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Transitivity


Johanna Seibt


Pages 570 - 579



Transitivity is a formal property of relations. A relation R is transitive just in case for arbitrary instantiations of variables x, y, z the following inference schema holds: if x is Rrelated to y and y is R-related to z, then it can be inferred that x is Rrelated to z. There are two ways in which a relation can fail to be transitive – it may be non-transitive or intransitive. For example, the relation ‘x is shorter than y’ is a transitive relation, while the relation ‘x is a friend of y’ is a non-transitive relation – for some but not all instantiations of x, y, z the inference is valid, in contrast, the relation ‘x is the biological mother of y’ is an intransitive relation, since there are no instantiations of x, y, z for which the inference is valid. Whether our reasoning about part-whole relationships is transitive, non-transitive, or intransitive is a significant issue since “failure of transitivity as a general part-whole principle would appear to have important philosophical ramifications” (Varzi 2006: 141). For example, philosophical analyses of material constitution and emergence, of spatio-temporal coincidence and transtemporal identity directly depend on this question. Yet in the contemporary philosophical discussion of formalisations of partwhole relationships the axiom of transitivity has received much less critical attention than other and less fundamental issues such as the extensionality of parthood or the existence of arbitrary sums. This may be the indication that transitivity of parthood is indeed a ‘law of thought’ of common sense reasoning; alternatively, it might reflect the contingent restrictions of a research tradition.




1Department for Philosophy and the History of Ideas, University Aarhus



1 Artale, A.; Franconi, E; Guarino, N; Pazzi, L., (1996), “Part-Whole Relations in Object-centered Systems: An Overview”, Data & Knowledge Engineering 20: 347-383.

2 Calosi, C.; Graziani, P. (eds), (2014), Mereology and the Sciences: Parts and Wholes in the Contemporary Scientific Context, Heidelberg: Springer.

3 Casati, R.; Varzi, A., (1999), Parts and Places – The Structures of Spatial Representation, Cambridge, MA: MIT Press.

4 Cotnoir, A., (2010), “Anti–Symmetry and Non–Extensional Mereology”, The Philosophical Quarterly 60: 396-405.

5 Cruse, D., (1979), “On the Transitivity of the Part-Whole Relation”, Journal of Linguistics 15: 29-38.

6 Fiorini, S.; Gärdenfors, P.; Abel, M., (2014), ”Representing Part-Whole Relations in Cognitive Spaces”, Cognitive Processing 15: 127-142.

7 Gerstl, P.; Pribbenow, S., (1995), “Midwinters, Endgames, and Body Parts – A Classification of Part-Whole Relations”, International Journal for Human-Computer Studies 43: 865-889

8 Gerstl, P.; Pribbenow, S., (1996), “A Conceptual Theory of Part-Whole Relations and its Applications”, Data & Knowledge Engineering 20: 305-322.

9 Guizzardi, G., (2009), “The Problem of Transitivity of Part-Whole Relations in Conceptual Modeling Revisited” in Van Eck, P.; Gordijn, J.; Wieringa, R. (eds.), International Conference on Advanced Information Systems Engineering, Heidelberg: Springer, 94-109.

10 Iris, M. A.; Litowitz, B. E.; Evens, M., (1988), “Problems of the Part-Whole Relation”, in Evens, M. (ed.), Relational Models of the Lexicon: Representing Knowledge in Semantic Networks, Cambridge: Cambridge University Press, 261-288.

11 Johansson, I., (2004), “On the Transitivity of the Parthood Relations”, in Hochberg, H.; Mulligan, K. (eds.), Relations and Predicates, Frankfurt: Ontos Verlag, 161-181.

12 Johansson, I.; Smith, B., (2005), “Functional Anatomy: A Taxonomic Proposal”, Acta Biotheoretica 53: 153-166.

13 Johansson, I., (2006), “The Constituent Function Analysis of Functions”, in H. J. Koskinen, et al. (eds.), Science – A Challenge to Philosophy, Frankfurt: Peter Lang, 35-45.

14 Johansson, I., (2006), “Formal Mereology and Ordinary Language–Reply to Varzi”, Applied Ontology 1: 157-161.

15 Kearns, S., (2011), “Can a Thing Be Part of itself?”, American Philosophical Quarterly 48: 87-93.

16 Lyons, J., (1977), Semantics I., Cambridge: Cambridge Univ Press.

17 Mani, A. (2012) “Axiomatic Granular Approach to Knowledge Correspondences”, in Li T. et al. (eds) Rough Sets and Knowledge Technology. RSKT 2012. Lecture Notes in Computer Science, vol 7414, Berlin: Springer, 482-487.

18 Moltmann, F., (1997). Parts and Wholes in Semantics, Oxford: Oxford University Press.

19 Motschnig-Pitrik, R., (1993). “The Semantics of Parts versus Aggregates in Data/Knowledge Modelling”, in Rolland, C.; Bodart, F.; Cauvet, C. (eds.), CAiSE ’93: Proceedings of Advanced Information Systems Engineering, Berlin: Springer, 352–373.

20 Miller, G. A.; Johnson-Laird, P. N., (1976), Perception and Language, London: Cambridge University Press.

21 Pietruszczak, A., (2014), “A General Concept of Being a Part of a Whole”, Notre Dame Journal for Formal Logic 55: 359-381.

22 Pribbenow, S., (2002), “Meronymic Relationships: From Classical Mereology to Complex Part-Whole Relations”, in Green, R.; Bean, C. A.; Sung Hyon Myaeng (eds.), The Semantics of Relationships, Berlin: Springer, 35-50.

23 Rescher, N., (1955), “Axioms for the Part Relation”, Philosophical Studies 6: 8-11.

24 Seibt, J., (2000), “The Dynamic Constitution of Things”, Poznan Studies in the Philosophy of the Sciences and the Humanities 76: 241-278.

25 Seibt, J., (2009), “Forms of Emergent Interaction in General Process Theory”, Synthese 166: 479-512.

26 Seibt, J., (2015), “Non-transitive Parthood, Leveled Mereology, and the Representation of Emergent Parts of Processes”, Grazer Philosophische Studien 91: 165-190.

27 Varzi, A., (2006), “A Note on the Transitivity of Parthood”, Applied Ontology 1: 141-146.

28 Vieu, L., (2006), “On the Transitivity of Functional Parthood”, Applied Ontology 1: 147–155.

29 Vieu, L.; Aurnague, M., (2007), “Partof Relations, Functionality and Dependence”, in: Aurnague, M.; Hickmann, M.; Vieu, L. (eds.), Categorization of Spatial Entities in Language and Cognition. Amsterdam: John Benjamins, 307-336.

30 Winston, M. E.; Chaffin, R.; Herrmann, D., (1987), “A Taxonomy of Part-Whole Relations”, Cognitive Science 11: 417-444.

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