Toward the Formulation of Presupposition by Illative Combinatory Logic

Model-theoretic semantics, which originates with Tarski, and proof-thoretic semantics, which originates with Gentzen, are two views in semantics of logic that are distinct from but closely related to each other. Each has advantages over the other in investigating a certain aspect of logic, and it is more or less commonly accepted that rather than being a matter of methodological choice, utilizing them gives us diversified standpoints on various issues. A good example of this is the two proofs of the consistency of LK: one proof is based on soundness and the other on cut-elimination, which makes use of different resources but also together reveals what the consistency of predicate calculus indeed depends on.