Partially Ordered Connectives and Sum11 on Finite Models

In this paper we take up the study of Henkin quantifiers with boolean variables [4], also known as partially ordered connectives [19]. We consider first-order formulae prefixed by partially ordered connectives, denoted D, on finite structures. D is characterized as a fragment of second-order existential logic ${\sum^1_1\heartsuit}$, whose formulae do not allow existential variables as arguments of predicate variables. By means of a game theoretic argument, it is shown that ${\sum^1_1\heartsuit}$ harbors a strict hierarchy induced by the arity of predicate variables, and that it is not closed under complementation. It is further shown that allowing at most one existential variable to appear as an argument of a predicate variable, already yields a logic coinciding with full ∑$^{\rm 1}_{\rm 1}$.