Why are variables not considered part of the lexicon of LR($\sigma$)?

Question

I am reading through Mathematical Logic by Chiswell and Hodges and just finished section 5.3, which introduces LR, the Language of Relations, for first-order logic. In this section the following notions are defined:

1. A first-order signature
2. Variables
3. A parsing tree for terms of LR($\sigma$)
4. A parsing tree for formulas of LR($\sigma$)
5. Compositional definitions for each of the above trees mentioned.

My question is on exercise 5.3.1, which asks:

Suppose $\sigma$ is a signature. What are the symbols in the lexicon of LR($\sigma$)?

The answer in the back of the book states that the lexicon consists of the symbols in $\sigma$ together with the twelve symbols '$\lnot$' '$\land$' '$\lor$' '$\to$' '$\leftrightarrow$' '$\bot$' '$=$' '$\forall$' '$\exists$' '$($' '$)$' '$,$'. This is what I had gotten as my answer too, based on the details of the above notions, except I also inlcuded the variables (which in section 5.3 are defined as the infinitely many symbols $x_0$, $x_1$, $x_2$ $...$).

So, why are variables not considered part of the lexicon for LR($\sigma$)?

Furthermore, there doesn't seem to be an explicit definition of a lexicon in this chapter (like there was for the earlier chapter on propositional logic, in which the authors stated explicitly what symbols were in the lexicon for LP($\rho$) for some propositional signature $\rho$). However, it seems clear enough that the lexicon is just the set of symbols we want to use to make strings of our language. But based on that loose definition, again, it would seem that the variables should be included in the lexicon. For example, the authors give $\exists x_0 (F(F(x_0)) = x_0)$ as a formula of LR, so wouldn't we consider the variable $x_0$ to be part of the lexicon?

Answer 1

Remarkably, this is the only textbook I am aware of that treats propositional calculus in a formal language setting; probably because propositional calculus is often the very initial subject with which teaching of logic begins. Let us review the relevant definition from that part:

Definition 3.1.1 For each signature $\sigma$:

(a) The lexicon of $\mathrm{LP}(\sigma)$ is the set of symbols consisting of the truth function symbols (3.1), the parentheses (3.2) and the symbols in $\sigma$.

(b) An expression of $\mathrm{LP}(\sigma)$ is a string of one or more symbols from the lexicon of $\mathrm{LP}(\sigma)$. The length of the expression is the number of occurrences of symbols in it. (Often the same symbol will occur more than once.)

Lexicon (or, “vocabulary”) of a formal language $\mathcal{L}$ is composed of two segments: a logical lexicon and a non-logical lexicon. Non-logical lexicon is also called signature, for it is the distinctive part of the language. Indeed, it is common among mathematicians to leave logical vocabulary out of the definition of language altogether, since a particular system of logic (first-order predicate logic in many cases) is assumed.

$Lxn_{LP} =\{\underbrace{\wedge,\vee,\rightarrow,\leftrightarrow,\neg,\bot, (, )}_{logical\;lexicon},\underbrace{p_{0}, p_{1}, p_{2},\ldots}_{non-logical\; lexicon}\}$

Definition 5.3.1 A first-order signature (shortened to signature when there is no danger of confusion with propositional signatures) is a 4-tuple $(Co,Fu, Re, r)$ where:

(1) $Co$ is a set (possibly empty) of symbols called the constant symbols;

(2) $Fu$ is a set (possibly empty) of symbols called the function symbols;

(3) $Re$ is a set (possibly empty) of symbols called the relation symbols;

(4) $Co, Fu, Re$ are pairwise disjoint;

(5) $r$ is a function taking each symbol $s$ in $Fu\cup Re$ to a positive integer $r(s)$ called the rank (or arity) of $s$. We say a symbol is $n$-ary if it has arity $n$; binary means $2$-ary.

Let us denote the elements of $Co, Fu, Re$ by the sequences of symbols, $c_{0}, c_{1}, c_{2}, \ldots$, $f_{0}, f_{1}, f_{2}, \ldots$) and $R_{0}, R_{1}, R_{2}, \ldots$, respectively.

We shall combine the preceding considerations with the statement on p. 104 (bold font added):

The language $\mathrm{LR}(\sigma)$ will have symbols to stand as variables. It will also have a symbol $\forall$ to stand for the expression ‘for every’, and a symbol $\exists$ for the expression ‘there is’.

$Lxn_{LR} =\{\underbrace{\wedge,\vee,\rightarrow,\leftrightarrow,\neg,\bot, =,\forall,\exists, x_{0}, x_{1}, x_{2},\ldots, (, ),`\text{,'} }_{logical\;lexicon},\underbrace{c_{0}, c_{1}, c_{2}, \ldots, f_{0}, f_{1}, f_{2}, \ldots,R_{0}, R_{1}, R_{2}, \ldots}_{non-logical\; lexicon}\}$

as stated in the answer:

5.3.1. The lexicon of $\mathrm{LR}(\sigma)$ consists of the symbols in $\sigma$ together with the twelve symbols ‘$\neg$’, ‘$\wedge$’, ‘$\vee$’, ‘$\rightarrow$’, ‘$\leftarrow$’, ‘$\bot$’, ‘$=$’, ‘$\forall$’, ‘$\exists$’, ‘$($’, ‘$)$’ and ‘,’.

However, a question remains: Why do the “variables” shift from the non-logical lexicon of propositional calculus to the logical lexicon of predicate logic?

Notice that the authors talk about “propositional symbols,” never “propositional variables.” They are not actually variables, but the names of propositions. For example $p_{0}$ designates ‘snow is white’, $p_{1}$ designates ‘grass is green’, $p_{2}$ designates ‘$i^{2}=-1$’ and so on. So, they can be immediately evaluated:

Definition 3.5.3 Let $\sigma$ be a signature. By a $\sigma$-structure we mean a function $A$ with domain $\sigma$, that assigns to each symbol $p$ in $\sigma$ a truth value $A(p)$.

Compare to the case of predicate logic: The individual variables of a formula are first assigned to semantic relata in the universe, thereafter the formula is evaluated.