Let $X$ be a set. For example, $X$ may be a set of strings consisting only of specific characters. What we explain here is the phrasing peculiar to mathematics used when recursively defining subsets of $X$ and the like.

For example, using an element $c$ of $X$, non-negative integers $n_0,m_0,n_1,m_1$, and maps $F_0 \colon X^{n_0} \times X^{m_0} \to X$ and $F_1 \colon X^{n_1} \times X^{m_1} \to X$, one may write as follows:

We recursively define a subset $X_0$ of $X$ as follows:

• $c \in X_0$.
• For any $(a,t) \in X_0^{n_0} \times X^{m_0}$, we have $F_0(a,t) \in X_0$.
• For any $(a,t) \in X_0^{n_1} \times X^{m_1}$, we have $F_1(a,t) \in X_0$.

This is only an example; here we used $3$ conditions, but in general there need not be $3$, and relations may be used instead of maps. The general formulation of recursive definitions^{[1] 6.1.1 Definition} is quite abstract, so here we consider only the examples needed to solve this problem.

What is important here is that this definition does not simply mean that $X_0$ is an arbitrary subset of $X$ satisfying all $3$ conditions, but represents the following definition:^{[1] see 6.1.2 Definition and the paragraph immediately following it}

We define the subset $X_0$ of $X$ as the intersection of all subsets $Y$ of $X$ satisfying all of the following $3$ conditions:

• $c \in Y$.
• For any $(a,t) \in Y^{n_0} \times X^{m_0}$, we have $F_0(a,t) \in Y$.
• For any $(a,t) \in Y^{n_1} \times X^{m_1}$, we have $F_1(a,t) \in Y$.

Taking $Y = X$ indeed satisfies the above $3$ conditions, so the set of all $Y$ satisfying the conditions is non-empty and the intersection is defined. It then follows as a result that $X_0$ is the smallest among the $Y$ satisfying the $3$ conditions.

Note that, in this way, a recursive definition of a subset is by convention understood to imply minimality as well, rather than being a mere list of conditions. However, this convention applies only to recursive definitions; an ordinary definition of course does not imply minimality, and in the first place, making a self-reference like that of a recursive definition in an ordinary definition often results in circular reasoning, and the definition breaks down.

Now, since a $1$-ary relation on $X$ is equivalent to a subset of $X$, the same rule applies to recursive definitions of $1$-ary relations on $X$. Specifically,

We recursively define what it means for an element of $X$ to be a good term as follows:

• $c$ is a good term.
• For any $(a,t) \in X^{n_0} \times X^{m_0}$, if every component of $a$ is a good term, then $F_0(a,t)$ is also a good term.
• For any $(a,t) \in X^{n_1} \times X^{m_1}$, if every component of $a$ is a good term, then $F_1(a,t)$ is also a good term.

when a recursive definition of the $1$-ary relation "is a good term" is written like this, it does not mean that the predicate "is a good term" represents an arbitrary $1$-ary relation satisfying the $3$ conditions. The above definition is equivalent to the following definition:

We define an element $x$ of $X$ to be a good term if $x \in Y$ holds for every subset $Y$ of $X$ satisfying all of the following $3$ conditions:

• $c \in Y$.
• For any $(a,t) \in Y^{n_0} \times X^{m_0}$, we have $F_0(a,t) \in Y$.
• For any $(a,t) \in Y^{n_1} \times X^{m_1}$, we have $F_1(a,t) \in Y$.

Based on the above, the recursive definition of good propositions in this problem is equivalent to the following definition:

We define a string $s$ consisting only of propositional variables, parentheses $(, )$, and $\land, \lor, \Rightarrow$ to be a good proposition if, for every set $Y$ consisting only of strings consisting only of propositional variables, parentheses $(, )$, and $\land, \lor, \Rightarrow$, it holds that $s \in Y$ whenever $Y$ satisfies all of the following $3$ conditions:

• Every propositional variable is an element of $Y$.
• For any elements $a$ and $b$ of $Y$, the string $(a \land b)$ obtained by concatenating the strings $($, $a$, $\land$, $b$, and $)$ in this order is an element of $Y$.
• For any elements $a$ and $b$ of $Y$, the string $(a \lor b)$ obtained by concatenating the strings $($, $a$, $\lor$, $b$, and $)$ in this order is an element of $Y$.
• For any elements $a$ and $b$ of $Y$, the string $(a \Rightarrow b)$ obtained by concatenating the strings $($, $a$, $\Rightarrow$, $b$, and $)$ in this order is an element of $Y$.

This is considerably more complicated than the original phrasing. To avoid such complicated phrasing, mathematics defines and adopts the concise phrasing introduced here.

For example, in the references listed later, the following

• the recursive definitions of the $1$-ary relations "be an $\mathcal{L}$-term" and "be an $\mathcal{L}$-formula" in [1], etc.
• the recursive definitions of the set $T$ and the $1$-ary relation "be a principal term" in [2], etc.
• the recursive definitions of the sets $B(\alpha,\beta)$ and $\textrm{K}_{\kappa}(\gamma)$ in [3], etc.

actually adopt such phrasing.