Syntax vs Semantics: Comparing Consistency Proofs for Minimal Propositional Logics

Consistency is a key property of any logical system. However, proofs of consistency usually rely on heavy proof theory notions like admissibility of cut. A more semantics-based approach to consistency proofs explores the correspondence between a logic and its relationship with the evaluation in a λ-calculus, known as Curry-Howard isomorphism. In this work, we present a comparison between two formalizations of consistency for minimal propositional logic: one using a semantic-based approach and another following the (traditional) syntactic, proof-theoretical approach in both Coq proof assistant and Agda programming language. We conclude by discussing the lessons learned during the cerfication of these results in both languages.