 Atomic Formula
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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.

Atomic formula in first-order logic

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

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

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 tk. 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 tk, 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 ,..., tn) for P a predicate, and the tn 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

• $P(x)$ • $Q(y,f(x))$ • $R(z)$ 