Difference between inductive definition and recursive definition as used in "A concise introduction to mathematical logic" by W. Rautemberg. I'm trying to make sense of the definition of propositional formulas in "A concise introduction to mathematical logic" by W. Rautenberg. This is the definition given by Rautenberg (page 4):

"... we define a propositional language $F$ of formulas built up from the symbols $($, $)$, $\land$, $\lor$, $\lnot$, $p_1$, $p_2$, $\dots$    inductively as follows:

($F_1$) The atomic strings $p_1$, $p_2$, $\ldots$ are formulas, called prime formulas, also called atomic formulas, or simply prime.

($F_2$) If the strings $\alpha$, $\beta$ are formulas, then so too are the strings $(\alpha \land \beta)$, $(\alpha \lor \beta)$, and $\lnot \alpha$.

This is a recursive (somewhat sloppily also called inductive) definition
in the set of strings on the alphabet of the mentioned symbols..."

My question is, if he defines formulas inductively, why he then says that this is a recursive definition somewhat sloppily called inductive? Is that an inductive or recursive definition? What is the difference between defining inductively and a recursive definition?

Thanks,
At least some people feel that strictly speaking it's proofs that are inductive, while a definition may be recursive. If so the question is not the difference between an inductive definition and a recursive definition, the point is there's simply no such thing as an "inductive definition", strictly speaking.
I think the answer to your question is in the author's "somewhat sloppily called".

This definition is recursive because it depends on knowing about multiple predecessors. The set of formulas can be represented as a directed graph with sources the atomic formulas.

Strictly speaking, an inductive definition would depend on a single previously defined formula. The set of formulas would then be a chain starting with an atomic formula, or perhaps a union of disjoint chains.

Speaking sloppily, the author conflates the two ideas, since he probably never needs to distinguish and his readers may prefer "inductive".