What are the differences between logical equivalence and "if and only if"?

Asked Dec 18 '23 at 14:11

Active Dec 18 '23 at 14:21

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I am confused. If $q,p$ are logically equivalent ($q\equiv p$) it means that they both have exactly the same truth values.

The connector "if and only if", $q\iff p$ is a true statement when both $q$ and $p$ have the same truth value.

So why is $\equiv$ and $\iff$ used to denote the same thing?
Second question, what is the difference between a logical symbol and a meta-logical symbol?

edited Dec 18 '23 at 14:17

asked Dec 18 '23 at 14:11

mawaior

1

Does this answer your question? is \iff the same as \equiv? When to use which? – soupless Dec 18 '23 at 14:33
The natural language "if and only if" may mean both the sentential connective Logical bi-conditional as well as the relation of Logical equivalence: two notions that are different but strictly linked. – Mauro ALLEGRANZA Dec 18 '23 at 14:46
No, it does not. @soupless – mawaior Dec 18 '23 at 14:46
"The formula q⟺p is a true statement when both q and p have the same truth value." Correct, but the formula does not asserts that they have, while the "meta" expression "$\phi$ and $\psi$ are logical equiv" instead asserts it. – Mauro ALLEGRANZA Dec 18 '23 at 15:26
See also this post with further link and see the post Distinction between language and metalanguage. – Mauro ALLEGRANZA Dec 18 '23 at 15:26
I agree that $q \iff p$ is not necessarily true. So if I use $\equiv$ I assert that both $q,p$ have the same truth value and if I use $\iff$ I don't mean it. Am I right? @MauroALLEGRANZA – mawaior Dec 18 '23 at 16:19
Exactly. If you write formula $\lnot p$ what is its truth-value? Answer: it depends. – Mauro ALLEGRANZA Dec 18 '23 at 16:50

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