TITLE: Are mathematical definitions logical equivalences or material equivalences?
QUESTION [5 upvotes]: 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?
$\lim_{x\rightarrow c} f(x) = L \longleftrightarrow \forall \, \epsilon > 0, \exists \, \delta > 0 \; s.t. \;0<|x - c| < \delta \longrightarrow |f(x) - L| < \epsilon .\text{)}$
or
$\lim_{x\rightarrow c} f(x) = L \equiv \forall \, \epsilon > 0, \exists \, \delta > 0 \; s.t. \;0<|x - c| < \delta \longrightarrow |f(x) - L| < \epsilon .\text{)}$
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.

REPLY [3 votes]: A logical equivalence is something like $$\forall\epsilon{>}0 \;    \exists\delta{>}0 \; \big(0<|x - c| < \delta \longrightarrow |f(x) -    L| < \epsilon \big)\\≡\\     \forall\delta{>}0 \;0<|x - c| < \delta    \longrightarrow \forall\epsilon{>}0 \; |f(x) - L| < \epsilon,$$    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.)

Since a mathematical definition is a meta sentence (mathematics is a    metalanguage), it isn't a material biconditional either.

The equivalence statement $$\lim_{x\to c} f(x) = L    \iff\forall\epsilon{>}0 \; \exists\delta{>}0 \; \big(0<|x - c| <    \delta \longrightarrow |f(x) - L| < \epsilon \big)$$ 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 $$\lim_{x\to c} f(x) = L    \overset{\text{def}}{\iff}\forall\epsilon{>}0 \; \exists\delta{>}0    \; \big(0<|x - c| < \delta \longrightarrow |f(x) - L| < \epsilon    \big)$$ or as $$\lim_{x\to c} f(x) = L    \overset{\text{def}}{\iff}\forall\epsilon{>}0 \; \exists\delta{>}0    \; \big(0<|x - c| < \delta \implies |f(x) - L| < \epsilon    \big).$$



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

$\overset{\text{def}}{\iff}$ connects statements while $\::=\:$ or $\:\overset{\text{def}}=\:$ connects logical terms or mathematical expressions.




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 equivalent, while ⟺ might mean that two statements are universally equivalent. They're both in the metalanguage, and are neither logic nor arithmetic symbols. By “mathematical equivalence”, I mean an equivalence under the assumption of mathematical axioms and definitions.
Logical equivalence (also ≡), on the other hand, is a stronger concept.
The material biconditional ⟷ is a symbol in the object language.