Atomic formula

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" ("t"1, …, "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" ("t"1, …, "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" ("t"1, …, "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

*


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