Ranks in positive logic (KIRBYJ2_U26EMP)

Model theory is traditionally done with “classical first-order logic”, the logic which allows unlimited use of the operators AND, OR, NOT, with the EXISTS and FOR ALL quantifiers. More recently, positive logic has emerged as a useful generalisation. For a theory in positive logic, one can specific how much you are allowed to use the NOT operator and the FOR ALL quantifier: either without restriction (to get the classical case) or much less.

Positive logic is more appropriate to use directly for some applications in algebra, such as for modules, and for theories which do not admit quantifier elimination. Theories in classical logic can be classified according to their combinatorial complexity, via a number of dividing lines which are mostly due to Shelah. A map of this classification appears at http://forkinganddividing.com.

Many of these dividing lines are related to the existence of a suitable rank notion on formulas such as Morley rank, U-rank, dp-rank. These are ordinal-valued ranks which explain the complexity of a formula, not as a piece of syntax but in terms of how it relates to other formulas within the context of the particular theory. For example, in vector spaces, any formula has an associated vector subspace and all these ranks will give the dimension of that subspace.

These ranks are then often the most useful model-theoretic tool to apply to the theories. We know how to extend some of these dividing lines to positive logic, but so far ranks have not been studied