This paper enlarges classical syllogistic logic with assertions having to do with comparisons between the sizes of sets. So in addition to assertions like All x are y and Some x are y, we also have There are at least as many x as y, and There are more x than y. Our work also allows all nouns to be complemented. We thus obtain sentences equivalent to No x are y and At least half of the universe are x. We work on finite models exclusively. We formulate a syllogistic logic for our language. The main result is a soundness/completeness theorem. The logic has a rule of ex falso quodlibet, and reductio ad absurdum is admissible. There are efficient algorithms for proof search and model construction, and the logic has been implemented.
Moss, L. S. (2016). Syllogistic Logic with Cardinality Comparisons. In Outstanding Contributions to Logic (Vol. 8, pp. 391–415). Springer. https://doi.org/10.1007/978-3-319-29300-4_18