An Investigation of the Negationless Fragment of the Rescher-Härtig quantifier

The Rescher quantifier and Härtig quantifier have been the subject of much theoretical inquiry, due to the ability of their respective languages to capture much of second-order logic. This paper deals with a hybrid of the two, which for clarity I have termed the Rescher-Härtig quantifier, the first-order language of which is expressively equivalent to that of the Rescher quantifier but contains language fragments with interesting properties. Specifically, the negationless fragment has the property of satisfiability of all sentences within a single model, yet the question of entailment in the language fragment has equal complexity to the question of entailment in the whole language. This can be demonstrated by embedding problems of entailment from the whole language into the language fragment. By additionally embedding the problem into propositional logic, in which entailment is known to be decidable, and reducing the first problem of entailment to the two embedded problems, we show that entailment in the whole language is no more complex than in the language fragment. The paper concludes with a discussion of the different notions of logical compactness and various open questions.