Physics:Atomic formula
In mathematical logic, an atomic formula (also known as an atom or a prime formula) 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, a propositional variable is often more briefly referred to as an "atomic formula", but, more precisely, a propositional variable is not an atomic formula but a formal expression that denotes an atomic formula. 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:
- [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].
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
- ↑ Hodges, Wilfrid (1997). A Shorter Model Theory. Cambridge University Press. pp. 11–14. 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.
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