# What is the difference between an atom and a term? (logic)

When I read on Wikipedia about atomic formula, it reads that an atomic formula can also be called an atom. As I interpret the text, the atom is something that is evaluated to be true or false such as "there is 1 bike with x amount of fuel left", while a term is simply something that exists e.g. 1, x or bike. I.e. an atom consists of terms.

Maybe an atom could also be considered a term if it is part of another atomic formula. I find the terminology complex and confusing...

Have I understood the difference correctly? If not: What is the difference between a term and an atom?

• I'd say an atom is just a constant term. If you think of a term as a tree, then the atoms are the leaves. It has nothing to do with being a proposition. May 21, 2020 at 12:07
• @Couchy311 No, that's not true. An atom is a formula, not a term. May 29, 2020 at 12:24

A term is a "name": variables and constants are terms.

And terms can be manufactured using function symbols.

Example: $$n$$ is a variable, $$0$$ is a constant and $$+$$ is a (binary) function symbol.

Thus, $$n,0$$ and $$n+0$$ are terms.

Formulas are statements.

Atomic formulas are the basic building blocks for manufacturing statements.

They are formulas that have no sub-parts that are formulas.

They are manufactured using predicate symbols, like e.g. $$\text {Even}(x)$$, equality and terms.

Thus, $$\text {Even}(n), 0=0$$ and $$n+0=n$$ are atomic formulas.

With connectives and quantifiers we can write more complex formulas, like: $$\forall n (n+0=n)$$ and $$0=0 \to \forall n (n+0=n)$$.

Regarding the example, due to the fact that ""there is 1" is a numerical quantifier and its treatment is a little bit tricky, I'll use: "there is at least one bike with x amount of fuel left".

We can parse it with the predicates $$\text {Bike}(y)$$ expressing "y is a bike" and $$\text {FuelLeft}(y,x)$$, expressing "y has an amount x of fuel left".

The complete statement will be written using the existential quantifier for "there is at least one" ($$\exists$$) and the connective "and" ($$\land$$):

$$\exists y \ (\text {Bike}(y) \land \text {FuelLeft}(y,x))$$.

In this formula, $$\text {Bike}(y)$$ and $$\text {FuelLeft}(y,x)$$ are atomic formulas, while $$(\text {Bike}(y) \land \text {FuelLeft}(y,x))$$ is a non-atomic formula.

• $0=0 \to \forall n (n+0=n)$ is not an atomic formula then, since it consists of other formulas? May 21, 2020 at 14:33
• It would be nice if you could explain which parts that are atoms and terms in that example and the example with $\exists y \ (\text {Bike}(y) \land \text {FuelLeft}(y,x))$ at the end. (still a bit confused) May 21, 2020 at 14:38
• Very nice! This makes sense to me now. Thanks! May 22, 2020 at 15:08
• @dekuShrub - you are welcome :-) May 22, 2020 at 16:45