Physics:Atomic formula

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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]

Atomic formula in first-order logic

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

Terms:

  • [math]\displaystyle{ t \equiv c \mid x \mid f (t_{1},\dotsc, t_{n}) }[/math],

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:

  • [math]\displaystyle{ 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 }[/math],

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 ,…, tn) for P a predicate, and the tn 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

  • [math]\displaystyle{ P (x) }[/math]
  • [math]\displaystyle{ Q (y, f (x)) }[/math]
  • [math]\displaystyle{ R (z) }[/math]

See also

References

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

Further reading

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

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