Atomic formula

This article is about a concept in mathematical logic. For the concept from chemistry, see chemical formula.

In mathematical logic, an atomic formula (also known as an atom or a prime formula) 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, a propositional variable is often more briefly referred to as an "atomic formula", but, more precisely, a propositional variable is not an atomic formula but a formal expression that denotes an atomic formula. For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a term. In model theory, atomic formulas 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:

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

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

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

As there are no quantifiers appearing in an atomic formula, all occurrences of variable symbols in an atomic formula are free. ^[2]

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. Hodges, Wilfrid (1997). A Shorter Model Theory. Cambridge University Press. pp. 11–14. ISBN   0-521-58713-1.
2. W. V. O. Quine, Mathematical Logic (1981), p.161. Harvard University Press, 0-674-55451-5

Further reading

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