DIVIDING LINES BETWEEN POSITIVE THEORIES

Abstract

Abstract We generalise the properties $\mathsf {OP}$ , $\mathsf {IP}$ , k- $\mathsf {TP}$ , $\mathsf {TP}_{1}$ , k- $\mathsf {TP}_{2}$ , $\mathsf {SOP}_{1}$ , $\mathsf {SOP}_{2}$ , and $\mathsf {SOP}_{3}$ to positive logic, and prove various implications and equivalences between them. We also provide a characterisation of stability in positive logic in analogy with the one in full first-order logic, both on the level of formulas and on the level of theories. For simple theories there are the classically equivalent definitions of not having $\mathsf {TP}$ and dividing having local character, which we prove to be equivalent in positive logic as well. Finally, we show that a thick theory T has $\mathsf {OP}$ iff it has $\mathsf {IP}$ or $\mathsf {SOP}_{1}$ and that T has $\mathsf {TP}$ iff it has $\mathsf {SOP}_{1}$ or $\mathsf {TP}_{2}$ , analogous to the well-known results in full first-order logic where $\mathsf {SOP}_{1}$ is replaced by $\mathsf {SOP}$ in the former and by $\mathsf {TP}_{1}$ in the latter. Our proofs of these final two theorems are new and make use of Kim-independence.