Reasoning About the Sizes of Sets: Progress, Problems, and Prospects

We discuss what is known about logical reasoning concerning the sizes of sets, including expressions like there are at least as many x as y, there are more x than y, and most x are y. It turns out that reasoning with expressions like this can be done efficiently, that formal proofs can be obtained which do not employ translation to standard logic, and that counter-models can also be generated. The paper also contains a new result, a completeness theorem for syllogistic reasoning involving the sentences in our fragment, and adding sentential negation. So we are not done with the project of getting a complete logics for reasoning about sizes of sets. At the same time, there are some open questions. We show one implementation. We mention briefly some very new work which allows one to do logical reasoning on sentences as they come (not from a toy grammar), but at some cost. Finally, we discuss connections to cognitive science, many of which are waiting to be made.