This paper is a step toward showing what is achievable using non-classical metatheory—particularly, a substructural paraconsistent framework. What standard results, or analogues thereof, from the classical metatheory of first order logic(s) can be obtained? We reconstruct some of the originals proofs for Completeness, Löwenheim-Skolem and Compactness theorems in the context of a substructural logic with the naive comprehension schema. The main result is that paraconsistent metatheory can ‘re-capture’ versions of standard theorems, given suitable restrictions and background assumptions; but the shift to non-classical logic may recast the meanings of these apparently ‘absolute’ theorems.

这篇论文是向展示使用非经典元理论（尤其是子结构的同构框架）可以实现的目标迈出的一步。可以从一阶逻辑的经典元理论中获得哪些标准结果或类似结果？我们在具有朴素理解模式的子结构逻辑的上下文中重建了完整性、Löwenheim-Skolem 和紧致性定理的一些原始证明。主要结果是，在给定适当的限制和背景假设的情况下，超一致元理论可以“重新捕获”标准定理的版本；但是向非经典逻辑的转变可能会改变这些明显“绝对”定理的含义。