Can variable-containing statements (propositional functions) contain propositional variables?

Question

My understanding of a propositional function (like P(x)) is that it is a declaration that contains one or more variables, so that when values are substituted in for the variables, a proposition (a statement that is either true or false) is produced. This definition made me wonder whether or not it is valid for a propositional function to contain propositional variables as its variables. For example, can I validly define C(q,r) to be the conjunction "q ^ r" and I(q,r) to be the implication "q -> r" where q and r represent unspecified propositions since when q and r are replaced with propositions, the statements become propositions or is my provided definition/reasoning flawed?

Answer 1

In first order logic, "variables" represent individuals of a domain that you may want to consider, so not "truth values" as in propositional logic. It's not very pedagogical that two very different concepts share the same name, but that's what's happening here.

Therefore if you have variables $x,y$ in first order logic, $x\land y$ doesn't a priori make sense (it might make sense if you have a $2$-ary symbol in your language called $\land$, but then calling it $\land$ is a good way to ensure confusion).

If you want to simulate propositional variables in a first order language, you can add $0$-ary relation symbols: an interpretation of such a symbol will be a subset of $M^0=\{*\}$ (where $M$ is the universe), that is $\emptyset$ or $\{*\}$, so true or false : you can simulate propositional variables this way.