# Are mathematical definitions logical equivalences or material equivalences?

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.

• 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 $>$ vs symbols at the logical level such as $\land$ or $\lnot$ vs at the metalogical level like $\equiv$? Jun 4, 2022 at 11:33
• I've expanded my initial comments into an answer below. A small note: your proposed formal definition of limit has omitted "for all x(" immediately after the "s.t.". Jun 5, 2022 at 9:58
• I guess I'll try it here for now, as I'm not comfortable with the question yet: Assuming that $P(x) \Leftrightarrow Q(x)$ means that every element in the truth set of $P(x)$ is in the truth set of $Q(x)$, or equivalently, $\forall x, P(x) \leftrightarrow Q(x)$. Also, considering that S. Epp doesn't seem to use different symbols to differentiate between a definition and an invocation of a definition. Is it possible to write a completely symbolic definition/invocation of the limit of a function using only the symbols defined by S.Epp? And if so, how? Jul 6, 2022 at 15:45
• Oh, then that means that I still don't understand what mathematical equivalence is... Jul 6, 2022 at 16:06
• I'll try and re-read. If you know of references that touch the topic (difference between equivalences, I guess) while not being too formal could you share? Sometimes continuous text, as opposed to SE posts, seems to help me. I found this book, that mentions meta-statements. But he doesn't mention mathematical equivalence, and besides that I like to read various sources. Jul 6, 2022 at 17:08

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.

• Thanks. Is there a difference between $\overset{\text{def}}{=}$ and $\overset{\text{def}}{\iff}$ ? Jun 3, 2022 at 23:31
• "It is ... purely syntactical" -- FWIW, this only takes us so far. For instance if we have some recursively defined predicate like $\text{isEven}(n) \Leftrightarrow n \in \mathbb N \land (n = 0 \lor \text{isEven}(-n) \lor \text{isEven}(n - 2)$, then suddenly we're doing a whole lot of work in the metalanguage Jun 4, 2022 at 4:24
• @Quelklef Good point. We often define things as "the unique $x$ with a given property". So $\text{isEven}\overset{\text{def}}{=}\text{unique } R\subseteq\mathbb N\text{ s.t. }n\in R\leftrightarrow n=0\lor -n\in R\lor n-2\in R$, with $\overset{\text{def}}{=}$ rather than $\overset{\text{def}}{\iff}$. It is as if we are saying $\text{isEven}=R\overset{\text{def}}{\iff}R\subseteq\mathbb N\land(\forall n)n\in R\leftrightarrow(n=0\lor -n\in R\lor n-2\in R)$. Note the same trick can be applied to $\lim$: $\lim_{x\to c}f(x)\overset{\text{def}}{=}\text{the unique }L\text{ s.t.}\lim_{x\to c}f(x)=L$.
– user700480
Jun 4, 2022 at 7:41
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.

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.

• can I suggest that you edit your answer to explain that main connective convention that justifies the use of ⟹? Feb 15 at 13:20
• @fire-bee → ⇒ ⊨ ⊢ Feb 15 at 14:53
• I still find it confusing that you can use two meta symbols (⟺ and ⟹) in the equivalence statement, as opposed to (⟺ and →) like Stinkin Bishop does. How would you read each version, and why do you feel that yours is more suitable? May 19 at 11:03
• May 19 at 21:32