atomic formula

Let $\Sigma$ be a signature and $T(\Sigma)$ the set of terms over $\Sigma$ . The set $S$ of symbols for $T(\Sigma)$ is the disjoint union of $\Sigma$ and $V$ , a countably infinite set whose elements are called variables. Now, adjoin $S$ the set $\lbrace =, (, )\rbrace$ , assumed to be disjoint from $S$ . An atomic formula $\varphi$ over $\Sigma$ is any one of the following:

1. either $(t_1=t_2)$ , where $t_1$ and $t_2$ are terms in $T(\Sigma)$ ,
2. or $(R(t_1,...,t_n))$ , where $R\in\Sigma$ is an $n$ -ary relation symbol, and $t_i\in T(\Sigma)$ .

Remarks.

1. Using atomic formulas, one can inductively build formulas using the logical connectives $\vee$ , $\neg$ , $\exists$ , etc... In this sense, atomic formulas are formulas that can not be broken down into simpler formulas; they are the building blocks of formulas.
2. A literal is a formula that is either atomic or of the form $\neg \varphi$ where $\varphi$ is atomic. If a literal is atomic, it is called a positive literal. Otherwise, it is a negative literal.
3. A finite disjunction of literals is called a clause. In other words, a clause is a formula of the form $\varphi_1 \vee \varphi_2 \vee \cdots \vee \varphi_n$ , where each $\varphi_i$ is a literal.
4. A qunatifier-free formula is a formula that does not contain the symbols $\exists$ or $\forall$ .
5. If we identify a formula $\varphi$ with its double negation $\neg (\neg \varphi)$ , then it can be shown that any quantifier-free formula can be identified with a formula that is in conjunctive normal form, that is, a finite conjunction of clauses. For a proof, see this link