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I'm studying real analysis from Terence Tao's book, Analysis 1, and was familiarizing myself with mathematical logic that Tao explains in the appendix. In it, he covers the biconditional, or "if and only if" statements. From what I understand, a biconditional is only true when both side are held to be true, or "logically equivalent". The examples he gives as a biconditional which evaluates to true and one that evaluates to false were:

if $x$ is a real number, then the statement “$x = 3$ if and only if $2x = 6$” is true: this means that whenever $x = 3$ is true, then $2x = 6$ is true, and whenever $2x = 6$ is true, then $x = 3$ is true. On the other hand, the statement “$x = 3$ if and only if $x^2 = 9$” is false; while it is true that whenever $x = 3$ is true, $x^2 = 9$ is also true, it is not the case that whenever $x^2 = 9$ is true, that $x = 3$ is also automatically true

From what I see, these biconditional statements contain statements that seem to be logically related, or logically relevant to each other: by being given that $x = 3$, we are then able to assess the truth of the statement $2x = 6$ for example.

My question is: is it necessary for the statements to be logically relevant to each other? For example, if I have the statement "It is sunny today if and only if it is a Tuesday", and I were given that the statements "it is sunny" and "it is a Tuesday" were both true statements, would the biconditional statement hold true, despite the fact that the truth of these statements are determined independently from each other, and have no logical correlation? Is it necessary in a biconditional that each statement be logically related to the other, where each statement holds relevant information which is then utilized to assess the truth of the other statement?

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    $\begingroup$ No, it is not. The statement $\forall x\in\Bbb R\,(x^2\ge 0)\leftrightarrow\forall x\in\Bbb R\exists y\in\Bbb R\,(x<y)$ is true, even though the two component statements have nothing to do with each other, simply because they are both true. $\endgroup$ Commented Jun 15, 2020 at 2:49
  • $\begingroup$ @MattAPelto actually it does follow. You can write $T\land S$ for it's sunny and Tuesday, and then use an OR introduction rule to derive $T\land S\lor (\lnot T)\land (\lnot S)$, which is equivalent to the biconditional. $\endgroup$
    – user400188
    Commented Jun 15, 2020 at 3:32

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The biconditional ($\iff$) is a logical connective that is true when both operands are true, or both are false $(0)$, it is false otherwise.

$$(\color{red}{\lnot A}\land\color{red}{\lnot B})\lor(\color{blue}{A}\land \color{blue}{B})\tag{0}$$

The truth table of the biconditional is the following:

\begin{array}{c|cc} &\lnot A&A\\\hline \lnot B&\color{red}{1}&0\\ B&0&\color{blue}{1} \end{array}


Looking at the truth table alone, it is clear that the biconditional is true whenever both its constituents are true. So given that it is Tuesday, and it is sunny, the statement "It is sunny today if and only if it is a Tuesday", is true by virtue of the biconditionals definition $(0)$.

So to answer your question, it is not necessary for the statements to be relevant to another. A subtle point is that the two statements "It is Tuesday" and "It is sunny" are correlated, because they both have the same truth value (today at 3:37 am UTC in my (and possibly your) location at least).

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