I'm learning about the two truth-functional connectives "material conditional" and "material biconditional".
I came across this particular snippet in a book:
An important fact about the biconditional symbol is that two sentences P and Q are logically equivalent if and only if the biconditional formed from them, P ↔ Q, is a logical truth. Another way of putting this is to say that P ⟺ Q is true if and only if the first-order logic sentence P ↔ Q is logically necessary.
This is a little confusing, because the biconditional symbol represents the English expression "if and only if", and this expression is used to express what it means to be logically equivalent in terms of a biconditional relationship.
Here is an example with one of the DeMorgan laws:
My doubt is what exactly does this mean: "P ⟺ Q is true if and only if the first-order logic sentence P ↔ Q is logically necessary".
Does this English sentence also represent a biconditional relationship? If so, how do I show it in a truth table?
It seems to me that there is a new statement "P ↔ Q is logically necessary", which is true always because P ↔ Q is true always. And there is another statement "P is logically equivalent to Q", which is always true because as P is logically equivalent to Q, this is the case always.
Therefore, both statements are actually tautologies, so they follow from each other; in other words "P ↔ Q is logically necessary" iff "P is logically equivalent to Q".