first order language

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

1. the set $S(\mathrm{\Sigma })$ of symbols of $\mathrm{FO}(\mathrm{\Sigma })$, which is the disjoint union of the following sets:

   1. $\mathrm{\Sigma }$ (the non-logical symbols),

   2. a countably infinite set $V$ of variables,

   3. the set of logical symbols $\{\wedge ,\vee ,\mathrm{\neg },\Rightarrow ,\iff ,\forall ,\exists \}$,

   4. the singleton consisting of the equality symbol $\{=\}$, and

   5. the set of parentheses (left and right) $\{(,)\}$;

2. the set $T(\mathrm{\Sigma })$ of terms of $\mathrm{FO}(\mathrm{\Sigma })$, which is built inductively from $S(\mathrm{\Sigma })$, as follows:

   1. Any variable $v\in V$ is a term;

   2. Any constant symbol in $\mathrm{\Sigma }$ is a term;

   3. If $f$ is an $n$-ary function symbol in $\mathrm{\Sigma }$, and $t_1,\mathrm{\dots },t_n$ areterms, then $f(t_1,\mathrm{\dots },t_n)$ is a term.

3. the set $F(\mathrm{\Sigma })$ of formulas of $\mathrm{FO}(\mathrm{\Sigma })$, which is built inductively from $T(\mathrm{\Sigma })$, 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 $t_1,\mathrm{\dots },t_n$ areterms, then $(R(t_1,\mathrm{\dots },t_n))$ is a formula;

   3. If $\phi$ is a formula, then so is $(\mathrm{\neg }\phi )$;

   4. If $\phi$ and $\psi$ are formulas, then so is $(\phi \vee \psi )$;

   5. If $\phi$ is a formula, and $x$ is a variable, then $(\exists x(\phi ))$ is a formula.

In other words, $T(\mathrm{\Sigma })$ and $F(\mathrm{\Sigma })$ are the smallest sets, among all sets satisfying the conditions given for terms and formulas, respectively.

Formulas in 3(a) and 3(b), which do not contain any logical connectives, are called the atomic formulas.

For example, in the first order language of partially ordered rings, expressions such as

$$0,x^2,\text{ and }\mathit{\hspace{1em}}y+zx$$

are terms, while

$$(x=xy),(x+y\le yz),\text{ and }\mathit{\hspace{1em}}(\exists x((x\le 0)\vee (0\le x)))$$

are formulas, and the first two of which are atomic.

Remarks.

1. Generally, one omits parentheses in formulas, when there is no ambiguity. For example, a formula $(\phi )$ can be simply written $\phi$. As such, the parentheses are also called the auxiliary symbols.

2. The other logical symbols are obtained in the following way :

   $\phi \wedge \psi$$\stackrel{def}{:=}\mathrm{\neg }(\mathrm{\neg }\phi \vee \mathrm{\neg }\psi )$$\mathrm{\hspace{1em}\hspace{1em}}\phi \Rightarrow \psi$$\stackrel{def}{:=}\mathrm{\neg }\phi \vee \psi$

   $\phi \iff \psi$$\stackrel{def}{:=}(\phi \Rightarrow \psi )\wedge (\psi \Rightarrow \phi )$$\mathrm{\hspace{1em}\hspace{1em}}\forall x(\phi )$$\stackrel{def}{:=}\mathrm{\neg }(\exists x(\mathrm{\neg }\phi ))$

   where $\phi$ and $\psi$ are formulas. All logical symbols are used when building formulas.

3. In the literature, it is a common practice to write ${\mathrm{\Sigma }}_{\omega \omega }$ for $\mathrm{FO}(\mathrm{\Sigma })$. The first subscript means that every formula in $\mathrm{FO}(\mathrm{\Sigma })$ 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, ${\mathrm{\Sigma }}_{\alpha \beta }$ denotes a language built from $\mathrm{\Sigma }$ 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 $\mathrm{\infty }$ in place of $\alpha$ (or $\beta$). Clearly, if $\alpha$ and $\beta$ are not $\omega$, we get a language that is not first-order.

First Order Languages as Formal Languages

If the signature $\mathrm{\Sigma }$ and the set $V$ of variables are countable, then $S(\mathrm{\Sigma }),T(\mathrm{\Sigma })$, and $F(\mathrm{\Sigma })$ can be viewed as formal languages over a certain (finite) alphabet $\mathrm{\Gamma }$. The set $\mathrm{\Gamma }$ should include all of the logical connectives, the equality symbol, and the parentheses, as well as the following symbols

$$R,F,V,I,\mathrm{\#},$$

where they are used to form words for relation, formula, and variable symbols. More precisely,

• $V{I}^n\mathrm{\#}$ stands for the variable $v_n$,

• $R{I}^n\mathrm{\#}{I}^m\mathrm{\#}$ stands for the $m$-th relation symbol of arity $n$, and

• $F{I}^n\mathrm{\#}{I}^m\mathrm{\#}$ stands for the $m$-th function symbol of arity $n$,

where $m,n\ge 0$ are integers. The symbol $\mathrm{\#}$ is used as a delimiter or separator. Note that the constant symbols are then words of the form $F\mathrm{\#}{I}^m\mathrm{\#}$. It can shown that $S(\mathrm{\Sigma }),T(\mathrm{\Sigma })$ and $F(\mathrm{\Sigma })$ are context-free over $\mathrm{\Gamma }$, and in fact unambiguous.