Paraconsistent Metatheory: New Proofs with Old Tools

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