In mathematical logic, an atomic formula (also known simply as an atom) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic. Compound formulas are formed by combining the atomic formulas using the logical connectives.

The precise form of atomic formulas depends on the logic under consideration; for propositional logic, for example, the atomic formulas are the propositional variables. For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a term. More precisely, the well-formed terms and propositions of ordinary first-order logic have the following syntax:

The formulae of the form P (t₁, …, t_n) are the atomic formulas. Any well-formed formula—for example, ∀x. P (x) ∧ ∃y. Q (y, f (x)) ∨ ∃z. R (z)— comprises the atoms

• P (x)
• Q (y, f (x))
• R (z)

and the syntax rules.

See also

• In model theory, structures assign an interpretation to the atomic formulas.
• In proof theory, polarity assignment for atomic formulas is an essential component of focusing.
• Atomic sentence

References

• Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-568-81262-0.