TITLE: Difference between biconditional and logical equivalence
QUESTION [8 upvotes]: A biconditional can also be stated as "P is equivalent to Q," whereas a logical equivalence can also be stated as "P is logically equivalent to Q." I never understood intuitively what the difference between them was. I get that biconditional is true when both P and Q are true or false, and logical equivalence means P and Q have all of the same possible truth values, but I cannot grasp intuitively what they really mean and what the difference between them is.

REPLY [2 votes]: Think of the material biconditional $\LARGE\leftrightarrow$ as a logical operation on two sentences rather than as equivalence. Unlike the latter, it is not making an assertion nor in a metalangauge.

The statement $\Large P\equiv Q$ asserts the logical equivalence of $P$ and $Q.$ It means that $\large P\leftrightarrow Q$ is a (logical) validity, i.e., is true regardless of interpretation.
What's the difference between $↔$ and $≡$ ?

The statement $\Large P {\iff} Q$ asserts the (not necessarily logical) equivalence of $P$ and $Q.$ This means that that $\large P \leftrightarrow Q$ is either analytically (perhaps in an axiom system) or synthetically true , i.e., that $P$ and $Q$ have the same truth value in the given interpretation.
$\Large P {\iff} Q$ can be read as ‘$P$ is true if and only if $Q$ is true’ or ‘$P$ being true is a necessary and sufficient condition for $Q$ to be true’.


P.S. Symbolic logic is an area rife with conflicting notation, terminology and even notions; my understanding is eclectically evolving.