Submission date: 15 April 2023

Abstract:

We give definitions of the properties OP, IP, k-TP, TP_1, k-TP_2, SOP_1, SOP_2 and SOP_3 in 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 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 OP iff it has IP or SOP_1 and that T has TP iff it has SOP_1 or TP_2, analogous to the well-known results in full first-order logic where SOP_1 is replaced by SOP in the former and by TP_1 in the latter. Our proofs of these final two theorems are new and make use of Kim-independence.

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Full text arXiv 2304.07557: pdf, ps.