Atomic formula

 ( Predicate Logic ) 

In mathematical logic, an atomic formula (also known as an atom or a prime formula) is a formula with no deeper 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.

---

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

Term algebra:

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

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 of objects to objects.

Propositions:

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

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

• $P\; (x)$
• $Q\; (y,\; f\; (x))$
• $R\; (z)$.

As there are no quantifiers appearing in an atomic formula, all occurrences of variable symbols in an atomic formula are free.W. V. O. Quine, Mathematical Logic (1981), p.161. Harvard University Press, 0-674-55451-5

---

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

---

Further reading

• P. Hinman (2025). 9781568812625, A K Peters. ISBN 9781568812625

Post Comment