Question regarding the notation of $\equiv$ and $\iff$

Question

What difference does the notation in those two terms make? $$ =:⟺∀:(∈\iff∈) $$ $$ =:⟺∀:(∈\equiv∈) $$

Answer 1

The formula can be read as

$X=Y \equiv_{Def}\forall(x) ( x\in X \leftarrow\rightarrow x\in Y)$

( In words : saying that X = Y is logically equivalent ( by definiiton) to saying that X and Y happen to have exactly the same elements).

with

• $\equiv_{Def}$ denoting logical equivalence ( more precisely, equivalence-by-definition)

and

• $\leftarrow\rightarrow$ denoting material equivalence or material bi-implication, which is a truth fonctional operator.

The first relation is a metalogical relation; the second belongs to the object language.

The relation between logical equivalence and material equivalence is as follows : formulae $\phi$ and $\psi$ are logically equivallent when the material conditional $ (\phi\leftarrow\rightarrow\psi)$ is true in all logically possible cases.

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• The iff that is in the middle is a logical equivalence, more precisely, an equivalence-by-definition. equivalence-by-definition works in the same way as ordinary logical equivalence ( that is, two propositions are equivalent just in case it is impossible for them not to have different truth values, whatever possible case is considered).

Note : equivalence is interesting since it allows to substitute the LHS for the RHS ( and vice versa).

• The iff that is on the left side is not a logical equivalence, but a material bi-implication. Two propositions are materially equivalent just in case it factually happens that they have the same truth value, or if you prefer, just in case it factually happens that we do not have the first true and the second false, and reciprocally) .

• Consider this application of the extensionality principle.

Let H be the set of animal that have a heart and K the set of animal that have kidneys.

The material conditional $\forall(x) ( x\in H \leftarrow\rightarrow x\in K)$ is true.

By the definition of set equality, the formula just above is logically equivalent to saying that the two sets are equal, that is, it is a logicall impossibility ( once the definition is set forth) that the material conditional holds while $H=K$ does not, and vice versa.

But this is not to claim that having a heart is logically equivalent to having kidneys. It simply happens factually that , as a matter of fact, the two sets have exactly the same elements, but a world in which an animal has a heart without having kidneys ( or the other way round) is still logically possible.

To put it briefly: set identity is logically equivalent to co-extensionality; but , by itself, coextensionality holds even when it is only factual or contingent. Hence the material conditional on the LHS.

Answer 2

One way of reading it is as a sigle formula, meaning: two sets are equal iff "condition".

In this case, it is an inconsistency to use two different symbols for the same concept: the bi-conditional.

Another reading is to consider it as an "abbreviation", meaning: we write $X=Y$ exactly when "condition" holds.

In this case, there is no benefit from translating the leftmost "iff" with a symbol. The abbreviation is not a formula of the object language but a statement in the meta-language, and it is not necessary to "formalize" it.