Atomic formula

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

Terms:

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

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 t_k. Functions map tuples of objects to objects.

Propositions:

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

that is, a proposition is recursively defined to be an n-ary predicate P whose arguments are terms 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

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

• 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

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