Book I: Chapter 1

3. The three layers of the mathematical language

1. Mathematical language has three "layers":
• The first layer consists of variables, usually they are Latin letters in italic typeset ($a,b,..., A, B, ...$), which are supposed to represent objects that we want to talk about: these can be numbers, statements, or any other predefined class of objects;
• The second layer consists of expressions (statements / claims) involving these objects, which are called formulas, and which are usually represented by Greek letters $\alpha,\beta,\gamma,...$; formulas describe a meaningful connection between objects;
• And the third layer consists of the so-called connectives $$\wedge, \vee, \neg, \Rightarrow$$ and the quantifiers $$\forall, \exists,$$ each of which has a specific meaning and use (that will be explained as we proceed), that allow to combine the formulas to form new formulas.
2. In the ordinary language, we can talk about anything we like, and we may not care to specify in the beginning of conversation exactly what are we going to be talking about. In mathematics, we do not have such freedom: it is important to assume an initial agreement as to which objects can the Latin letters represent.
3. As an illustration, suppose we agree that we want to talk about cats. Then, upon this agreement, Latin letters will always represent cats. This describes the first layer of the language in which we will talk about cats.
4. Now, the second layer: let us list some examples of formulas, i.e. expressions about cats:
$x$ has yellow fur.
$x$ is a kitten.
$x$ enjoys drinking milk in the company of $y$.
$x$, $y$ and $z$ always fight with each other.
5. Suppose $\alpha$ represents the first formula above. To indicate the fact that there is a first-layer variable $x$ used in the formula $\alpha$, we can write $\alpha(x)$ for $\alpha$.
6. Then, putting $\alpha(y)$ would mean substituting $y$ in the place of $x$ in the formula $\alpha$, so that $\alpha(y)$ would represent the following formula:
$y$ has yellow fur.
7. When there are two variables in a formula, like in the third expression about cats above, we write $\beta(x,y)$.
8. Then, $\beta(x,x)$ would mean substituting $x$ in the place of $y$, so that the formula becomes
$x$ enjoys drinking milk in the company of $x$
which is quite different from the original formula, as now it only talks about one cat $x$, and previously it talked about possibly two different cats $x,y$.
9. Indeed, while $\beta(x,y)$ is very rarely true, $\beta(x,x)$ is probably always true. The rest of Book I describes the third layer of the mathematical language.