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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?

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  • $\begingroup$ 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. $\endgroup$
    – Couchy
    May 21, 2020 at 12:07
  • $\begingroup$ @Couchy311 No, that's not true. An atom is a formula, not a term. $\endgroup$
    – lemontree
    May 29, 2020 at 12:24

1 Answer 1

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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.

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  • $\begingroup$ $0=0 \to \forall n (n+0=n)$ is not an atomic formula then, since it consists of other formulas? $\endgroup$
    – dekuShrub
    May 21, 2020 at 14:33
  • $\begingroup$ 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) $\endgroup$
    – dekuShrub
    May 21, 2020 at 14:38
  • $\begingroup$ Very nice! This makes sense to me now. Thanks! $\endgroup$
    – dekuShrub
    May 22, 2020 at 15:08
  • $\begingroup$ @dekuShrub - you are welcome :-) $\endgroup$ May 22, 2020 at 16:45

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