Logics for the Relational Syllogistic

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.

Interactively Illustrating the Context-Sensitivity of Aristotelian Diagrams

“…The traditional classification according to quality is thus exclusively based on whether the statements' predicates are negated. Over the course of history, however, logicians have also become interested in the effects of subject negation [7,8,16,21,22,25,35,37], thereby obtaining the new statements all(¬A, B), some(¬A, B), all(¬A, ¬B), some(¬A, ¬B). The 8-formula fragment F ‡ is defined to contain exactly these 4 new statements, together with the 4 original statements of F † .…”

An Object-Oriented Approach to Counter-Model Constructions in A Fragment of Natural Language

“…A modern completeness theorem was given by Moss [7]. Some complexity results of syllogistic sentences of English, completeness results of some syllogistic logics and algorithms and completeness results of some relational syllogistic logics were given in order of by [9], [7] and [10]. The fragment we consider in this paper is contained by [9] in view of complexity and by [7] in view of completeness.…”

No Syllogisms for the Numerical Syllogistic

“…Since the standard system of syllogisms presented in Aristotle's Prior Analytics can be shown-with a few relatively minor adjustments-to license exactly the valid arguments in the traditional syllogistic [1][2][3][4], it is natural to ask whether a similar situation holds for the numerical syllogistic.…”