A Binary Quantifier for Definite Descriptions in Nelsonian Free Logic
Yaroslav Petrukhin
TL;DR
This work studies Kürbis' binary definite-descriptions quantifier $\mathtt{I}$ in Nelson's paraconsistent logics $\mathbf{N4}$ and $\mathbf{N4^{NF}}$, aiming for a constructive treatment of both truth and falsehood and addressing Russellian paradoxes. It develops Gentzen-Prawitz–style natural deduction systems with explicit rules for $\mathtt{I}$, constructs Kripke-style semantics and paradefinite valuations, and proves embedding theorems that connect Nelsonian logics to intuitionistic counterparts via a translation $\tau$. The paper also establishes syntactic and semantic embeddings, showing interderivability between $\mathtt{I}$ and its existential Russell form and obtaining completeness results for the base logics. It further outlines how these embeddings lay groundwork for extending to other free-logical variants and for future work on normalisation and broader logical extensions. Overall, the framework provides a constructive, paraconsistent approach to definite descriptions with potential to mitigate Russellian contradictions without explosion.
Abstract
The method Kürbis used to formalise definite descriptions with a binary quantifier I, such that I$x[F,G]$ indicates `the F is G', is examined and improved upon in this work. Kürbis first looked at I in intuitionistic logic and its negative free form. It is well-known that intuitionistic reasoning approaches truth constructively. We also want to approach falsehood constructively, in Nelson's footsteps. Within the context of Nelson's paraconsistent logic N4 and its negative free variant, we examine I. We offer an embedding function from Nelson's (free) logic into intuitionistic (free) logic, as well as a natural deduction system for Nelson's (free) logic supplied with I and Kripke style semantics for it. Our method not only yields constructive falsehood, but also provides an alternate resolution to an issue pertaining to Russell's interpretation of definite descriptions. This comprehension might result in paradoxes. Free logic, which is often used to solve this issue, is insufficiently powerful to produce contradictions. Instead, we employ paraconsistent logic, which is made to function in the presence of contradicting data without devaluing the process of reasoning.