Atomic Formula

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## Atomic formula in first-order logic

## See also

## References

## Further reading

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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]}

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

- ,

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:

- ,

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

- 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

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

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

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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