Exact meaning of terms in First-Order Logic

Question

I have recently studied first-order logic, but I am confused on a very basic idea: Terms. Is the term the value inputed in $f(x)$ for $f(t_1....t_n)$, for example, it is $1$ in $f(1)$? Or is it, for example, $f(1)$, the whole thing? Thank you in advance.

Answer 1

A term is something that represents an element of a structure. For instance, if we're working in the language of arithmetic, a term is something like $(1+1)*x + y$. It is built from variables, constant symbols, and function symbols, and again let's notice that (if we choose values for $x$ and $y$) our term represents an element of our structure.

This is contrasted with a relation, which asks a question about elements of our structure. For instance, $x < y$ is a question about two terms: $x$ and $y$. We can also glue relations together in order to ask more complicated questions, like $(x < y) \lor (y < x) \lor (x = y)$.

So for your question, if $f$ is a function symbol, then both $1$ and $f(1)$ are terms! Indeed they both represent elements of your structure. To go back to the specific example of arithmetic, where we might take $f = +$, it's true that $1$ is a term, and so is $x$, and then $1+x$ is also a term.

---

I hope this helps ^_^

Answer 2

All constants and variables are terms. So, $1$ is a term.

With $t_1, ... t_n$ being terms, and $f$ a function symbol, $f(t_1, ... t_n)$ is also a term. So, $f(1)$ is also a term.

So, both $1$ and $f(1)$ are terms. However, focusing on term $f(1)$, only the $1$ is a term that goes into (or gets 'inputed' as you say) $f(1)$.

So, while $f(1)$ is a term, it is not a term that goes into or is an argument of $f(1)$