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Binary Definite-Description Quantifier in Nelson's Paraconsistent Free Logic

A treatment of Russellian definite descriptions ('the F is G') using a binary quantifier Ix[F,G] within Nelson's paraconsistent constructive logic N4 and its negative free variant, rather than intuitionistic or free logic alone. The work supplies a Gentzen-Prawitz natural deduction system, Kripke-style semantics, and syntactic/semantic embedding theorems into intuitionistic (free) logic, yielding completeness and making both truth and falsity constructive while blocking Russell's descriptions from generating paradox.