Atomic formula
Part of Formal languages |
In
The precise form of atomic formulas depends on the logic under consideration; for
Atomic formula in first-order logic
The well-formed terms and propositions of ordinary first-order logic have the following syntax:
- ,
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:
- ,
that is, a proposition is recursively defined to be an n-ary
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
- .
As there are no quantifiers appearing in an atomic formula, all occurrences of variable symbols in an atomic formula are free.[2]
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
- ISBN 0-521-58713-1.
- ^ W. V. O. Quine, Mathematical Logic (1981), p.161. Harvard University Press, 0-674-55451-5
Further reading
- Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-56881-262-0.