Propositional Logic meta-variable notation abuse

Question

When defining Formation Sequence, van Dalen (4th edition page 9) says:

A sequence $(\varphi_0,\varphi_1,...,\varphi_n)$ is called a formation sequence of $\varphi$ if $\varphi_n=\varphi$ and:

(i)$ \varphi_i $ is atomic;

(ii)$\varphi_i = (\varphi_j \square \varphi_k) $ with $j,k<i \quad$ (where $\square \in \{\land, \lor, \rightarrow, \leftrightarrow\})$

(iii)$\varphi_i = (\neg \varphi_j) $ with $j<i$

But he then says:

Observe that in this definition we are considering strings $\varphi$ of symbols from the given alphabet; this mildly abuses our notational convention

But the notation convention was to use $\varphi$ as meta-variable for propositions, and this is what my eyes see here.

Am I supposed to make a distinction between propositions and strings of propositional logic alphabet symbols? Aren´t propositions a subset of propositional alphabet strings?

Answer 1

The explanation is in the paragraph following Definition 1.1.4 of formation sequence.

Examples. (a) $⊥, p_2, p_3, (⊥ \lor p_2), (¬(⊥ \lor p_2)), (¬p_3)$ and $p_3, (¬p_3)$ are both formation sequences of $(¬p_3)$. Note that formation sequences may contain ‘garbage’ [emphasis added].

As you noted [page 8] : "$\varphi$ and $\psi$ are used as variables for propositions".

In the example above of the formation sequence, the meta-variables $\varphi_i$ stand for strings of symbols.