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atomic formula
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{{about|a concept in mathematical logic|the concept from chemistry|chemical formula}}{{short description|mathematical logic concept}}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 formula are merely strings of symbols with a given signature, which may or may not be satisfiable with respect to a given model.BOOK, Wilfrid Hodges, A Shorter Model Theory, 1997, Cambridge University Press, 0-521-58713-1, 11–14,

Atomic formula in first-order logic

The well-formed terms and propositions of ordinary first-order logic have the following syntax:Terms:
  • t equiv c mid x mid f (t_{1},dotsc, t_{n}),
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 tk. Functions map tuples of objects to objects.Propositions:
  • 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 ,
that is, a proposition is recursively defined to be an n-ary predicate P whose arguments are terms tk, 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 (t1 ,…, 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
  • P (x)
  • Q (y, f (x))
  • R (z)

See also

References

{{reflist}}

Further reading

  • BOOK, Hinman, P., Fundamentals of Mathematical Logic, A K Peters, 2005, 1-56881-262-0,


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