Atomic formula

The precise form of atomic formulas depends on the logic under consideration; for propositional logic, for example, a propositional variable is often more briefly referred to as an "atomic formula", but, more precisely, a propositional variable is not an atomic formula but a formal expression that denotes an atomic formula. For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a term. In model theory, atomic formulas are merely strings of symbols with a given signature, which may or may not be satisfiable with respect to a given model.

Atomic formula in first-order logic

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

Terms:

• ${\displaystyle t\equiv c\mid x\mid f(t_1,\dots ,t_n)}$,

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:

• ${\displaystyle A,B,...\equiv P(t_1,\dots ,t_n)\mid A\wedge B\mid \mathrm{\top }\mid A\vee B\mid \mathrm{\perp }\mid A\supset B\mid \mathrm{\forall }x.A\mid \mathrm{\exists }x.A}$,

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

• ${\displaystyle P(x)}$
• ${\displaystyle Q(y,f(x))}$
• ${\displaystyle R(z)}$.

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