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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Notes
- 1.
See Appendix B.1.2.
- 2.
- 3.
Cf. principle (\(\mathrm {M}'_{\scriptscriptstyle \mathrel {\mathsf {sum}}}\)).
- 4.
- 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.
) does not have to be identical with 
). See, e.g., Models References
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Pietruszczak, A. (2020). Theories Without the Assumption of Transitivity. In: Foundations of the Theory of Parthood. Trends in Logic, vol 54. Springer, Cham. https://doi.org/10.1007/978-3-030-36533-2_4
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