Are mathematical definitions logical equivalences or material equivalences?

Question

Lets consider representing logical equivalence using the symbol $\equiv$ and material equivalence with the symbol $\longleftrightarrow$.

I know that the formulas $P$ and $Q$ are logically equivalent if and only if the statement of their material equivalence $P \longleftrightarrow Q$ is a tautology.

My question is: When mathematicians define something, which of the equivalences is being used. For example, when defining the limit of a function $f$ we can write:

Let $I$ be an open interval containing $c$, and let $f$ be a function defined on $I$, except possibly at $c$. The limit of $f(x)$, as $x$ approaches $c$, is $L$, denoted by $$ \lim_{x\rightarrow c} f(x) = L, $$

means that given any $\epsilon>0$, there exists $\delta>0$ such that for all $x \neq c$, if $|x−c| < \delta $, then $|f(x)−L| < \epsilon$.

If we translate this to symbols, which one the correct?

$\lim_{x\rightarrow c} f(x) = L \longleftrightarrow \forall \, \epsilon > 0, \exists \, \delta > 0 \; \textrm{s.t.} \; \forall x, \;0<|x - c| < \delta \longrightarrow |f(x) - L| < \epsilon .\text{)}$

or

$\lim_{x\rightarrow c} f(x) = L \equiv \forall \, \epsilon > 0, \exists \, \delta > 0 \; \textrm{s.t.} \;\forall x, \;0<|x - c| < \delta \longrightarrow |f(x) - L| < \epsilon .\text{)}$

This question arose when reading the book Discrete Mathematics with Applications by Susanna S. Epp, because the author defines both $\equiv$ and $\longleftrightarrow$, but then uses $\iff$ (which is never defined in the book) when writing definitions.

EDIT After progressing through the book, I found out that the author does indeed define the symbol, it just happens to be the case that in my edition of the book she uses it before writing its definition. She uses the notation $P(x) \iff Q(x)$ to mean $\forall x, P(x) \longleftrightarrow Q(x)$. When writing definitions she writes them in English and then restates them symbolically (where $\iff$ might be used). Nonetheless both the answers given by @Stinking Bishop and @ryang are valid and useful.

Answer 1

It is neither, and that is why it is just good that your book uses a third symbol.

The definition introduces a new symbol. That symbol appears on one (left) side of the equivalence sign. On the other (right) side of the equivalence sign, you write what this new symbol represents. This is not about what is true, or what can be derived within your language. Instead, it is extending the language itself.

In the following, I am happy to use the symbol $\Leftrightarrow$ for this “equivalence by definition”. You may also encounter something like $\overset{\text{def}}{\iff}$.

So, if you write:

$$\lim_{x\to c}f(x)=L\Leftrightarrow(\forall\epsilon>0)(\exists\delta>0)(\forall x)|x-c|<\delta\rightarrow|f(x)-L|<\epsilon$$

this means that, wherever you see, in the subsequent text, the symbol $\lim_{x\to c}f(x)=L$ (for some $f,c,L$), you can replace this with the whole sentence on the right of the symbol $\Leftrightarrow$.

The symbol $\Leftrightarrow$ therefore does not carry any of the meanings associated with $\equiv$ (“you can derive the left side from the right side and vice versa”) or with $\leftrightarrow$ (“whenever the left side is true, the right side is true and vice versa”). It is a purely syntactical, almost "search and replace" tool. However, once you’ve had the definition stated, it immediately implies both of those equivalences. Namely:

$$\lim_{x\to c}f(x)=L\leftrightarrow(\forall\epsilon>0)(\exists\delta>0)(\forall x)|x-c|<\delta\rightarrow|f(x)-L|<\epsilon$$

means (by the virtue of “whenever you see... replace it with...”) the following thing:

$$(\forall\epsilon>0)(\exists\delta>0)(\forall x)|x-c|<\delta\rightarrow|f(x)-L|<\epsilon\leftrightarrow(\forall\epsilon>0)(\exists\delta>0)(\forall x)|x-c|<\delta\rightarrow|f(x)-L|<\epsilon$$

which is obviously true (even if you don’t know a thing about mathematical analysis) - simply because the statements on the left and on the right side of $\leftrightarrow$ have become identical.

The same logic applies to $\equiv$ too.

Answer 2

1. A logical equivalence is something like $$\lnot\forall\epsilon{>}0 \; \exists\delta{>}0 \; \forall x{\in}D \;\big(0<|x - c| < \delta \implies |f(x) - L| < \epsilon \big)\\≡\\ \exists\epsilon{>}0 \; \forall\delta{>}0 \; \exists x{\in}D \big(0<|x - c| < \delta \quad\text{and}\quad |f(x) - L| \ge \epsilon \big),$$ which a mathematical definition is not.

   (As suggested by Stinking Bishop, ≡ is also used to write mathematical identities like $$1-\sin^2x\equiv \cos^2x,$$ but these are not logical equivalences either.)

2. Neither is a mathematical definition a material biconditional, since it isn't a truth function.

3. The equivalence statement $$\lim_{x\to c} f(x) = L \iff\forall\epsilon{>}0 \; \exists\delta{>}0 \; \forall x{\in}D \;\big(0<|x - c| < \delta \implies |f(x) - L| < \epsilon \big)$$ can mean either mathematical equivalence (when invoking, in an argument, the definition of continuity) or equivalence by definition; the latter can be specifically written as $$\lim_{x\to c} f(x) = L \overset{\text{def}}{\iff}\forall\epsilon{>}0 \; \exists\delta{>}0 \; \forall x{\in}D\; \big(0<|x - c| < \delta \implies |f(x) - L| < \epsilon \big).$$

4. Is there a difference between $\overset{\text{def}}{=}$ and $\overset{\text{def}}{\iff} $ ?

   $\::=\:$ and $\:\overset{\text{def}}=\:$ and $\overset{\text{def}}{\iff}$ informally and interchangeably signify the assignment of a value or denotation. If you are being fastidious, perhaps use the first two for subatomic logic formulae (e.g. mathematical expressions), and the third for propositional functions and propositions.

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Addendum

Do you know of any books where mathematical equivalence is discussed and compared to the others? Are you saying that $\iff$ is a symbol of mathematics just like $=$ or $\leq $ or $>$ versus symbols at the logical level such as $\land$ or $\lnot $ versus at the metalogical level like $\equiv$?

In mathematics, ≡ might mean that two expressions are universally equal (i.e., an equation is universally true), while ⟺ might mean that two statements are universally equivalent. As such, they are indeed mathematics symbols.

By “mathematical equivalence”, I mean an equivalence under the assumption of mathematical axioms/definitions.

Logical equivalence of sentences (also ≡), on the other hand, is a stronger concept.

The material biconditional ⟷ is a logic symbol.