Osman Gazi Birgül

The article aims to uncover the underlying cognitive foundations of the logical and semantic paradoxes. It is divided into two sections. The first part explores the cognitive and logical dimensions to unravel the reasons behind our vulnerability to paradoxes. By focusing on two established triggers—allowing excessively large sets in a system and resorting to impredicative definitions—it is argued that Kant’s insights into reason’s pursuit of unity and an unconditioned condition shed light on paradoxes arising from treating too large multiplicities as sets. Concerning impredicative definitions, it is argued that Kant’s analysis of the qualitative negation of qualities explains why pseudo-sets resulting from such definitions are not representable thought entities. Drawing on Kantian insights, this section reinterprets Cantor’s distinction between completed sets and inconsistent multiplicities and examines two famous solutions: Zermelo’s axiom schema of separation and von Neumann’s nonset class V. The second part contends that Kantian diagnoses are relevant to both logical and semantic paradoxes, elucidating the structures of sample paradoxes for each type. The Burali-Forti paradox and Russell’s paradox are examined as logical paradoxes, which result in pseudo-sets, illustrating Kant’s concept of nihil negativum—the empty object without a concept. The semantic paradoxes considered are Grelling’s and Berry’s paradoxes, whereby how self-reference and the use of nonapplicable predicates contribute to their paradoxical nature are explained. The study concludes by asserting that while current solutions effectively address paradoxes, additional support is needed in their cognitive dimensions, or areas where Kant’s epistemology provides profound insights.

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.

This paper aims at contributing to contemporary literature on nothings and absences from a Kantian perspective. To achieve its aim, the article is structured in two main sections. The first investigates nihil privativum’s metaphysical-epistemological, cognitive, and linguistic aspects, shedding light on its enduring relevance and multifaceted nature. The analysis begins by elucidating nihil privativum’s negativity, highlighting the distinctions between its material and formal senses, and explores the epistemological intricacies of accessing knowledge concerning nothings and absences. Engaging with contemporary perspectives within the framework of Kant’s philosophy, it demonstrates the enduring applicability of Kantian framework in addressing contemporary philosophical debates. Furthermore, the section delves into the linguistic dimension of nihil privativum, examining the distinctions between various forms of nothings and absences and the classification of terms denoting contradictory nothings. The second section begins with exploring the cognitive aspect of nihil privativum, yet this time as a general and abstract concept, unveiling the step-by-step process involved in its formation. The second section ends by an analysis of the linguistic aspect of ‘nihil privativum’ as a rigid general term, arguing for its rigidity as a general abstract term by logical necessity, which is shown through two thought experiments within the discourse of possible worlds. The paper concludes that in the material sense, nothings and absences are not perceivable, in their formal sense they are conceptually representable, and as a general and abstract term ‘nihil privativum’ is rigid, containing all particular nihil privativa in its extension and designating the property of ‘being non-existent or absent’ in all possible worlds.

This paper consists of two parts and has two aims. The first is to elaborate on the historical-philosophical background of Cantor’s notions of infinity in the context of Spinoza’s influence on him. To achieve this aim, in the first part I compare Spinoza’s and Cantor’s conceptions of actual infinity along with their mathematical implications. Explaining the metaphysical, conceptual, and methodological aspects of Cantor’s expansion of the orthodox finitist conception of number of his time, I discuss how he adopts Spinoza’s motifs to overcome the challenges on his way to proposing his transfinite infinites. To achieve the second aim, i.e., to investigate the philosophical foundations of Cantor’s transfinite infinites, I follow a three-step outline analyzing Cantor’s three principles along with their metaphysical and epistemological-linguistic aspects. I analyze the strengths and weak sides of Lakoff and Núñez’s Basic Metaphor of Infinity and Pantsar’s Process → Object Metaphor, then compare their interpretations with mine. I conclude that the metaphors can be improved when their source domains are limited to the target domains they apply to and when supported by a thorough analysis of the historical-philosophical background of their target domains.

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.