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The Syllogistic with Unity

Ian Pratt-Hartmann

TL;DR

The paper analyzes the syllogistic extended with unity, showing that bounding counting quantifiers to 1 yields proof-theoretic and complexity-theoretic properties akin to the full numerical syllogistic: no finite sound-and-complete syllogistic system exists even with reductio, while satisfiability remains NP-time complete. It develops a formal framework for languages $\mathcal{S}_z$ and $\mathcal{S}^\dagger_z$, defines direct and indirect syllogistic derivations, and constructs a counterexample family $\Gamma^n$ to prove noncompleteness. The core result generalizes to all $z>0$, indicating a fundamental limitation of finite syllogistic-style calculi for these counting logics. These results illuminate the boundary between classical syllogistic reasoning and counting-based logics in terms of proof theory and complexity.

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.

The Syllogistic with Unity

TL;DR

The paper analyzes the syllogistic extended with unity, showing that bounding counting quantifiers to 1 yields proof-theoretic and complexity-theoretic properties akin to the full numerical syllogistic: no finite sound-and-complete syllogistic system exists even with reductio, while satisfiability remains NP-time complete. It develops a formal framework for languages and , defines direct and indirect syllogistic derivations, and constructs a counterexample family to prove noncompleteness. The core result generalizes to all , indicating a fundamental limitation of finite syllogistic-style calculi for these counting logics. These results illuminate the boundary between classical syllogistic reasoning and counting-based logics in terms of proof theory and complexity.

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.

Paper Structure

This paper contains 4 sections, 5 theorems, 28 equations, 1 table.

Table of Contents

  1. Introduction
  2. Syntax and semantics
  3. Syllogistic proof systems
  4. No sound and complete syllogistic systems for $\mathcal{S}_z$ or $\mathcal{S}^\dagger_z$

Key Result

Theorem 2.1

For all $z > 0$, the problem of determining the satisfiability of a given set of $\mathcal{S}_z$-formulas is NPTime-complete, and similarly for $\mathcal{S}^\dagger_z$-formulas.

Theorems & Definitions (20)

  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 4.1
  • proof
  • Claim 4.2
  • proof
  • ...and 10 more