# Is the sentence “all apples are red” an atomic sentence?

In An Introduction to Logic by Patrick Suppes, an atomic sentence is defined as a sentence that contains no sentential connective. However, in a later chapter, a sentence is defined as a formula which contains no free variables.

Now if we take a sentence like all apples are red, it could be symbolized as (x)( A(x) => R(x) ) which does contain an implication. So is that sentence an atomic sentence or not?

## 2 Answers

In propositional logic sentence and formula are synonyms.

An atomic sentence is a formula without propositional connectives [see Suppes, page 12].

In predicate logic, sentence and formula have different meaning: a sentence is a formula with no occurrences of free variables [see Def. page 54].

The formal definition of formula for predicate logic is at page 52:

a predicate followed by the appropriate number of terms as arguments is an atomic formula [examples: Px,Rxy].

Formulas are: atomic formula or formulas produced using connectives and quantifiers.

Thus, "Socrates is a philosopher", which can be symbolized with Phil(S), is an example of atomic sentence, while "Every Greek is a philosoper" [ ∀x ( Greek(x) → Phil(x)) ] is not.

It is a sentence but not an atomic sentence.

It is a sentence because the all-quantifier binds the free variables and so closes the formula. It is not an atomic sentence because it contains an implication.