Positive Logic: An Introduction for Model Theorists
Mark Kamsma
TL;DR
This work develops a self-contained framework for model theory in positive logic, a negation-free fragment that nevertheless supports compactness and a broad model-theoretic toolkit. It systematically builds from basics (p.c. models, homomorphisms, type spaces) to advanced topics (monster models, simplicity, stability) under mild thickness assumptions, culminating in a Kim–Pillay style synthesis. The notes then apply the developed theory to hyperimaginaries via an (-)^{heq} construction and to continuous logic by embedding its monster models into the positive framework, illustrating the robustness and reach of positive logic. The resulting treatment provides a coherent, self-contained path for translating core model-theoretic results into positive logic, with clear routes to practical applications and further study.
Abstract
Positive logic is a generalisation of full first-order logic that does not have negation built in. Still, many model-theoretic ideas, tools and techniques work perfectly fine in positive logic. Importantly, there is a compactness theorem. With some care, many classical results hold in the generality of positive logic without giving up any strength. In these self-contained notes we give an introduction to model theory in positive logic. We give a complete treatment of the basics of positive model theory and then we move on to deeper model-theoretic concepts. First, we discuss countable categoricity, where we work towards a theorem that characterises countably categorical positive theories. After that, we briefly discuss how the convenient formalism of monster models goes through in positive logic as usual. This is helpful in the remainder of the notes, where we discuss simple and stable theories. The main aim in those chapters is to develop dividing independence and prove Kim-Pillay style theorems. For a smoother treatment we assume thickness, which is the relatively mild assumption that being an indiscernible sequence is type-definable. We finish by discussing two big applications of positive logic: hyperimaginaries and continuous logic. For the former we define an $(-)^{\text{heq}}$ construction, analogous to the $(-)^{\text{eq}}$ construction for imaginaries in full first-order logic. Where the $(-)^{\text{heq}}$ construction is problematic in full first-order logic, it does stay within the framework in positive logic and it preserves many nice properties. For the latter we explain how continuous logic can be studied as a special case of positive logic, making it so that all abstract model-theoretic results in positive logic apply to continuous theories. In the appendix we provide a quick guide to the material covered in these notes, including very brief proof sketches.