# Propositional Logic meta-variable notation abuse

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?

• not sure what you mean (where does the box stand for) also does (iii) really have brackets? (and there is no k in (iii) – Willemien Oct 19 '14 at 13:38
• to me it looks a bit like the list on how to decide if a formula is wellformed, only now upside down, so starting with atomic sentences make all possible formulas hope this helps – Willemien Oct 19 '14 at 13:42
• sorry edited, the square was defined before. The brackets are necessary. – villasv Oct 19 '14 at 13:42
• It does read rather chaotic if you follow it exactly as given $\varphi \to \varphi$ can be part of the formation sequence of $\varphi \lor \varphi$ maybe the author means something like "don't create subformulas of $\varphi_n$ this way, it is only a theoretical way to create all possible formulas of the language" – Willemien Oct 19 '14 at 13:52
• The definition is permissive indeed, one (not the) formation sequence can contain "garbage". – villasv Oct 19 '14 at 13:55

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.
• So the abuse is using $\varphi$ for expression variable instead of proposition variable? wow, this is what I call a mild abuse. – villasv Oct 19 '14 at 15:16
• @Victor - yes; he "abuse" of $\varphi$, that has been introduced to refer to propositions, using it to refer also to strings of symbols which are not propositions. – Mauro ALLEGRANZA Oct 19 '14 at 15:22
• @MauroALLEGRANZA why is $(\bot \lor p_2)$ not wellformed? (or is $\bot$ not wellformed anyway) – Willemien Oct 20 '14 at 3:29