Abstract
This article discusses the quiddity of the empty set from its epistemological and linguistic aspects. It consists of four parts. The first part compares the concept of nihil privativum and the empty set in terms of representability, arguing the empty set can be treated as a negative and formal concept. It is argued that, unlike Frege’s definition of zero, the quantitative negation with a full scope is what enables us to represent the empty set conceptually without committing to an antinomy. The second part examines the type and scope of the negation in the concept of nihil privativum and the empty set. In the third part the empty set is interpreted as a rigid abstract general term. The uniqueness of the empty set is explained via a widened version of Kripke’s notion of rigidity. The fourth part proposes a construction for the pure singleton, comparing it with Zermelo’s conception of singletons with the Ur-elements. It is argued that the proposed construction does not face the criterion and ontological inflation problems. The first conclusion of the article is that the empty set can be construed as a negative, formal and unique abstract general term, with quantitative negation full in scope. The second conclusion is that the pure singleton constructed out of the empty set construed in this way overcomes the criterion and ontological inflation problems.
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Notes
The interpretations that identify sets with concepts or note their similarities are neither new nor based on mere analogies. There are some theoretical and textual grounds that either allow such interpretations or can be used in such interpretations. See Parsons (1984), especially pp. 111–115.
As is known, the Critique of Pure Reason is quoted according to the German paginations of the first and second editions, respectively. Kant’s other works are quoted by volume and the page number of the Akademie-Ausgabe. For these works, the following abbreviations are used: HS = Kant’s Handschriftlicher Nachlass, LM = Lectures on Metaphysics, NS = Natural Science, P = Prolegomena to any Future Metaphysics, TP = Theoretical Philosophy, 1755–1770. Regarding the translation of the quotes, when available, the Cambridge edition is used (sometimes with minor modifications), or I translate the quoted passages.
Zermelo (1908a) treats the empty set as fictitious: “Axiom II. (Axiom of elementary sets.) There exists a (fictitious) set, the null set, 0, that contains no element at all” (p. 202). Notice that the fictitious names are linguistically empty terms and they have no designatum. And epistemologically, they are cases for the concept of nihil privativum. Unlike Zermelo’s treatment, I identify the empty set with the concept of nihil privativum and argue that the concept of nihil privativum is not itself fictitious, but an abstract general term by which the fictitious terms can be explained. The relevant details will be presented in due course.
For coherence in usage, from here on I write the terms absence and nihil privativum in single quote marks when they are mentioned. In due course, there are some usages where ‘nihil privativum’ is mentioned as ‘‘nihil privativum’’. Please note that they are all single quote marks.
The purity of the singleton {\(\emptyset ~\)} needs some clarification. In the Kantian terminology, the pure concepts are the categories and unlike the empirical concepts they are not derived from experience. I use the adjective pure in a slightly different sense. Accordingly, ‘nihil privativum’ is derived from experience and it is an empirical concept in terms of how it is derived. However, it does not contain any object of intuition or experience in it, but the absence itself. It is called pure only in this respect.
Of course one may claim that it is possible to represent two identical thought entities, such as two equilateral triangles. However, these triangles are not counterexamples because they are imaginary in essence and they are not qualitatively identical, i.e., one of them is imagined on the right and the other one on the left. The thought entities concerned here are concepts. The claim is that if there are any two qualitatively identical concepts, they cannot be represented as numerically distinct. As an example, one cannot represent two equilateral triangle concepts because the concepts are not triangular figures. Due to their qualitative identity, they eventually collapse into one single concept, i.e., the concept equilateral triangle.
This point gets clearer when the axiom of infinity is considered. In the axiom of infinity, \(\exists I~\left( {~\emptyset ~ \in ~I~ \wedge ~\forall x~ \in I~\left( {~\left( {~x \cup \left\{ x \right\}} \right) \in I~} \right)~} \right)~)\), the iteration through union can be carried out only after constructing the empty set.
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Birgül, O.G. On the Onto-Epistemological Status of the Empty Set and the Pure Singleton. Axiomathes 32, 1111–1128 (2022). https://doi.org/10.1007/s10516-021-09571-6
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DOI: https://doi.org/10.1007/s10516-021-09571-6