# Definition: Terms in Predicate Logic

Let $$\Sigma = (V_\Sigma,F_\Sigma,P_\Sigma)$$ be a signature of a predicate logic. A set of predicate logic terms is defined recursively:

• Every variable $$x\in V_\Sigma$$ is a term.
• Every constant (i.e. a function with arity $$0$$) $$f\in F_\Sigma$$ is a term.
• If $$s_1,s_2,\ldots, s_n$$ are terms $$f\in F_\Sigma$$ is a function with arity $$n\ge 1$$, then $$f(s_1,s_2,\ldots,s_n)$$ is a term.

### Example

Take real numbers as the domain of discourse, and consider the $$\epsilon-\delta$$ definition of continuous real functions:

A real function $$f:D\to\mathbb R$$ is continuous at the point $$a\in D$$, if for every $$\epsilon &gt; 0$$ there is a $$\delta &gt; 0$$ such that $$|f(x)-f(a)| &lt; \epsilon$$ for all $$x\in D$$ with $$|x-a| &lt; \delta.$$

This proposition can be codified using a formula like this:

$\forall\epsilon\,(\epsilon &gt; 0)\,\exists\delta\,(\delta &gt; 0)\,\forall x\,(x\in D)\,(|x-a|&lt;\delta\Longrightarrow|f(x)-f(a)|&lt;\epsilon).$

In this formula, the strings $$"x"$$, $$"\delta"$$, $$"\epsilon"$$ and $$"a"$$ are variables, thus they are terms. Also the constant $$"0"$$ as well as the functions $$"f(x)"$$, $$"f(a)"$$, $$"x-a"$$, $$"f(x)-f(a)"$$ and $$"|x-a|"$$, and $$"|f(x)-f(a)|"$$ are terms, because they are nullary, unary and binary functions.

are terms.

### Other Examples of Terms in Predicate Logic

• $$0$$
• $$1$$
• $$x$$
• $$y$$
• $$f(x,x)$$
• $$f(0,1)$$
• $$f(x,y)$$
• $$f(f(x,x),f(0,1))$$
• ...

Definitions: 1

### References

#### Bibliography

1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011