Syllogistic Logic with Cardinality comparisons, on Infinite Sets

This article enlarges classical syllogistic logic with assertions having to do with comparisons between the sizes of sets. So it concerns a logical system whose sentences are of the following forms: All x are y and Some x are y , There are at least as many x as y , and There are more x than y . Here x and y range over subsets (not elements) of a given infinite set. Moreover, x and y may appear complemented (i.e., as $\bar{x}$ and $\bar{y}$ ), with the natural meaning. We formulate a logic for our language that is based on the classical syllogistic. The main result is a soundness/completeness theorem. There are efficient algorithms for proof search and model construction.