Abstract
In everyday speech, the expression ‘part’ is usually understood as having the sense of the expressions ‘fragment’, ‘bit’, and so forth.
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- 1.
See Footnote 18 in Appendix A.
- 2.
It is well-known that a relation is asymmetric iff it is irreflexive and antisymmetric. All necessary information on the subject of relations (including dependencies between the properties of a given relation) may be found in Appendix A.3.
- 3.
Recall that for binary relations \(R_1\) and \(R_2\) on U the binary relation \(R_1\circ R_2\) on U may be defined by the following condition: \(x\mathrel {R_1\circ R_2} y \iff \exists _{z\in U}(x\mathbin {R}_1 z\wedge z\mathbin {R}_2 y)\).
- 4.
The broadening of the extension of the word ‘part’ can sometimes lead to “philosophical misunderstandings”. The strange-sounding term ‘ingrediens’ is in this case an “ally”, as it reminds us that it is an “artificial” concept. Let us also note here that a misunderstanding arises from the choice of ‘ingredient’ as the translation of ‘ingredjens’ in the English translations of Leśniewski’s works (cf. Leśniewski 1991). The word ‘ingredient’ has a ready translation in Polish as ‘składnik’ and Leśniewski deliberately distanced himself from this word. It is therefore better to use the word ‘ingrediens’ in translation, thus preserving continuity with its (philosophico-logical) use in contemporary Polish.
- 5.
- 6.
We explain why in Sect. 2.13.
- 7.
This is a special case, as we could also recognise the element u as the zero and so add nothing.
- 8.
By starting from the strictly partially-ordered set \(\langle U,\prec \rangle \) with the zero \(\varnothing \) and putting \(\mathop {\preceqq } :\!=\mathord {\prec }\cup \mathrm {id}_{U}\), we can also obtain a partially-ordered set with zero \(\varnothing \).
- 9.
Such concepts we call coextensive (cf. Remark A.2.13).
- 10.
It is, however, different in the case of distributive classes (cf. Remark A.2.13).
- 11.
A name which has exactly one referent we will call monoreferential. A name which has at least two referents we will call polyreferential.
- 12.
- 13.
We have used here figuratively the word ‘true’ instead of the word ‘proper’, because the latter word could be used in connection with the word ‘part’ whereas here we are concerned with ‘truth’ in the quasi sense.
- 14.
See Appendices A.3.3 and B.9.
- 15.
It is not now permissible to adopt the definition ‘\(\forall \!_{x,y\in U}(x<y \;\mathrel {\mathord {:}\mathord {\Longleftrightarrow }}\;x\lesssim y \wedge x\ne y)\)’, so long as one wants to have an asymmetric relation. Under this second definition, the relation is asymmetric if and only if \(\lesssim \) is antisymmetric.
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Pietruszczak, A. (2020). An Introduction to the Problems of the Theory of Parthood. In: Foundations of the Theory of Parthood. Trends in Logic, vol 54. Springer, Cham. https://doi.org/10.1007/978-3-030-36533-2_1
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