Quantifiers and Predicates Chapter
Introduction: Ordinary language is frequently imprecise and ambiguous. These
traits, while adequate for relaxed and informal discussion, and even for poets and politicians, are totally inadequate for conveying mathematical ideas from person to person. In some instances it is perfectly appropriate to intentionally mislead, be vague, or to want many possible interpretations. For example, in the case of a best-selling murder mystery or a diplomatic response to a question involving sensitive international issues. But mathematicians, scientists, and others require precision and clarity. There is a need to be able to rely on mathematical results, not only in mathematics, but in many other disciplines. A principal means for attaining the objective of having reliable results is through the process of deductive or inferential reasoning. The Derivation method, developed in previous chapters, as well as the algebraic manipulation of statements, also discussed previously, provide efficient ways to tell whether certain arguments are valid. In addition, a list of valid arguments is given in Table 2.5. Those rules for deductive reasoning have long been used to assure that certain results in mathematics cannot get called into question. Those rules will be expanded in this chapter to allow us to deal with a greater variety of arguments.