2.1 Introduction to logic

Mathematics is largely concerned with statements about mathematical objects (numbers, sets, functions and so on), and with whether such statements are true or false. Thus for example,

• $p\colon$

  ‘‘an even number plus an odd number is an odd number’’

is a true statement, whereas

• $q\colon$

  ‘‘there are natural numbers $a,\,b,\,c$ and $n$ such that $n\geqslant 3$ and $a^{n}+b^{n}=c^{n}$’’

is a false statement. (It is easy to see that $p$ is true. On the other hand, $q$ is the negation of Fermat’s Last Theorem; Wiles and Taylor finally proved Fermat’s Last Theorem in 1997, so $q$ is false; that is, no such solutions exist.)

We shall spend some time examining the structure of statements and, in the next chapter, some general methods to establish their truth or falsity. In order to do so we shall work with a mathematical model (a piece of mathematics intended to simulate something); this will make the structure more apparent.

Before we describe this ‘‘logic model’’, we shall describe another model with a very similar structure, but where the mathematics is very simple. This will make it easier to understand the logic model, which is what really interests us.