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Published in:
An Introduction to the Problems of the Theory of Parthood Read chapter Read first chapter

2020 | OriginalPaper | Chapter

4. Theories Without the Assumption of Transitivity

Author : Andrzej Pietruszczak

Published in: Foundations of the Theory of Parthood

Publisher: Springer International Publishing

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Abstract

We have already observed (in Sect. 1.​2) that, in the literature, the transitivity of the relation is a part of is often called into question. We cited there a number of works which take different and interesting views on the matter. In this chapter, we will introduce some general approaches which will hopefully delight both defenders and opponents of the assumed transitivity of the relation. We will introduce the concept of the local transitivity  of a given relation which will be such that every transitive relation is also locally transitive.

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previous chapter “Existentially Involved” Theories of Parts
Footnotes
1
See Appendix B.1.2.
 
2
Observe that \(\mathrel {\varnothing ^M}\) (resp. \(\mathrel {(\!\!)^M}\), https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-36533-2_4/436544_1_En_4_IEq166_HTML.gif ) does not have to be identical with \(\varnothing |_M\) (resp. \(\mathord {(\!\!)}|_M\), https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-36533-2_4/436544_1_En_4_IEq169_HTML.gif ). See, e.g., Models 4.4 and 4.5.
 
3
Cf. principle (\(\mathrm {M}'_{\scriptscriptstyle \mathrel {\mathsf {sum}}}\)).
 
4
As in the case of Fact 4.5.7, we make use here of Model 4.5. Therefore, in addition, let us add that 12 and 23 belong to \(\min _{\sqsubset }(M_1)\), but \(12\varnothing 23\), which contradicts condition (a) from Lemma 4.8.1.
 
5
In this sense, axiom \((A4 _{\mathrm {w}}) \) is a “partial version” of axiom (A4).
 
6
In order not to complicate matters (see the point above), we will not introduce other “partial versions” of the conditions we have examined.
 
Literature
Pietruszczak, A. (2012). Ogólna koncepcja bycia częścią całości. Mereologia a nieprzechodnia relacja bycia częścią (A general concept of being a part of a whole. Mereology and the non-transitive relation of being a part.). In J. Golińska-Pilarek & A. Wójtowicz (Eds.), Identyczność znaku czy znak identyczności? (The identify of a sign or the sign of identity?) (pp. 157–179). Warszawa: Wydawnictwa Uniwersytetu Warszawskiego.
Metadata
Title
Theories Without the Assumption of Transitivity
Author
Andrzej Pietruszczak
Copyright Year
2020
Book
Foundations of the Theory of Parthood

Print ISBN: 978-3-030-36532-5

Electronic ISBN: 978-3-030-36533-2

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-36533-2_4

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