It is ironic that intuitionism, whose origins are rooted in the concept of “proofs”, should produce so many (apparently) different kinds of models: Kripke models, Beth models, topological models, realizability, Swart models, and so on. Furthermore there appears to be a general view that most of the modellings are equivalent, although occasionally it is observed that they are not! In this talk we consider the concept of an abstract semantics for a logic L which we believe satisfies the minimum requirements in order to be called a “truth-value semantics” for L. We then discuss possible notions of equivalence between different semantics for L and in particular we catalogue just about all the truth-value semantics for intuitionistic logic and some of its extensions. We conclude with a Beth-like modelling for the extension CD (constant domains) of intuitionistic logic. © 1980, North-Holland Publishing Company.
López-Escobar, E. G. K. (1980). Semantical models for intuitionistic logics. Studies in Logic and the Foundations of Mathematics, 99(C), 191–207. https://doi.org/10.1016/S0049-237X(09)70486-0