Facilitating the precise interpretation of statements first made in natural language — this has been the role of mathematical notations like predicate calculus and lambda calculus. Their use has been for the clarification of argument. They were faulted for inadequate precision or the introduction of paradox, never for forcing too much precision. Formal languages are precise, but selective in what can be easily expressed. It has always been understood that quantifiers like ∀ and ∃ only modeled some aspects of a system of natural language quantifiers. However, they captured the most useful properties very neatly.