Atomic formula

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 logically invalid 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. In model theory, atomic formula are merely strings of symbols with a given signature, which may or may not be satisfiable with respect to a given model.^[1]

Atomic formula in first-order logic

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

Terms:

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

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:

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

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 formulae 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

• $P(x)$
• $Q(y,f(x))$
• $R(z)$

When all of the terms in an atom are ground terms, then the atom is called a ground atom or ground predicate.

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

1. Wilfrid Hodges (1997). A Shorter Model Theory. Cambridge University Press. pp. 11–14. ISBN 0-521-58713-1.

Further reading

• Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-56881-262-0.