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DIVIDING LINES BETWEEN POSITIVE THEORIES
- Part of
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- Journal:
- The Journal of Symbolic Logic , First View
- Published online by Cambridge University Press:
- 06 December 2023, pp. 1-25
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- Article
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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.
