Abstract
Mereology is the theory based on a binary relation “being a part of”. Intuitively, using such a relation, notions such as “being an atom” and “being composed of” can be defined, and indeed, there are relevant axioms which can be found in the literature, such as the atomicity axiom and the fusion axiom schema. In this light, it might be hoped that some axiomatized mereological theory can secure an atomic domain, that is, can guarantee that everything in the domain is composed of atoms only. However, several negative results have been shown previously. This paper will give more negative results by showing that the strongest first-order atomic mereological theory which can be generated by axioms found in the literature, that is, General Extensional Mereology with the atomicity axiom, as well as a natural second-order extension of such a theory, that is, Classical Mereology with the atomicity axiom, cannot secure an atomic domain either.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The definition of “being composed of” here, which is in effect also the definition of “being a fusion of”, is slightly different from the definition of “being a sum of” mentioned in (Tsai and Varzi 2016) in that the former does not require objects in the collection to be pairwise disjoint whereas the latter does. But this does not matter for my purpose, for atoms are of course pairwise disjoint.
- 2.
An atomic domain contains a collection of “atoms” (an atom is something which has no proper parts) and each member there is composed of some atoms and nothing else. If any model of a mereological theory has an atomic domain, we say that such a theory can “secure an atomic domain” or can “guarantee that everything in the domain is composed of atoms only”.
- 3.
An atomic mereological theory is a theory whose language has only one binary predicate P and whose axioms contain the atomicity axiom. \(\mathbf {GEM}+\text {(A)}\) is the strongest first-order atomic theory that can be axiomatized by axioms found in the literature, for \(\mathbf {GEM}\) can prove all other traditional axioms which are compatible with (A). For this issue, please refer to the ordering of theories mentioned above and to the axioms listed in (Simons 1987) and in (Casati and Varzi 1999). By the way, after showing this result, I realized that I had made a careless mistake in (Tsai 2015) by saying that \(\mathbf {GEM}+\text {(A)}\) can secure an atomic domain.
- 4.
The consistency of \(\mathbf {GEM}+\text {(A)}+\text {(Infinity)}\) can be easily checked. Also see \(M_4\) below.
- 5.
Here the construction of \(M_2\) is based on an idea similar to that in (Tsai and Varzi 2016).
- 6.
The name “classical mereology” follows the nomenclature in Pietruszczak (2015) and the formulation here is a rephrasing of the definition of classical mereology given there.
- 7.
An anonymous referee seems to suggest that a finite model can also have members with gunky parts. But this can hardly make sense. Consider the following finite model whose domain contains three members \(\{2, 3\}\cup (0, 1), \{2\}\) and \(\{3\}\) and whose interpretation of P is the set inclusion. In this model, \(\{2, 3\}\cup (0, 1)\) is composed of \(\{2\}\) and \(\{3\}\), or in other words, \(\{2, 3\}\cup (0, 1)\) is the “fusion” of \(\{2\}\) and \(\{3\}\). One might think that the fusion of \(\{2\}\) and \(\{3\}\) should be \(\{2, 3\}\) instead of \(\{2, 3\}\cup (0, 1)\), but \(\{2, 3\}\) is not in the domain (or to put it more drastically, it does not “exist”) and hence in that model the fusion will be \(\{2, 3\}\cup (0, 1)\). Furthermore, one cannot sensibly say that \(\{2, 3\}\cup (0, 1)\) is a member with a gunky part, for (0, 1) has not been divided indefinitely in that model. It is not tenable either to say that (0, 1) is “necessarily” a piece of gunk, for it won’t be a piece of gunk in a model whose domain contains all nonempty subsets of the set of real numbers. In short, whether something is a piece of gunk depends on the model where it is situated. In my opinion, here an important issue is: what kind of models can be accepted by metaphysicians as counterexamples to the traditional definition of composition. A model as simple as the one mentioned above can hardly be acceptable. Some philosophical concerns must be considered. Even though we have made a point previously that the traditional definition of composition seems to be defective (Tsai and Varzi 2016), for me it is still unclear how far a mathematical model can go when it comes to a metaphysical issue. This paper is aimed at giving two further technical results and I do not intend to address the aforementioned philosophical issue, but I will probably deal with it somewhere else.
References
Casati, R., & Varzi, A. C. (1999). Parts and Places. Cambridge: MIT Press.
Pietruszczak, A. (2015). Classical mereology is not elementarily axiomatizable. Logic and Logical Philosophy, 24, 485–98.
Simons, P. (1987). Parts: A Study in Ontology. Oxford: Clarendon Press.
Tsai, H.-C. (2013). Decidability of general extensional mereology. Studia Logica, 101(3), 619–36.
Tsai, H.-C. (2015). Notes on models of first-order mereological theories. Logic and Logical Philosophy, 24, 469–82.
Tsai, H.-C., & Varzi, A. C. (2016). The Limits of ‘Composition’. Erkenntnis, 81, 231–35.
Acknowledgements
This paper is a product of a research project (104-2410-H-194 -098 -MY3) funded by Ministry of Science and Technology, Taiwan.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Tsai, HC. (2017). Infinite “Atomic” Mereological Structures. In: Yang, SM., Lee, K., Ono, H. (eds) Philosophical Logic: Current Trends in Asia. Logic in Asia: Studia Logica Library. Springer, Singapore. https://doi.org/10.1007/978-981-10-6355-8_14
Download citation
DOI: https://doi.org/10.1007/978-981-10-6355-8_14
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-10-6354-1
Online ISBN: 978-981-10-6355-8
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)