The Syllogistic with Unity

TL;DR

This paper extends classical syllogistic logic with new quantifier-like sentence forms and proves that the resulting logic cannot be finitely axiomatized with sound and complete syllogism-like rules, even with reductio ad absurdum.

Contribution

It introduces new sentence-forms to classical syllogistic logic and demonstrates the impossibility of finite sound and complete rule sets for this extended logic.

Findings

The extended logic includes 'At most 1 p is a q' and 'More than 1 p is a q' forms.

No finite set of syllogism-like rules can be both sound and complete for this logic.

Reductio ad absurdum does not enable finite axiomatization of the extended logic.

Abstract

We extend the language of the classical syllogisms with the sentence-forms "At most 1 p is a q" and "More than 1 p is a q". We show that the resulting logic does not admit a finite set of syllogism-like rules whose associated derivation relation is sound and complete, even when reductio ad absurdum is allowed.