Distinction between equality, logical equivalence and biconditionality.

Solution 1:

1. The identity sign is standardly used for a two-place relation, and to get a well formed sentence, a two-place relation needs to be combined with two terms (referring expressions denoting objects), not two propositions. So $P = Q$ is ill-formed in a standard syntax where $P$, $Q$ are propositional variables.

2. Conventions differ. One common convention is that $\leftrightarrow$ belongs to the formal language of propositional or predicate logic (to express "if and only if"), while $\Leftrightarrow$ is shorthand added to mathematician's English, and means "entails and is entailed by" or "is logically equivalent to".

3. Conventions differ. But perhaps most common is that $\equiv$ belongs to the formal language of propositional or predicate logic (to express "if and only if") — so is just a stylistic variant of $\leftrightarrow$.

4. "Compound statements are like functions that deal with truth values rather than numerical values". Applying the square of function to a number gives us a number. Applying the logical operator $\neg$ to the proposition $P$ (grass is blue) gives us another proposition $\neg P$ (grass is not blue). True, the compound proposition has a truth-value, but it isn't happy to say that it deals with a truth-value (it deals with the colour of grass!). But to be sure, the logical operator is such that the truth-value of $P$ fixes the truth-value of $\neg P$ — so, as they say, it is truth-functional.