In the wikipedia article on logical terms it is written:
In analogy to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact, in mathematical logic, a term denotes a mathematical object and a formula denotes a mathematical fact. In particular, terms appear as components of a formula.
I am not sure to understand exactly what that means, and the exact difference between:
- a term
- a formula
- an expression
If authors adopt different conventions, I would like the standpoint of one that makes a clear distinction between the terms. In particular things that are not clear not me are questions like:
- is every term a formula?
- is every term an expression?
- is every formula a term?
- is every formula an expression?
- is every expression a term?
- is every expression a formula?
Examples of things that are terms, formulas, and expressions are welcome.
A term is a "name": variables and constants are terms.
In addition, "complex" 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 $0=0 \to \forall n (n+0=n)$.
Expression can be a "generic" category: it may mean a string of symbols.
We may call well-formed expression a string of symbols that satisfies the rule of the syntax.
If so, it is either a term or a formula.