Let $\Sigma$ be a signature. The first order language $\operatorname{FO}(\Sigma)$ on $\Sigma$ contains the following:

1. the set $S(\Sigma)$ of symbols of $\operatorname{FO}(\Sigma)$ , which is the disjoint union of the following sets:
   1. $\Sigma$ (the non-logical symbols),
   2. a countably infinite set $V$ of variables,
   3. the set of logical symbols $\lbrace \And, \Or, \neg, \Implies, \Iff, \forall, \exists \rbrace$ ,
   4. the singleton consisting of the equality symbol $\lbrace =\rbrace$ , and
   5. the set of parentheses (left and right) $\lbrace (, )\rbrace$ ;
2. the set $T(\Sigma)$ of terms of $\operatorname{FO}(\Sigma)$ , which is built inductively from $S(\Sigma)$ , as follows:
   1. Any variable $v\in V$ is a term;
   2. Any constant symbol in $\Sigma$ is a term;
   3. If $f$ is an $n$ -ary function symbol in $\Sigma$ , and $t_1,...,t_n$ are terms, then $f(t_1,...,t_n)$ is a term.
3. the set $F(\Sigma)$ of formulas of $\operatorname{FO}(\Sigma)$ , which is built inductively from $T(\Sigma)$ , as follows:
   1. If $t_1$ and $t_2$ are terms, then $(t_1=t_2)$ is a formula;
   2. If $R$ is an $n$ -ary relation symbol and $t_1,...,t_n$ are terms, then $(R(t_1,...,t_n))$ is a formula;
   3. If $\varphi$ is a formula, then so is $(\neg\varphi)$ ;
   4. If $\varphi$ and $\psi$ are formulas, then so is $(\varphi\Or\psi)$ ;
   5. If $\varphi$ is a formula, and $x$ is a variable, then $(\exists x(\varphi))$ is a formula.

For example, in the first order language of partially ordered rings, expressions such as $$0,\quad x^2,\quad\mbox{ and } \quad y+zx$$ are terms, while $$(x=xy),\quad (x+y \le yz),\quad \mbox{ and }\quad (\exists x ((x\le 0) \Or (0\le x)))$$ are formulas.

Remarks.

1. Generally, one omits parentheses in formulas, when there is no ambiguity. For example, a formula $(\varphi)$ can be simply written $\varphi$ . As such, the parentheses are also called the auxiliary symbols.
2. The first two types of formulas, not involving logical connectives, are the atomic formulas. It is evident that formulas are inductively built from atomic formulas.
3. The other logical symbols are obtained in the following way :
   
   where $\varphi$ and $\psi$ are formulas. All logical symbols are used when building formulas.
4. In the literature, it is a common practice to write $\Sigma_{\omega \omega}$ for $\operatorname{FO}(\Sigma)$ . The first subscript means that every formula in $\operatorname{FO}(\Sigma)$ contains a finite number of $\vee$ 's (less than $\omega$ ), while the second subscript signifies that every formula has a finite number of $\exists$ 's. In general, $\Sigma_{\alpha\beta}$ denotes a language built from $\Sigma$ such that, in any given formula, the number of occurrences of $\vee$ is less than $\alpha$ and the number of occurrences of $\exists$ is less than $\beta$ . When the number of occurrences of $\vee$ (or $\exists$ ) in a formula is not limited, we use the symbol $\infty$ in place of $\alpha$ (or $\beta$ ). Clearly, if $\alpha$ and $\beta$ are not $\omega$ , we get a language that is not first-order.
5. Also a common practice in the literature, $\Sigma$ is used to identify a signature and the first-order language it uniquely determines.

Bibliography

1

W. Hodges, A Shorter Model Theory, Cambridge University Press, (1997).

2

D. Marker, Model Theory, An Introduction, Springer, (2002).