Classical Logic I: First-Order Logic

The word 'logic' in the title of this chapter is ambiguous. In its first meaning, a logic is a collection of closely related artificial languages. There are certain languages called first-order languages, and together they form first-order logic. In the same spirit, there are several closely related languages called modal languages, and together they form modal logic. Likewise second-order logic, deontic logic and so forth. In its second but older meaning, logic is the study of the rules of sound argument. First-order languages can be used as a framework for studying rules of argument; logic done this way is called first-order logic. The contents of many undergraduate logic courses are first-order logic in this second sense. This chapter will be about first-order logic in the first sense: a certain collection of artificial languages. In Hodges (1983), I gave a description of first-order languages that covers the ground of this chapter in more detail. That other chapter was meant to serve as an introduction to first-order logic, and so I started from arguments in English, gradually introducing the various features of first-order logic. This may be the best way in for beginners, but I doubt if it is the best approach for people seriously interested in the philosophy of first-order logic; by going gradually, one blurs the hard lines and softens the contrasts. So, in this chapter, I take the opposite route and go straight to the first-order sentences. Later chapters have more to say about the links with plain English. The chief pioneers in the creation of first-order languages were Boole, Frege and C. S. Peirce in the nineteenth century; but the languages became public knowledge only quite recently, with the textbook of Hilbert and Ackermann (1950), first published in 1928 but based on lectures of Hilbert in 1917–22. (So first-order logic has been around for about 70 years, but Aristotle's syllogisms for well over 2000 years. Will first-order logic survive so long?) From their beginnings, first-order languages have been used for the study of deductive arguments, but not only for this – both Hilbert and Russell used first-order formulas as an aid to definition and conceptual analysis. Today, computer 9