Syllogistic Logic with Cardinality Comparisons, On Infinite Sets

Abstract

This paper 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: {\sf All $x$ are $y$} and {\sf Some $x$ are $y$}, {\sf There are at least as many $x$ as $y$}, and {\sf There are more $x$ than $y$}. Here $x$ and $y$ range over subsets (not elements) of a given \emph{infinite} set. Moreover, $x$ and $y$ may appear complemented (i.e., as $\overset{-}{x}$ and $\overset{-}{y}$), 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.