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Published in:
An Introduction to the Problems of the Theory of Parthood Read chapter Read first chapter

2020 | OriginalPaper | Chapter

3. “Existentially Involved” Theories of Parts

Author : Andrzej Pietruszczak

Published in: Foundations of the Theory of Parthood

Publisher: Springer International Publishing

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Abstract

Theories bearing this title are those for which at least one defining condition forces the existence of mereological sums. On Sect. 2.​6.​2 we said that a theory deserving the name ‘theory of parthood’ is one in which condition \((\mathrm {U}_{\scriptscriptstyle \mathrel {\mathsf {sum}}})\) is in force.

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previous chapter “Existentially Neutral” Theories of Parts
next chapter Theories Without the Assumption of Transitivity
Footnotes
1
Let us remember that Theorem 2.​6.​7 says that, in the class \({}_{{\varvec{\mathsf {s}}}}{\varvec{\mathsf {POS}}}\) (resp. \({\varvec{\mathsf {QOS}}}\)), sentences \((\mathrm {U}_{\scriptscriptstyle \mathrel {\mathsf {sum}}})\) and (\(\mathrm {ext}_{\scriptscriptstyle \bigcirc }\)) are equivalent.
 
2
We are using the term ‘product’ because—by virtue of (3.2.2)—it is the mereological sum of all common ingredienses of x and y. This product must also be at the same time their product in an algebraic sense, i.e., their greatest lower bound.
 
3
The phrase ‘for all and only those x and y’ we use on the strength of condition (3.2.1).
 
4
It is easy to see that this model arises from the four-element Boolean algebra from which the zero and the unity have been removed. Other models of this type are obtained in the same way from other finite Boolean algebras. For example, from the eight-element Boolean algebras we obtain Model 3.7.
 
5
In Sect. 3.10—where we investigate the case of finite structures—we will show that in the proof of this fact we had to take an infinite model, because \({}_{{\varvec{\mathsf {s}}}}{\varvec{\mathsf {PPOS}}}_{\mathrm {fin}}{+}(\exists _{{\scriptscriptstyle \mathrm {pair}}}\mathord {\mathrel {\mathsf {sum}}})={\varvec{\mathsf {MEM}}}_{\mathrm {fin}}{+}(\exists _{{\scriptscriptstyle \mathrm {pair}}}\mathord {\mathrel {\mathsf {sum}}})\).
 
6
Note that \(v=z\), because we have (\(\mathrm {U}_{\scriptscriptstyle \mathrel {\mathsf {sup}}}\)) and \(\mathord {\mathrel {\mathsf {sum}}}\subseteq \mathord {\mathrel {\mathsf {sup}}}\), from (\(\mathrm {SSP}\)). One could also use (\(\diamond \)), which is equivalent to (\(\mathrm {WSP}\)).
 
7
The universe U of the model of the class \({\varvec{\mathsf {MEM}}}\) in which \((\mathrm {c}\exists _{{\scriptscriptstyle \mathrm {pair}}}\mathord {\mathrel {\mathsf {sup}}})\) is false must be infinite, because the proof of Fact 3.10.1 shows that \({\varvec{\mathsf {MEM}}}_{\mathrm {fin}}{+}(\mathrm {c}\exists _{{\scriptscriptstyle \mathrm {pair}}}\mathord {\mathrel {\mathsf {sup}}})={\varvec{\mathsf {MEM}}}_{\mathrm {fin}}\).
 
8
The operation \(\mathbin {{\mathbf {\mathsf{{+}}}}}\) can also be introduced in the class \({}_{{\varvec{\mathsf {s}}}}{\varvec{\mathsf {PPOS}}}{+}(\mathrm {c}\exists _{{\scriptscriptstyle \mathrm {pair}}}\mathord {\mathrel {\mathsf {sup}}})\). Structures of this class with the operation \(\mathbin {{\mathbf {\mathsf{{+}}}}}\) are conditional semi-lattices.
 
9
The operation \(\sqcup \) can also be entered in the class \({}_{{\varvec{\mathsf {s}}}}{\varvec{\mathsf {PPOS}}}{+}(\mathrm {c}\exists _{{\scriptscriptstyle \mathrm {pair}}}\mathord {\mathrel {\mathsf {sum}}})\). Structures of this class with the operation \(\sqcup \) are also conditional semi-lattices.
 
10
We do not want to use the expression ‘conditional distributivity’ because it is used for “full” lattices. Here we are dealing with “full” distributivity in “partial” lattices.
 
11
The operation \(\mathbin {{\mathbf {\mathsf{{+}}}}}\) can also be introduced in the class \({}_{{\varvec{\mathsf {s}}}}{\varvec{\mathsf {PPOS}}}{+}(\exists _{{\scriptscriptstyle \mathrm {pair}}}\mathord {\mathrel {\mathsf {sup}}})\). Structures of this class with the operation \(\mathbin {{\mathbf {\mathsf{{+}}}}}\) are semi-lattices.
 
12
The operation \(\sqcup \) can also be entered in the class \({}_{{\varvec{\mathsf {s}}}}{\varvec{\mathsf {PPOS}}}{+}(\exists _{{\scriptscriptstyle \mathrm {pair}}}\mathord {\mathrel {\mathsf {sum}}})\). Structures of this class with the operation \(\sqcup \) are semi-lattices.
 
13
What was said in footnote 10 in the context of the theory \({\varvec{\mathsf {CMM}}}\) also applies to here the strengthened theorem.
 
14
In Sect. 3.10—where we investigate the case of finite structures—we will show that in the proof of this fact we had to take an infinite model, because \({\varvec{\mathsf {CMM}}}_{\mathrm {fin}}{+}({\ddag }_{\scriptscriptstyle \varnothing })={\varvec{\mathsf {CMM}}}_{\mathrm {fin}}\) and \({\varvec{\mathsf {MEM}}}_{\mathrm {fin}}{+}(\exists _{{\scriptscriptstyle \mathrm {pair}}}\mathord {\mathrel {\mathsf {sum}}}){+}({\ddag }_{\scriptscriptstyle \varnothing })={\varvec{\mathsf {MEM}}}_{\mathrm {fin}}{+}(\exists _{{\scriptscriptstyle \mathrm {pair}}}\mathord {\mathrel {\mathsf {sum}}})\).
 
15
In some of the literature, \((\mathrm {WRP})\) is actually called the ‘weak remainder axiom’; see, e.g., (Cotnoir and Varzi 2018) and references therein.
 
16
In Sect. 3.7 we will show that principle \((\mathrm {SSP+})\) has a connection with axiom M4 used by Grzegorczyk (1955).
 
17
In some of the literature, \((\mathrm {RP})\) is actually called the ‘remainder axiom’ or ‘complementation’; see, e.g., (Varzi 2016; Cotnoir and Varzi 2018) and references therein.
 
18
The results described above concerning the partial operations of mereological relative complementation and product make reference to facts established in Sect. B.6, which concerns Grzegorczykian lattices from (Grzegorczyk 1955). We derive these results in an analogous way (with an insignificant modification). Such operations are not introduced in (Grzegorczyk 1955).
 
19
We will find two such elements x and y when and only when \(\langle U,\sqsubset \rangle \) is non-degenerate.
 
20
We can also get the simple implication from (3.6.1). If \(x\ne 1\ne y\) and \(x\sqsubseteq y\), then \(1\not \sqsubseteq x\), \(1\not \sqsubseteq y\) and https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-36533-2_3/436544_1_En_3_IEq1310_HTML.gif .
 
21
This fact cannot be proven by using finite models, as we will show in Sect. 3.10.
 
22
In a topological space \(\langle U,\mathrm {Int}\rangle \) a set X is regularly open iff \(X=\mathrm {Int}\mathrm {Cl}(X)\), where \(\mathrm {Cl}(X)=U\setminus \mathrm {Int}(U\setminus X)\), i.e., X is equal to the interior of its closure. The family of regularly open sets in \(\langle U,\mathrm {Int}\rangle \) creates a complete Boolean algebra with the operations \(X\cdot Y:=X\cap Y\), \(X+ Y:=\mathrm {Int}\mathrm {Cl}(X\cup Y)\) and \(-X:=\mathrm {Int}(U\setminus X)=U\setminus \mathrm {Cl}(X)\), in which \(\varnothing \) is the zero and U is the unity. We can therefore put \(X - Y:=X\cdot -Y\). We have \(X-Y=\varnothing \) iff \(X\subseteq Y\).
 
23
The fact below is obviously a strengthening of Fact 3.6.9. However, the model used in the proof of Fact 3.6.9 is simpler than the one used below.
 
24
If S is an infinite family included in \(\mathcal {O}\), then S has no least upper bound.
 
25
Classical mereological structures will be discussed in Sect. 3.9.
 
26
However, this plain way of putting things is ultimately risks being rather misleading. Since a Boolean algebra by definition has a zero, talk of a Boolean algebra without zero is rather like talking of a “larger half” or a “round square”.
 
27
We have the reciprocal definability of the greatest lower and the least upper bounds. It is, however, essential to assume that the least upper (resp. greatest lower) bound should exist for an arbitrary set of elements. A greatest lower (resp. least upper) bound of some infinite set may, however, correspond to the least upper (resp. greatest lower) bound of a finite set of elements. We do not, therefore, have the reciprocal definability of the operations of sum and product, because they are operative only on finite sets of elements.
 
28
Leśniewski’s original mereology was presented in Sect. 1.​4.
 
29
An elementary proof of this fact can be found (for example) in (Pietruszczak 2000, p. 125); cf. also (Pietruszczak 2018, pp. 155–156).
 
30
In (Pietruszczak 2000, 2018), this theory is signified by ‘\({\varvec{\mathsf {TS}}}\)’.
 
31
The problem here was that \((\exists ^1\mathord {\mathrel {\mathsf {sum}}})\) postulates the uniqueness of the relation \(\mathrel {\mathsf {sum}}\) only for non-empty sets, but \((\mathrm {U}_{\scriptscriptstyle \mathrel {\mathsf {sum}}})\) for all sets. Since, however, the relation \(\mathrel {\mathsf {sum}}\) does not overlap \(\varnothing \), condition \((\mathrm {U}_{\scriptscriptstyle \mathrel {\mathsf {sum}}})\) is also satisfied.
 
32
Obviously, instead of the last condition we could use (\(\mathrm {irr}_{\scriptscriptstyle \sqsubset }\)) whilst also admitting any of the conditions that define the class \({}_{{\varvec{\mathsf {s}}}}{\varvec{\mathsf {POS}}}\).
 
33
In (Pietruszczak 2000, 2018), this theory is signified by ‘\({\varvec{\mathsf {MS}}}\)’.
 
34
Another of Leśniewski’s axiomatisations guarantees the uniqueness of this collective class. We use the word ‘completeness’ in a metaphorical sense because there is no mereological sum for the empty set.
 
35
The first set of axioms arises simply through the addition of condition (\(\mathrm {compl}_{\mathrel {{\mathsf {sup}}_{\scriptscriptstyle {}}}}\)) to the second set.
 
36
We discussed this issue in detail there.
 
37
In (Tarski 1956b), this is footnote 1 on pp. 333–334.
 
38
We will show this later when we enter into a more detailed discussion of Grzegorczyk’s theory.
 
39
In this way arose those structures which we have called Grzegorczykian lattices and which are the subject of detailed discussion in Appendix B.6. All Grzegorczykian lattices with unity are, however, Boolean lattices.
 
40
Although in (Grzegorczyk 1955) the definition of ‘ingr’ is given with the help of a condition corresponding to our (\(\mathrm {df}\,\sqsubseteq \)), the truth is that it provides only an “intuitive” explanation of that concept. Otherwise, the axiom stating that ‘ingr’ is reflexive would clearly be superfluous, as it would follow from that definition. Furthermore, axioms for the predicate ‘is a part of’, to which corresponds the symbol ‘\(\sqsubset \)’, would also be missing.
 
41
The adoption of definition (\(\mathrm {df}\,\sqsubseteq \)) fits with Grzegorczyk’s intentions (see the previous footnote). Obviously, one can treat the presentation of Grzegorczyk’s theory as a description of an elementary theory whose models we would consider as structures (cf. point 2.​5.​1).
 
42
In (Pietruszczak 2000, 2018, p. 86 or p. 107), other proofs of the last three conditions are given. They were carried out, however, for the class \({\varvec{\mathsf {LMS}}}\) of classical mereological structures.
 
43
In (Grzegorczyk 1955), only this third part of this theorem appears, and without any proof.
 
44
What was said in the previous footnote applies here in the same way.
 
45
Observe that, in the corollary below, we must add the assumption of the irreflexivity of the relation \(\sqsubset \) in condition (b).
 
46
Below, by drawing on this corollary, we will prove that \({\varvec{\mathsf {LMS}}}={}_{{\varvec{\mathsf {s}}}}{\varvec{\mathsf {PPOS}}}{+}(\exists _{{\scriptscriptstyle \mathrm {pair}}}\mathord {\mathrel {\mathsf {sum}}}){+}(\mathrm {w}_{\scriptscriptstyle \mathrm {2}}\exists \mathord {\mathrel {\mathsf {sum}}})\), which entails \({\varvec{\mathsf {GMS}}}{+}(\mathrm {w}_{\scriptscriptstyle \mathrm {2}}\exists \mathord {\mathrel {\mathsf {sum}}})={\varvec{\mathsf {LMS}}}\) (see Theorem 3.9.6).
 
47
Model 3.7 arose from Model 2.​8 through the removal of the unity. Obviously, we had to use model without unity in the light of Lemma 3.8.3.
 
48
We are not drawing on here the following result proven by Tarski, namely that for structures of the form \(\langle U,\sqsubseteq \rangle \) conditions (\(\mathrm {r}_{\scriptscriptstyle \sqsubseteq }\)) and (\(\mathrm {antis}_{\scriptscriptstyle \sqsubseteq }\)) follow from (\(\mathrm {t}_{\scriptscriptstyle \sqsubseteq }\)) and \((\exists ^1\mathord {\mathrel {\mathsf {sum}}})\). For we are looking at structures of the form \(\langle U,\sqsubset \rangle \), and (\(\mathrm {r}_{\scriptscriptstyle \sqsubseteq }\)) follows from (\(\mathrm {df}\,\sqsubseteq \)). Therefore, we immediately get the result that there is no mereological sum for the empty set.
 
49
This is in any case perhaps only the only way of establishing this result, even if we were not to involve ourselves with \({}_{{\varvec{\mathsf {s}}}}{\varvec{\mathsf {PPOS}}}{+}(\mathrm {w}_{\scriptscriptstyle \mathrm {1}}\exists \mathord {\mathrel {\mathsf {sum}}})\). This is shown by the fact that, with (\(\mathrm {SSP}\)) holding, condition \((\mathrm {SSP+})\) can be derived from a weaker condition than \((\exists \mathord {\mathrel {\mathsf {sum}}})\).
 
50
See Theorem 2.​12.​4 and Fact 2.​12.​5.
 
51
See Corollary 2.​7.​5 and Theorem 2.​9.​5, respectively.
 
52
See Lemma 3.8.3.
 
53
It is possible to substitute various “particular” formulas for the schema ‘Fs’, such as ‘\(s=x\vee s=y\)’ etc. In (Simons 1987) an elementary counterpart of condition (\(\mathrm {as}_{\scriptscriptstyle \sqsubset }\)) is needlessly adopted, since it follows from (\(\mathrm {t}_{\scriptscriptstyle \sqsubset }\)) and (\(\mathrm {WSP}\)).
 
54
We remember that the following identity holds: \({\varvec{\mathsf {LMS}}}={}_{{\varvec{\mathsf {s}}}}{\varvec{\mathsf {POS}}}{+}(\mathrm {WSP}){+}(\exists \mathord {\mathrel {\mathsf {sum}}})\).
 
55
Observe that \(\bigcap \mathsf {I}[\varnothing ]=\bigcap \{\mathsf {I}(z): z\in \varnothing \}=\bigcap \varnothing =U\ne \varnothing \).
 
56
That, in the class \({\varvec{\mathsf {TMS}}}\), conditions (\(\mathrm {r}_{\scriptscriptstyle \sqsubseteq }\)) and (\(\mathrm {antis}_{\scriptscriptstyle \sqsubseteq }\)) hold was proven by Tarski by applying Theorem . We will give an elementary proof of this result.
 
57
Drewnowski’s notes are unpublished. I read them courtesy of Professor Kordula Świtorzecka of the Cardinal Wyszyński University in Warsaw.
 
58
Thus, we see that Leśniewski defined the concept of collective class differently than in (1927; 1928) (cf. Definition \({\mathrm {df}}\,\mathrm {cc}{} \texttt {P}\)).
 
59
Observe once again that, in the corollary below, we must add the assumption of the irreflexivity of \(\sqsubset \) to condition (b).
 
Literature
Breitkopf, A. (1978). Axiomatisierung einiger Begriffe aus Nelson Goodmans the structure of appearance (Axiomatization of some concepts from Nelson Goodman’s the structure of appearance). Erkenntnis, 12, 229–247.CrossRef
Casati, R., & Varzi, A. C. (1999). Parts and places. Cambridge: The MIT Press.
Gruszczyński, R., & Pietruszczak, A. (2014). The relations of supremum and mereological sum in partially ordered sets. In C. Calosi & P. Graziani (Eds.), Mereology and the sciences. Parts and wholes in the contemporary scientific context. Synthese library “Studies in epistemology, logic, methodology, and philosophy of science” (Vol. 371). Berlin: Springer. https://​doi.​org/​10.​1007/​978-3-319-05356-1_​6.
Leśniewski, S. (1927). O podstawach matematyki (On the foundations of mathematics). Przegląd Filozoficzny, 30, 164–206.
Leśniewski, S. (1928). O podstawach matematyki (On the foundations of mathematics). Przegląd Filozoficzny, 31, 261–291.
Leśniewski, S. (1930). O podstawach matematyki (On the foundations of mathematics). Przegląd Filozoficzny, 33, 77–105.
Szmielew, W. (1981). Od geometrii afinicznej do euklidesowej (From affine to Euclidean geometry). Warszawa: PWN.
Szmielew, W. (1983). From affine to Euclidean geometry: An axiomatic approach. Netherlands: Springer. English version of (Szmielew 1981).
Tarski, A. (1929). Les fondements de la géométrie des corps (Foundations of the geometry of solids). Księga Pamiątkowa Pierwszego Zjazdu Matematycznego (pp. 29–30). Kraków: Annales de la Societé Polonaise de Mathématique.
Tarski, A. (1956a). Foundations 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).
Tarski, A. (1956b). On the foundations of Boolean algebra. In J. H. Woodger (Ed.), Logic, semantics, metamathematics. Papers from 1923 to 1938 (pp. 320–341). Oxford: Oxford University Press. English version of (Tarski 1935).
Metadata
Title
“Existentially Involved” Theories of Parts
Author
Andrzej Pietruszczak
Copyright Year
2020
Book
Foundations of the Theory of Parthood

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

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

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-36533-2_3

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