# Syllogistic Logic with Cardinality Comparisons, On Infinite Sets

## TL;DR

This paper extends classical syllogistic logic to include set size comparisons on infinite sets, providing a formal logical system with soundness, completeness, and efficient proof algorithms.

## Contribution

It introduces a novel logical framework combining syllogistic reasoning with cardinality comparisons for infinite sets, along with soundness, completeness, and algorithmic proof methods.

## Key findings

- Established a soundness and completeness theorem.
- Developed efficient algorithms for proof search.
- Formalized a logic for set size 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.