# Physics: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. 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.^{[1]}

## Atomic formula in first-order logic

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

- [math]\displaystyle{ t \equiv c \mid x \mid f (t_{1},\dotsc, t_{n}) }[/math],

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:

- [math]\displaystyle{ 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 }[/math],

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*_{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

- [math]\displaystyle{ P (x) }[/math]
- [math]\displaystyle{ Q (y, f (x)) }[/math]
- [math]\displaystyle{ R (z) }[/math]

## 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

- ↑ Wilfrid Hodges (1997).
*A Shorter Model Theory*. Cambridge University Press. pp. 11–14. ISBN 0-521-58713-1.

## Further reading

- Hinman, P. (2005).
*Fundamentals of Mathematical Logic*. A K Peters. ISBN 1-56881-262-0.

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