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.
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