First Order Logic

The last chapter presented the general concept of a logic. In this chapter we consider a particular logic | rst order predicate calculus, also known as simply rst order logic. First order logic is by far the most widely studied logic. It was rst formulated by Guttlob Frege in 1879. However, the fundamental semantic properties of rst order logic were not really understood until the work of Skolem, Herbrand, and Godel in the late 1920s and early 1930s Davis, 1983]. The semantic properties of rst order logic contain an apparent paradox. On the one hand rst order logic can be shown to be expressively weak | there are a great many common mathematical concepts, such as the concept of transitive closure, or of a nite set, that can not expressed in rst order logic. On the other hand rst order logic is often viewed as a universal language which can be used to represent arbitrary mathematical statements. Both of these statements are in some sense true. To avoid, or at least ameliorate, the confusion brought on by this apparent paradox a few words can be said about how both statements can be true simultaneously. Although there are many concepts not expressible in rst order logic, rst order logic can be viewed as an adequate (Turing universal) programming language. Like rst order logic, programming lanuages are expressively week. For example, no programming language can express the concept of halting, i.e., one can not write a predicate Halts which takes another program as input determines whether the input program always halts. So expressively weak languages like rst order logic can still be adequate programming languages. Mathematicians have used rst order logic as a programming language in which to encode all the known acceptable principles of mathematical inference. The result is axiomatic set theory. Any mathematical proof can, in principle, be expressed as a proof in rst order set theory. In this sense axiomatic set theory is an adequate foundation for mathematics. However,