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. More precisely, the well-formed terms and propositions of ordinary first-order logic have the following syntax:
| (terms) | t | ::= | x | f (t1, …, tn) |
| (propositions) | A, B, … | ::= | P (t1, …, tn) | A ∧ B | ⊤ | A ∨ B | ⊥ | A ⊃ B | ∀x. A | ∃x. A |
The formulae of the form P (t1, …, tn) are the atomic formulas. Any well-formed formula—for example, ∀x. P (x) ∧ ∃y. Q (y, f (x)) ∨ ∃z. R (z)— comprises the atoms
- P (x)
- Q (y, f (x))
- R (z)
and the syntax rules.
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
- Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-568-81262-0.