Atomic sentence

In Logic, (sentential logic and predicate logic), an atomic sentence is an atomic formula in which no variable occurs free.

Explanation
An atomic formula is an expression consisting of either a sentential letter or an n-place predicate letter followed by n individual symbols or functions. An individual symbol is a variable or individual constant. A well-formed formula (wff) is either an atomic formula or built up from one or more atomic formulae using truth functional logical connectives or quantifiers or both in accordance with their rules of use.

A sentence is a wff in which no variable occurs free. An atomic sentence is an atomic formula in which no variable occurs free.

An occurrence of variable in a wff is bound if it is  within the scope of a quantifier otherwise it is free. (A wff which is not an atomic formula is a compound formula. A sentence which is not an atomic sentence is a compound sentence.)

Examples
As examples, let F, G, H be predicate letters; let a, b, c be individual constants; let x, y, z be variables; and let p be a sentential letter. Then the following wfs are atomic sentences: The following wfs are atomic formulae but not atomic sentences because they incude free variables: The following wfs are not atomic formulae but are built up from atomic formulae using logical connectives. They are not sentences because they contain free variables. (They are compound formulae): The following wfs are sentences but not atomic sentences (because they are not atomic formulae). (They are compound sentences):
 * p
 * F(a)
 * H(b,a,c)
 * F(x)
 * G(a,z)
 * H(x,y,z)
 * F(x)🇦🇩G(a,z)
 * G(a,z)H(x,y,z)
 * x(F(x))
 * z(G(a,z))
 * x'y'z(H(x,y,x))
 * xz(F(x)🇦🇩G(a,z))
 * x'y'z (G(a,z)H(x,y,z)

Interpretations
A sentence is either true or false under an interpretation which assigns values to the the logical variables. We might for example make the following assignments:

Individual Constants
 * a: Socrates
 * b: Plato
 * c: Aristotle

Predicates:
 * Fα: α is sleeping
 * Gαβ: α hates β
 * Hαβγ: α made β hit γ

Sentential variables:
 * p "It is raining."

Under this interpretation the sentences discussed above would represent the following English statements:
 * p: "It is raining."
 * F(a): "Socrates is sleeping."
 * H(b,a,c): "Plato made Socrates hit Aristotle."
 * x(F(x)): "Everybody is sleeping."
 * z(G(a,z)): "Socrates hates somebody."
 * x'y'z(H(x,y,z)): "Somebody made everybody hit somebody."
 * xz(F(x)🇦🇩G(a,z)): Everybody is sleeping and Socrates hates somebody.
 * x'y'z (G(a,z)H(x,y,z): Either Socrates hates somebody or everybody made everybody hit somebody.

Translating sentences from a natural language into an artificial language
Sentences in natural languasges can be ambiguous, whereas the languages of the sentetial logic and predicat logics are precise. Traslation can reveal such ambiguities and express precisely the intended meaning.

For example take the English sentence "Father Ted married Jack and Jill". Does this mean Jack married Jill? In translating we might make the following assignments: Individual Constants


 * a: Father Ted
 * b: Jack
 * c: Jill

Predicates: Using these assigments the sentence above could be translated as follows:
 * Mαβγ: α officiated at the marriage of β to γ

To establish which is the correct translation of "Father Ted married Jack and Jill", it would be necessay to ask the speaker exactly what was meant.
 * M(a,b,c): Father Ted officiated at the marriage of Jack and Jill.
 * xy((M(a,b,x)🇦🇩 (M(a,c,y)): ): Father Ted officiated at the marriage of Jack to somebody and Father Ted officiated at the marriage of Jill to somebody.
 * xy(M(x,a,b)🇦🇩M(y,a,c)): Somebody officiated at the marriage of Father Ted to Jack and somebody officiated at the marriage of Father Ted to Jill.