Type Theory and its Meaning Explanations
At the heart of intuitionistic type theory lies an intuitive semantics called the "meaning explanations"; crucially, when meaning explanations are taken as definitive for type theory, the core notion is no longer "proof" but "verification". We'll explore how type theories of this sort arise naturally as enrichments of logical theories with further judgements, and contrast this with modern proof-theoretic type theories which interpret the judgements and proofs of logics, not their propositions and verifications.