atomic formula

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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.{{cite book |last=Hodges |first=Wilfrid |year=1997 |title=A Shorter Model Theory |publisher=Cambridge University Press |isbn=0-521-58713-1 |pages=11–14}}

Atomic formula in first-order logic

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

Terms:

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

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:

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

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

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

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

atomic formula

Atomic formula in first-order logic

See also

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Further reading