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Erschienen in:
An Introduction to the Problems of the Theory of Parthood Kapitel lesen Erstes Kapitel lesen

2020 | OriginalPaper | Buchkapitel

2. “Existentially Neutral” Theories of Parts

verfasst von : Andrzej Pietruszczak

Erschienen in: Foundations of the Theory of Parthood

Verlag: Springer International Publishing

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Abstract

Let U be an arbitrary non-empty (distributive) set of objects (a so-called universe of discourse). Let \(\sqsubset \) be the extension of the relational concept being a part with respect to U. It is a binary relation in U which strictly partially orders the universe.

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Vorheriges Kapitel An Introduction to the Problems of the Theory of Parthood
Nächstes Kapitel “Existentially Involved” Theories of Parts
Fußnoten
1
For more on the concepts of a partial order and a strict partial order, see Appendices A.3, B.1 and B.2.
 
2
On the subject of quasi-orders, see Appendices A.3 and B.9.
 
3
In the diagram of Model 2.1 occur the names of conditions which will only be introduced later. This model will, however, also be used for those conditions. The same will be true of other models we will meet.
 
4
We shall always proceed in this way when the class \({\mathbf {\mathsf{{POS}}}}\) is our object of interest.
 
5
The very same concept is found under the names of ‘discreteness’ and ‘disjointness’ in (Leonard and Goodman 1940, p. 46) and (Simons 1987, p. 28), respectively. The symbol used here is modelled on the symbol used in (Simons 1987). In (Leonard and Goodman 1940), the symbol ‘\(\urcorner \llcorner \)’ is used. The terms ‘discreteness’ and ‘disjointness’ are, however, inaccurate, because it is possible to give an interpretation of mereology in point-free geometry, in which two objects are exterior to one another yet “touch”.
 
6
It follows also from the weak supplementation principle (see footnote 5 in Chap. 1 and Lemma 2.3.2 in this chapter).
 
7
Degenerate structures are of no interest to us, although all the conditions mentioned also hold in such structures if we accept that their single element is an atom.
 
8
Of course, in addition (2.3.4) does not follow from (\(\mathrm {WSP}\)) (cf. Remark 2.3.2). Moreover, (2.3.2) follows from (2.3.4); and so (2.3.2) follows also from (\(\mathrm {SSP}\)).
 
9
Obviously, it can only be the zero in degenerate structures.
 
10
In (Simons 1987) sentence (\(\mathrm {ext}_{\scriptscriptstyle \sqsubset }\mathrm {+}\)) is called the Proper Parts Principle. In Simons’ terminology ‘proper part’ corresponds to what we understand here by ‘part’.
 
11
Obviously—as in the case of Facts 2.1.1, 2.1.3 and 2.3.6—we also have counterparts of Facts 2.4.10 and 2.4.11 for the class \({\mathbf {\mathsf{{POS}}}}\). We will henceforth not emphasise this.
 
12
Simons (1987, p. 29) gives another proof of this result.
 
13
Recall that the simpler Model 2.3 from the class \({_{{\mathbf {\mathsf{{s}}}}}}{{\mathbf {\mathsf{{POS}}}}}\) suffices to show that (\(\mathrm {SSP}\)) does not follow from (\(\mathrm {ext}_{\scriptscriptstyle \bigcirc }\)). Obviously, another model simpler than Model 2.4 would likewise suffice to show that, in the class \({_{{\mathbf {\mathsf{{s}}}}}}{{\mathbf {\mathsf{{POS}}}}}\), condition (\(\mathrm {SSP}\)) does not follow from (\(\mathrm {ext}_{\scriptscriptstyle \sqsubset }\mathrm {+}\)).
 
14
The simpler Model 2.1 from \({_{{\mathbf {\mathsf{{s}}}}}}{{\mathbf {\mathsf{{POS}}}}}\) suffices to show that (\(\mathrm {WSP}\)) does not follow from (\(\mathrm {ext}_{\scriptscriptstyle \sqsubset }\mathrm {+}\)).
 
15
Here, the symbols ‘\(\sqsubset \)’, ‘\(\sqsubseteq \)’, ‘ https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-36533-2_2/436544_1_En_2_IEq930_HTML.gif ’, ‘\(\bigcirc \)’ and ‘\((\!\!)\)’ play a double role: as two-place predicates and the names of binary relations (more precisely: variable or binary relations). Obviously, it would be possible to have different symbols for the predicates and the names of relations.
 
16
It needn’t be a new class of structures. A given class may simply have various different names (through the addition of different names of conditions).
 
17
With reference to Sect. 2.1.2, let us recall that (\(\mathrm {r}_{\scriptscriptstyle \sqsubseteq }\)) is a direct consequence of definition (\(\mathrm {df}\,\sqsubseteq \)) and the reflexivity of identity. Therefore, in our theory we can have condition (\(\mathrm {r}_{\scriptscriptstyle \sqsubseteq }\)) without any additional assumptions. This holds for all the sentences in this lemma.
 
18
The proven sentence says that in the class \({_{{\mathbf {\mathsf{{s}}}}}}{{\mathbf {\mathsf{{POS}}}}}\) (resp. \({\mathbf {\mathsf{{QOS}}}}\)) the relation \(\mathrel {\mathsf {sum}}\) is included in the relation \(\mathrel {\mathsf {fu}}\) satisfying the following condition: \(x\mathrel {\mathsf {fu}}S\) iff \(\mathsf {O}(x)=\bigcup \mathsf {O}[S]\). The relation \(\mathrel {\mathsf {fu}}\) is another explication of the concept of being a collective set. In Sect. 2.12 of this chapter we will show that in the class \({_{{\mathbf {\mathsf{{s}}}}}}{{\mathbf {\mathsf{{PPOS}}}}}\) we obtain \(\mathord {\mathrel {\mathsf {sum}}}=\mathord {\mathrel {\mathsf {fu}}}\).
 
19
Henceforth, we will omit ‘and from our definitions’ from statements of entailments.
 
20
Obviously, to reject (\(\mathrm {S}_{\scriptscriptstyle \mathrel {\mathsf {sum}}}\)), (\(\mathrm {U}_{\scriptscriptstyle \mathrel {\mathsf {sum}}}\)) and (\(\mathrm {M}_{\scriptscriptstyle \mathrel {\mathsf {sum}}}\)) in the class \({_{{\mathbf {\mathsf{{s}}}}}}{{\mathbf {\mathsf{{POS}}}}}\), Model 2.1 suffices. To see this, \(1\not \sqsubseteq 0\) and thanks to (2.6.5) and (2.6.7): \(0\mathrel {\mathsf {sum}}\{0\}\) and \(1\mathrel {\mathsf {sum}}\{0\}\).
 
21
In (Leonard and Goodman 1940), the relation \(\mathrel {\mathsf {fu}}\) is also called the relation of sum-individual.
 
22
In (Leonard and Goodman 1940), the authors make reference to (Leśniewski 1931) several times.
 
23
Similar to how the empty set \(\varnothing \) is disjoint from every other set, including itself.
 
Literatur
Casati, R., & Varzi, A. C. (1999). Parts and places. Cambridge: The MIT Press.
Chisholm, R. M. (1992/93). Spatial continuity and the theory of part and whole. A Brentano study. Brentano Studien, 4, 11–23.
Leśniewski, S. (1931). O podstawach matematyki (On the foundations of mathematics). Przegląd Filozoficzny, 34, 142–170.
Libardi, M. (1990). Teorie delle parti e dell’intero. Mereolologie estensionali (Theories of parts and the whole. Extensional mereologies). Vol. II of Quaderni, Aprile–Dicembre.
Tarski, A. (1929). Les fondements de la géométrie des corps (Foundations of the geometry of solids). Księga Pamiątkowa Pierwszego Zjazdu Mate (pp. 29–30).
Tarski, A. (1956). Fundations of the geometry of solids. In J. H. Woodger (Ed.), Logic, semantics, metamathematics. Papers from 1923 to 1938 (pp. 24–29). Oxford: Oxford University Press. English version of (Tarski, 1929).
Metadaten
Titel
“Existentially Neutral” Theories of Parts
verfasst von
Andrzej Pietruszczak
Copyright-Jahr
2020
Buch
Foundations of the Theory of Parthood

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

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

Copyright-Jahr: 2020

DOI
https://doi.org/10.1007/978-3-030-36533-2_2