Atomic formula in first-order logic

Statement

The well-formed terms and propositions of ordinary first-order logic have the following syntax:

Terms:

• $$t \equiv c \mid x \mid f (t_{1},\dotsc, t_{n}),$$

that is, a term is recursively defined to be a constant c (a named object from the domain of discourse), or a variable x (ranging over the objects in the domain of discourse), or an n-ary function f whose arguments are terms t_k. Functions map tuples of objects to objects.

Propositions:

• $$A, B, ... \equiv P (t_{1},\dotsc, t_{n}) \mid A \wedge B \mid \top \mid A \vee B \mid \bot \mid A \supset B \mid \forall x.\ A \mid \exists x.\ A,$$

that is, a proposition is recursively defined to be an n-ary predicate P whose arguments are terms t_k, or an expression composed of logical connectives (and, or) and quantifiers (for-all, there-exists) used with other propositions.

An atomic formula or atom is simply a predicate applied to a tuple of terms; that is, an atomic formula is a formula of the form P (t₁ ,…, t_n) for P a predicate, and the t_n terms.

All other well-formed formulas are obtained by composing atoms with logical connectives and quantifiers.

For example, the formula ∀x. P (x) ∧ ∃y. Q (y, f (x)) ∨ ∃z. R (z) contains the atoms

• $$P (x)$$
• $$Q (y, f (x))$$
• $$R (z).$$

As there are no quantifiers appearing in an atomic formula, all occurrences of variable symbols in an atomic formula are free.