Clarification on logical equivalence, bi-conditionals, and operators

Question

What are the differences between $<-->$ vs $\iff$ vs $\equiv$ in terms of biconditionals and logical equivalence?

Kindly please let me know :)

Answer 1

There is a difference between material equivalence and logical equivalence.

Material equivalence is a truth-functional operator and as a symbol it is part of logical expressions. So it is a logic symbol.

On the other hand, logical equivalence is a relationship that holds between two logic expressions: it says something about those two expressions. As such, it is a meta-logic symbol.

Unfortunately there is no super strict standard as to which symbol is used for which.

Indeed, I have seen all those three symbols being used for the truth-functional operator as well as for the meta-logical relationship. Confusingly, some texts will even use the very same symbol for both concepts.

So, context of their usage should tell you how the author/text uses it.

But the most important thing is to understand the difference between these two ideas.

Answer 2

This is likely to be context dependent. Some authors will use $\implies$ for implication while others use $\rightarrow$. I have seen one is being used as a logical operator with the other being used as shorthand for "therefore". However this is not standard. Without further context, this question is unlikely to be answerable.