What are "parameters" in a formula of first-order logic?

Question

I've seen the term "parameter" being used in logic literature. For example, $p_1,\dots ,p_n$ are parameters in a formula $\varphi(x,p_1,\dots,p_n)$ (this is notation for parameters). I suspect that "parameters" are just free variables, taking into account that we denote that a variable $x$ is free in a formula $\varphi$ by writing $\varphi(x)$. But I'm not sure about that and I found no definite answer on the internet thus far. So, please, tell me the precise meaning of the term "parameter" in logic.

Answer 1

Parameters are more than just free variables; in short, it's a way of incorporating elements of (a subset of) an $\mathscr L$-structure in the study of the $\mathscr L$-structure itself. To illustrate this point, let $\mathscr L = \{+, \cdot, -, 0, 1\}$ be the language of rings and consider $\mathbb R$ as an $\mathscr L$-structure. One way of talking about the element $\sqrt \pi$ in $(\mathbb R; +, \cdot, -; 0,1)$ is to define it as the only element in $\mathbb R$ which satisfies the formula (with parameter $\pi$) $$\phi(x, \pi) := (x^2 = \pi) \wedge \exists u (u^2 =x).$$ Note that as written above, $\phi(x, \pi)$ is not an $\mathscr L$-formula as $\pi$ does not belong to our language; adding $\pi$ as a parameter enables us to define in a first order way the element $\sqrt \pi$ in $\mathbb R$.

The way we do this formally is by "adding a constant to our language". Taking our example above, we consider a new language $\mathscr L' = \mathscr L \cup \{\hat{\pi}\}$ where $\hat{\pi}$ is a new constant symbol, and we expand our $\mathscr L$-structure $(\mathbb R; +, \cdot, -; 0,1)$ to an $\mathscr L'$-structure $\mathscr R$ with universe $\mathbb R$ and which interprets all function symbols and constant symbols which belong to $\mathscr L$ just as $(\mathbb R; +, \cdot, -; 0,1)$, and furthermore it interprets the new constant symbol $\hat{\pi}$ as the element $\pi \in \mathbb R$; the result is that by "naming" such element syntactically (i.e. via a new constant symbol) we can include $\pi$ in the analysis of our original $\mathscr L$-structure.

In general, instead of adding a single constant symbol one adds a new set of constant symbols, each naming an element of a "parameter set" $A \subseteq M$ from an $\mathscr L$-structure $\mathscr M$.