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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:

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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".

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    The material conditional and material biconditional don't really represent any ordinary language expression like "if and only if", because of if-then statements in ordinary language often contain further implications beyond what can be read off a truth table, as discussed in the SEP article on the indicative conditional of ordinary language. For example, if-then statements may express claims about counterfactuals, like "if I had eaten lunch today then I wouldn't be so hungry now".
    – Hypnosifl
    Commented Jan 12, 2022 at 0:19
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    For any arbitrary pair of propositions P and Q that are either both true or both false, P<->Q is true, but in ordinary language it wouldn't make much sense to say something like "dogs have four legs if and only if Albany is the capital of New York state" even though the corresponding biconditional must be true given that both propositions are individually true.
    – Hypnosifl
    Commented Jan 12, 2022 at 0:19
  • Compare the two different definitions of Biconditional connective and Logical equivalence
    – Mauro ALLEGRANZA
    Commented Jan 12, 2022 at 8:10
  • Indeed @Hypnosifl you are addressing the main question I have which is the use of the expression "if and only if" in English when speaking of a biconditional which represents a relationship that is described as "if and only if".
    – xoux
    Commented Jan 12, 2022 at 23:04

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The material biconditional "P ↔ Q" expresses only that P and Q have the same truth value. It does not express logical equivalence, which is a much stronger relationship. Logical equivalence can be understood syntactically as P and Q are inter-derivable, or semantically as every model of P is a model of Q and vice versa. If your book is using "P ⟺ Q" to signify that P and Q are inter-derivable, then it is correct to say that "P ⟺ Q" is true if and only if "P ↔ Q" is a logical truth, or if you prefer, logically necessary.

If we use "P ⊢ Q" in the standard way to mean P proves Q, or Q is derivable from P, then we can use the deduction theorem to move from "P ⊢ Q" to "⊢ P → Q", and also from "Q ⊢ P" to "⊢ Q → P". Hence, by combining both, we can get from "P ⟺ Q" to "⊢ P ↔ Q". Conversely, we can use the rule of modus ponens to move from "⊢ P → Q" to "P ⊢ Q" and from "⊢ Q → P" to "Q ⊢ P". Again, by combining these we can get from "⊢ P ↔ Q" to "P ⟺ Q". So, taking the two together, we have that "P ⟺ Q" if and only if "⊢ P ↔ Q", which states formally that P and Q are inter-derivable, and thus logically equivalent, iff "P ↔ Q" is a theorem, or logical truth.

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