# Example: Examples of Variables in a Logical Calculus

(related to Definition: Variable in a Logical Calculus)

### Example 1

Let $s$ be the string "$x+2=5$", let $U$ be the domain of discourse of natural numbers $0,1,2,\ldots$ and let $I(s)$ be the interpretation of $s$ assigning it a meaning of adding natural numbers in $U$. Clearly, "$x$" is a substring of $s$. Also "$2$", "$5$" and "$+$" are substrings of $s$, but the interpretation $I$ means that only $x$ is a variable, since $2,5$ are interpreted as constant natural numbers and "$+$" is interpreted as the addition sign. Then the variable $x$ represents any of the (infinitely) many natural numbers $0,1,2,\ldots$. But only for one of these infinitely many possible meanings of $x$ the string $s$ will be valued as true: $[[s]]_I=1$ only for $x=3$.

### Example 2

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 $$\epsilon > 0$$ there is a $$\delta > 0$$ such that $$|f(x)-f(a)| < \epsilon$$ for all $$x\in D$$ with $$|x-a| < \delta.$$"

This proposition can be codified using a string like this:

"$\forall\epsilon\,(\epsilon > 0)\,\exists\delta\,(\delta > 0)\,\forall x\,(x\in D)\,(|x-a|<\delta\Longrightarrow|f(x)-f(a)|<\epsilon).$"

In this string, the substrings $$"x"$$, $$"\delta"$$, $$"\epsilon"$$ and $$"a"$$ are variables, since they represent real numbers.

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