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 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. 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_k 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.

References

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

Atomic formula

Atomic formula in first-order logic

See also

References

Further reading