The present paper is part of a large research programme investigating the nature and properties of the predicate logic inherent in natural language. The general hypothesis is that natural speakers start off with a basic-natural logic, based on natural cognitive functions, including the basic-natural way of dealing with plural objects. As culture spreads, functional pressure leads to greater generalization and mathematical correctness, yielding ever more refined systems until the apogee of standard modern predicate logic. Four systems of predicate calculus are considered: Basic-Natural Predicate Calculus (BNPC), Aritsotelian-Abelardian Predicate Calculus (AAPC), Aritsotelian-Boethian Predicate Calculus (ABPC), also known as the classic Square of Opposition, and Standard Modern Predicate Calculus (SMPC). (ABPC is logically faulty owing to its Undue Existential Import (UEI), but that fault is repaired by the addition of a presuppositional component to the logic.) All four systems are checked against seven natural logical intuitions. It appears that BNPC scores best (five out of seven), followed by ABPC (three out of seven). AAPC and SMPC finish ex aequo with two out of seven.