The singular square: Contrariety and double negation from Aristotle to homer

Abstract

While the traditional Square of Opposition was based on Aristotle’s logic, its first appearance postdates the Stagyrite by five centuries. In the Prior Analytics I, Aristotle himself lays out a somewhat different square, which I dub the Singular Square, to formalize his treatment of the interrelation of singular statements (it’s good, it isn’t good, it’s not-good, it isn’t not-good). Like the more familiar square, the Singular Square is based on the distinction between contradictory and contrary opposition. This paper focuses on the role of the Singular Square as a device for unmasking the conspiracy of MaxContrary, the natural language tendency for a formal contradictory (apparent wide-scope) negation ¬p to strengthen to a contrary of p in a variety of syntactic and lexical contexts. This conspiracy extends from the non-compositional narrow-scope readings of negation interacting with bare plurals, definite plurals, conjunctions, and neg-raising predicates to the prevalence of prohibitives and litotes, the contrary interpretations of affixal negation, and the seemingly illogical behavior of “logical” double negation.