# 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 $$j,k (where $$\square \in \{\land, \lor, \rightarrow, \leftrightarrow\})$$

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

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) Oct 19, 2014 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 Oct 19, 2014 at 13:42
• sorry edited, the square was defined before. The brackets are necessary. Oct 19, 2014 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" Oct 19, 2014 at 13:52
• The definition is permissive indeed, one (not the) formation sequence can contain "garbage". Oct 19, 2014 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. Oct 19, 2014 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. Oct 19, 2014 at 15:22
• @MauroALLEGRANZA why is $(\bot \lor p_2)$ not wellformed? (or is $\bot$ not wellformed anyway) Oct 20, 2014 at 3:29