Atomic formula
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In mathematical logic, an atomic formula, or atom, is a formula that has no subformulas. What formulas count as atoms depends on the logic being used. In propositional logic, for example, the only atomic formulas are the propositional variables.
The atoms are the "simplest" formulas in a logical system. The well-formed formulas in a logical system are typically defined recursively, by identifying all of the valid atomic formulas, and then giving rules for creating a well-formed formula out of other well-formed formulas. Formulas made from atomic formulas are compound formulas.
For example, in propositional logic, one has the following rules of formula construction:
- Any propositional variable p is a well-formed (atomic) formula.
- Given any well-formed formula A, the negation ¬A ("not A ") is a well-formed formula.
- Given any two well-formed formulas A and B, the conjunction A ∧ B ("A and B ") is a well-formed formula.
- Given any two well-formed formulas A and B, the disjunction A ∨ B ("A or B ") is a well-formed formula.
- Given any two well-formed formulas A and B, the implication A ⇒ B ("A implies B ") is a well-formed formula.
Thus, we can build arbitrarily complicated compound formulas, such as ((p ∧ ¬(q ⇒ r)) ∨ ¬p) , out of the simple atomic formulas p, q, r, and our construction rules.
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