In mathematical logic, an atomic formula (also known simply as an atom) is a formula with no deeper propositional structure, that is, a formula with no logical connectives or strict sub-formulas. 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 term arguments. More precisely, the well-formed propositions (A, B, …) 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 |
where P (t1, …, tn) are the atomic formulas. Any well-formed formula—for example, ∀x. P (x) ∧ ∃y. Q (y, f (x)) ∨ ∃z. R (z)—is formed from the relevant atoms (P (x), Q (y, f (x)) and R (z) in this case) 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.
