Atomic sentence

In the following examples:

• let F, G, H be predicate letters;
• let a, b, c be individual constants;
• let x, y, z be variables.

These wffs are atomic sentences; they contain no free variables or conjunctions:

• F(a)
• G(a, b)
• H(a, b, c)

These wffs are atomic formulae, but are not sentences (atomic or otherwise) because they include free variables:

• F(x)
• G(a, z)
• H(x, y, z)

These wffs are compound sentences. They are sentences, but are not atomic sentences because they are not atomic formulae:

• ∀x (F(x))
• ∃z (G(a, z))
• ∃x ∀y ∃z (H(x, y, z))
• ∀x ∃z (F(x) ∧ G(a, z))
• ∃x ∀y ∃z (G(a, z) ∨ H(x, y, z))

These wffs are compound formulae. They are not atomic formulae but are built up from atomic formulae using logical connectives. They are also not sentences because they contain free variables:

• F(x) ∧ G(a, z)
• G(a, z) ∨ H(x, y, z)

A sentence is either trueorfalse under an interpretation which assigns values to 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." (They may not have all hit the same person z, but they all did so because of the same person x.)
• ∀x ∃z (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 somebody made everybody hit somebody."

Sentences in natural languages can be ambiguous, whereas the languages of the sentential logic and predicate logics are precise. Translation 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:

• Mαβγ: α officiated at the marriage of β to γ

Using these assignments the sentence above could be translated as follows:

• M(a, b, c): Father Ted officiated at the marriage of Jack and Jill.
• ∃x ∃y (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.
• ∃x ∃y (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.

To establish which is the correct translation of “Father Ted married Jack and Jill”, it would be necessary to ask the speaker exactly what was meant.

Atomic sentences are of particular interest in philosophical logic and the theory of truth and, it has been argued, there are corresponding atomic facts.

An atomic sentence (or possibly the meaning of an atomic sentence) is called an elementary propositionbyLudwig Wittgenstein and an atomic propositionbyBertrand Russell:

• 4.2 The sense of a proposition is its agreement and disagreement with possibilities of existence and non-existence of states of affairs. 4.21 The simplest kind of proposition, an elementary proposition, asserts the existence of a state of affairs. — Wittgenstein, Tractatus Logico-Philosophicus.
• A proposition (true or false) asserting an atomic fact is called an atomic proposition. — Russell, "Introduction to Tractatus Logico-Philosophicus"
• See also^[2] and^[3] especially regarding elementary proposition and atomic proposition as discussed by Russell and Wittgenstein

Note the distinction between an elementary/atomic proposition and an atomic fact.

No atomic sentence can be deduced from (is not entailed by) any other atomic sentence, no two atomic sentences are incompatible, and no sets of atomic sentences are self-contradictory. Wittgenstein made much of this in his Tractatus. If there are any atomic sentences then there must be "atomic facts" which correspond to those that are true, and the conjunction of all true atomic sentences would say all that was the case, i.e., "the world" since, according to Wittgenstein, "The world is all that is the case". (TLP:1). Similarly the set of all sets of atomic sentences corresponds to the set of all possible worlds (all that could be the case).

The T-schema, which embodies the theory of truth proposed by Alfred Tarski, defines the truth of arbitrary sentences from the truth of atomic sentences.

• Logical constant

• Benson Mates, Elementary Logic, Oxford University Press, 1972.
• Elliott Mendelson, Introduction to Mathematical Logic, Van Nostrand Reinhold Company, 1964.