Positive Logic: An Introduction for Model Theorists

TL;DR

This paper introduces positive logic, a form of logic without negation, demonstrating that many classical model-theoretic results extend to it, including compactness, stability, and applications to hyperimaginaries and continuous logic.

Contribution

It provides a comprehensive introduction to positive model theory, extending key concepts like categoricity, independence, and hyperimaginaries within this negation-free framework.

Findings

Positive logic retains compactness and many classical results.

The $(-)^{ ext{heq}}$ construction works within positive logic, unlike in full first-order logic.

Continuous logic can be viewed as a special case of positive logic.

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.