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.

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 ... tn)
  ,

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 ... tn) || A ∧ B || →( || A ∨ B || &(er(; || A su(set B || ∀ x. A || eξsts 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.

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

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

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