Atomic formula

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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.

[edit] Atomic formula in first-order logic

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

Terms:

• t ::= c | x | f (t₁, …, 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, …  ::= P (t₁, …, t_n) | A ∧ B | ⊤ | A ∨ B | ⊥ | A ⊃ B | ∀x. A | ∃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.

[edit] 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

[edit] References

• Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-568-81262-0.