First-order intuitionistic and classical Nelson–Wansing and Arieli–Avron–Zamansky logics, which are regarded as paradefinite and connexive logics, are investigated based on Gentzen-style sequent calculi. The cut-elimination and completeness theorems for these logics are proved uniformly via theorems for embedding these logics into first-order intuitionistic and classical logics. The modified Craig interpolation theorems for these logics are also proved via the same embedding theorems. Furthermore, a theorem for embedding first-order classical Arieli–Avron–Zamansky logic into first-order intuitionistic Arieli–Avron–Zamansky logic is proved using a modified Gödel–Gentzen negative translation. The failure of a theorem for embedding first-order classical Nelson–Wansing logic into first-order intuitionistic Nelson–Wansing logic is also shown.

基于Gentzen式的序贯演算，研究了被认为是超定和连接逻辑的一阶直觉逻辑和经典Nelson-Wansing逻辑和Arieli-Avron-Zamansky逻辑。通过将这些逻辑嵌入一阶直觉逻辑和经典逻辑的定理，一致地证明了这些逻辑的割消定理和完备性定理。这些逻辑的修改后的 Craig 插值定理也通过相同的嵌入定理得到证明。此外，使用改进的 Gödel-Gentzen 否定翻译证明了将一阶经典 Arieli-Avron-Zamansky 逻辑嵌入一阶直觉 Arieli-Avron-Zamansky 逻辑的定理。还显示了将一阶经典 Nelson-Wansing 逻辑嵌入一阶直觉 Nelson-Wansing 逻辑的定理的失败。