What is the set of propositional formulas?

Question

What is the set of propositional formulas?

I am not sure if I understand this

Answer 1

From what I stated previously elsewhere:

I assume you are referring to a propositional language $L_V=\left<V,\rightarrow,\bot \right>$, where

• $V$ is a set, called the proposition symbols of $L_V$.

Now let $L_\Sigma^*$ denote the set of all strings of $L_V$.

The usual definition of the set of well-formed formula of $L_V$, call it $PROP_{L_V}$ is then:

Definition (PROP) The set PROP of well-formed formulas of $L_V$ is the set satisfying:

1. $\alpha \in PROP$ if $\alpha \in V$
2. $(\neg\alpha) \in PROP$ if $\alpha$ is a wff
3. $(\alpha \rightarrow \beta) \in PROP$ if $\alpha$ and $\beta$ are wff
4. No string that is not obtained by (1),(2) or (3) is in PROP.

Note that (4) assures that this set is unique. Indeed, there are infinitely many sets of strings over $L_V$ satisfying (1), (2) and (3), but still containing some "garbage" such as

'P(¬', '¬→→', '$Q→(¬$' ...

as elements of it. That is, satisfying the formation rules is a sufficient condition for being in $PROP$, but it does not uniquely define any set (Intuitively, it's kind of similar with the fact that from a sentence such, say, "there is a book on the table", one cannot conclude that neither there is one and only one book nor that there are no more objects on the table.)

In order to exclude those undesirable strings we state (4) for "closing the back door" saying that $PROP$ is the smallest set of strings over the rules (1),(2) and (3), which is actually the intersection of the family of all set of strings over $L_V$ satisfying the formation rules.