletons and the process of bracketing in the sense of Husserl (1985). This term describes the process of thinking away the natural interpretation of an experience to capture its intrinsic nature. The natural interpretation refers to the immediate perception, the intrinsic nature to its pure existence.

  8. 8.

    We take account of in the present paper only natural numbers, not arbitrarly ordinals. This restriction is sufficient since we consider elementary classifications of mereological systems, i.e. classifications based on the language of first order predicate logic.

  9. 9.

    The elementary classification of linear orderings is more complicated than for Boolean algebras because their are uncountably many elementary types of linear orderings.

  10. 10.

    In most cases the objects are individuals.

  11. 11.

    This remark does not contradict the supplementation principle because wholeness adds features to the entity that cannot captured by pure mereology.

  12. 12.

    There can be, of course, many other entities that belong to the domain which cannot be generated in this mereological way, for example, those entities which are individual properties that inhere in objects. These entities may be captured by considering relations which are different from the part-of relation.

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Acknowledgment

Many thanks to F. Loebe, and R. Hoehndorf for diverse discussions about several topics of mereology. I am grateful to R. Poli and anonymous reviewers their critical remarks that contributed to the quality of paper.

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  1. Research Group Onto-Med, IMISE, University Leipzig, Leipzig, Germany

    Heinrich Herre

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  1. Heinrich Herre
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Correspondence to Heinrich Herre .

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  1. Dipto. Sociologia e Ricerca Sociale, Universita di Trento, Via Verdi 26, Trento, 38100, Italy

    Roberto Poli

  2. Dept. Philosophy, University of Aarhus, Jens Chr. Skousvej 7, Aarhus C, 8000, Denmark

    Johanna Seibt

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Herre, H. (2010). The Ontology of Mereological Systems: A Logical Approach. In: Poli, R., Seibt, J. (eds) Theory and Applications of Ontology: Philosophical Perspectives. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-8845-1_3

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