Abstract

The relational syllogistic extends the classical syllogistic by allowing predicate phrases of the forms “rs every q”, “rs some q” and their negations, where q is a common noun and r a transitive verb. It is known that both the classical and relational syllogistic admit a finite set of syllogism-like rules whose associated derivation relation is sound and complete. In this article, we extend the classical and relational syllogistic by allowing ‘guarded’ predicate phrases of the form “rs onlyqs”, and their negations. We show that, in both cases, the resulting logic is pspace-complete. It follows, on the assumption that NPTIME≠PSPACE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textsc {nptime}\ne \textsc {pspace}$$\end{document}, that neither extension admits a finite set of syllogism-like rules whose associated derivation relation is sound and complete, even when reductio ad absurdum is allowed. We also show that further extending these systems with noun-complementation in sentence-subjects results in logics which are exptime-complete.