Continuous logic is an extension of classical logic, allowing to replace the set of truth values {0,1} with the real line, and Bolean logical operations with continuous real-valued functions. Affine logic is a restriction of the latter to affine, rather than continuous connectives. This was suggested by Bagheri in a series of papers (referring to it as linear logic).

The restriction of the expressive power has the effect of endowing type spaces in affine logic with the structure of convex spaces, raising new questions regarding extremality: extreme types, extremal models (realising only extreme types), and decomposition of arbitrary models as integrals of extremal models. The restricted expressive power also poses new obstacles in our path when attempting to construct canonical parameters, and / or canonical bases in the context of stability. Finally, it is interesting to observe that Keisler randomisations, which were advanced at the time as natural examples of something we can do in continuous logic better than in classical logic, fit in fact much better in affine logic.

This is a joint work with Tomás Ibarlucía and Todor Tsankov.