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What exactly is a parameter in the sense of first-order logic?

What "choices" are there for defining them formally?

My question is at least somewhat similar to this question, but is about how parameters work in general.

I find parameters sort of confusing since they seem to sit on the boundary between syntax and semantics. Parameters are used in well-formed formulas, but are drawn from the underlying model.

I think of a parameter as a new constant symbol in a new language created by attaching self-interpreting constant symbols to an existing language.

So, for example, if $M$ is an $L$-structure and $A$ is a subset of $M$, then I think of $L(A)$ as a new language with a bunch of new constant symbols that are sent to themselves by the interpretation function.

It seems like it also should be possible, at least in principle, to make a parameter a completely different type of entity from a constant or variable symbol and extend the definition of a term so that it can be a constant, a variable, a compound term headed by a function $f(t_1, t_2, \cdots, t_n)$ or a parameter.

I'm not sure what approach is most common or most natural for defining what a parameter is.

Greg Nisbet
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    Where would you use this nebulous notion of a "parameter" where variables, constant symbols, and terms don't work? – Mark Saving Sep 16 '21 at 00:02
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    There's a highly similar titled Post from one year ago in this site here, hope it's helpful. – cinch Sep 16 '21 at 00:38
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    @MarkSaving, so my question is largely about what the standard notion is. I think $L(A)$ is normally thought of as $L$ with some additional constant symbols that are interpreted as themselves in the underlying model. The "alternative" with "parameters as a weird thing that's another kind of term" is a straw man to some degree; I only intended to show that some choice appears to be possible when you want to sit down and define what a parameter is. – Greg Nisbet Sep 16 '21 at 00:42
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    No standard notion... In some treatment there are two sorts of "variables": those used only as bound: $x,y,z,...$ and those used only as free ones: $a,b,c,...$ sometimes called parameters. See Smullyan, First Order Logic. – Mauro ALLEGRANZA Sep 16 '21 at 07:10
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    @MarkSaving One instance where "parameter" is used is when we say that something is definable with parameters on some model. But I don't think it's a good idea to try to define a mathematical object called a "parameter". – Dabouliplop Sep 16 '21 at 11:15

2 Answers2

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A parameter is simply an element of a structure.

Usually, we use the word "parameter" in a context where we are intepreting some (but not necessarily all) of the free variables in a formula. For example, let $\varphi(x,y)$ be a formula with free variables $x$ and $y$ (or let $x$ and $y$ be finite tuples of free variables). Let $M$ be a structure, and $b\in M$ an element (or a tuple of the same length as $y$). Then $\varphi(x,b)$ is a formula with parameter $b$. Note that $\varphi(x,b)$ is no longer a purely syntactic object. But this is no problem: formally, we can view it as the formula $\varphi(x,y)$ together with a partial interpretation of the free variables by elements of $M$ (namely, $y$ is interpreted as $b$, but $x$ is left uninterpreted).

What do we do with formulas with parameters? We ask whether elements/tuples from $M$ satisfy them. And this question makes sense: Let $a\in M$ be an element (or a tuple of the same length as $x$). Then we say $a$ satisfies $\varphi(x,b)$ if and only if $M\models \varphi(a,b)$, i.e., if and only if $(a,b)$ satisfies $\varphi(x,y)$. Formally, we interpret the remaining free variables $x$ as $a$, and we apply the definition of satisfaction in $M$. The set defined by $\varphi(x,b)$ is $\varphi(M,b) = \{a\in M\mid M\models \varphi(a,b)\}$.

Now there are situations in which it can be conceptually or technically useful to view a formula with parameters as a purely syntactic object. For example, usually the compactness theorem is proven for theories (sets of sentences), and then we want to apply it to realize types (sets of formulas with parameters). In this sort of situation, we can (1) introduce a new constant symbol $c_b$ (or a tuple of constant symbols of the same length as $y$) to get a language $L(b)$, (2) expand $M$ to an $L(b)$-structure $M'$ by interpreting the new constant as $b$, and (3) substitute $c_b$ for $y$ in $\varphi(x,y)$, to get a new formula $\varphi(x,c_b)$, which is an $L(b)$-formula without parameters.

Some people might prefer to always do this, and to define a parameter to be a constant symbol which is interpreted as a designated element in some structure. It doesn't matter so much: these two points of view are essentially interchangeable, since for all $a\in M$, $$M\models \varphi(a,b)\iff M'\models \varphi(a,c_b).$$ I just find it easier to think directly about elements of structures, rather than changing the language all the time.

Alex Kruckman
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Just a footnote to Alex’s fine answer. The use of “parameter” in logic is analogous to the usage in other parts of (pure and applied) mathematics. For example, consider the formula for exponential growth, $Ae^{kt}$. You'd probably say this is function of $t$ with parameters $A$ and $k$, rather than just calling it a function of three variables. Also you think of $A$ and $k$ as being "fixed", while $t$ varies.

Likewise, logicians use "parameter" when one variable is thought of as the main protagonist in a formula, and the others play supporting roles. Or (a slight variant) when we give the other variables "fixed values" by substituting elements of a structure (as in Alex's answer). It’s not a formal term, but helps give insight.

Michael Weiss
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