Logics for the Relational Syllogistic

TL;DR

The paper investigates relational syllogistics by defining six fragments, $\mathcal{S}$, $\mathcal{S}^\dagger$, $\mathcal{R}$, $\mathcal{R}^\dagger$, $\mathcal{R}^*$, and $\mathcal{R}^{*\dagger}$, and analyzes the existence of sound and complete syllogistic proof systems for them, revealing distinct roles for reductio ad absurdum. It constructs direct systems for $\mathcal{S}$ and $\mathcal{S}^\dagger$, proves that $\mathcal{R}$ admits a refutation-complete direct system but not a sound-complete one, and shows that $\mathcal{R}^*$ requires indirect methods to achieve completeness, while $\mathcal{R}^\dagger$ and $\mathcal{R}^{*\dagger}$ admit no sound and complete indirect systems. The paper establishes sharp complexity bounds: validity in $\mathcal{S}$, $\mathcal{S}^\dagger$, and $\mathcal{R}$ is $\mathbf{NLogSpace}$-complete; in $\mathcal{R}^*$ it is $\mathbf{co}$-$\mathbf{NPTime}$-complete; and in $\mathcal{R}^\dagger$ or $\mathcal{R}^{*\dagger}$ it is $\mathbf{ExpTime}$-complete. Through these results, reductio ad absurdum is shown to significantly affect both proof-theoretic power and computational complexity, with connections to prior work by McAllester & Givan and Moss. Overall, the work delineates a landscape where limited expressiveness yields tractable yet nuanced relational logics, while more expressive fragments demand stronger proof principles and incur higher complexity.

Abstract

The Aristotelian syllogistic cannot account for the validity of many inferences involving relational facts. In this paper, we investigate the prospects for providing a relational syllogistic. We identify several fragments based on (a) whether negation is permitted on all nouns, including those in the subject of a sentence; and (b) whether the subject noun phrase may contain a relative clause. The logics we present are extensions of the classical syllogistic, and we pay special attention to the question of whether reductio ad absurdum is needed. Thus our main goal is to derive results on the existence (or non-existence) of syllogistic proof systems for relational fragments. We also determine the computational complexity of all our fragments.

Figures (5)

• Figure 1: The six syllogistic fragments: $\mathcal{S}$, $\mathcal{S}^\dagger$, $\mathcal{R}$, $\mathcal{R}^\dagger$, $\mathcal{R}^*$ and $\mathcal{R}^{*\dagger}$.
• Figure 2: Syntax of syllogistic fragments: quick reference guide.
• Figure 3: The structure $\mathfrak{A}^{(n)}$. Every element inside the dotted box is related by $r$ to $u_2$.
• Figure 4: The structure $\mathfrak{B}_i$. Every element inside either of the dotted boxes is related by $r$ to $u_2$.
• Figure 5: The six fragments studied in this paper together with the two-variable fragment $\hbox{FO}^2$ of first-order logic. The table shows strongest possible results on the existence of syllogistic systems, together with tight complexity bounds. See Section \ref{['sec:conclusions']} for an explanation.

Theorems & Definitions (72)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof