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Tao (Analysis I, 4e, p. 270):

If and only if (iff). If $X$ is a statement, and $Y$ is a statement, we say that “$X$ is true if and only if $Y$ is true”, whenever $X$ is true, $Y$ has to be also, and whenever $Y$ is true, $X$ has to be also (i.e., $X$ and $Y$ are “equally true”). Other ways of saying the same thing are “$X$ and $Y$ are logically equivalent statements”, or “$X$ is true iff $Y$ is true”, or “$X \leftrightarrow Y$”. Thus for instance, 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 (think of what happens when $x = −3$).

Statements that are equally true are also equally false: if $X$ and $Y$ are logically equivalent, and $X$ is false, then $Y$ has to be false also (because if $Y$ were true, then $X$ would also have to be true). Conversely, any two statements which are equally false will also be logically equivalent. Thus for instance $2 + 2 = 5$ if and only if $4 + 4 = 10$.

So (unless I misinterpreted), Tao says that $x=3$ is logically equivalent to $2x=6$. And also, $2+2=5$ is logically equivalent to $4+4=10$.

But this seems to contradict the answers at Is 1+1=2 logically equivalent to 99+1=100?:

The sentence $1+1=2$ is not logically equivalent with $99+1=100$.

No: counterexample: redefine + to mean 'double the product of the inputs', then the left statement is true while the right statement is false.

user182601
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  • They are equivalent if $+$ means $+$ and $2$ and $3$ and $6$ are considered as elements of $\Bbb Q$ or $\Bbb R$. I don't actually think the answers on that question are appropriate given that the OP was just trying to understand logical equivalence. It's a bit like telling a high school physics student their answer is wrong because they failed to include general relativity. – pancini Feb 14 '24 at 05:30
  • @pancini: Perhaps you'd like to add an answer to that Q. – user182601 Feb 14 '24 at 05:33
  • I would but I also see that OP also included "propositional calculus" and "first order logic" tags so I'm not really confident about the context there. – pancini Feb 14 '24 at 05:37
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    If Tao were writing for an audience of model theorist, he'd use the words "materially equivalent" instead of "logically equivalent". The answers to the linked question seem to assume that everyone is a model theorist. In practice, the words "materially equivalent" are rarely used by mathematicians, and the words "logically equivalent" are (ab)used to mean both. – Misha Lavrov Feb 14 '24 at 05:38
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    It might be a poor wording, but Tao definitely doesn't mean "first-order logic equivalent sentences". He simply talks about arithmetic sentences, where all symbols have well defined standard meaning. – freakish Feb 14 '24 at 06:03
  • Logical equivalence: "two statements are said to be logically equivalent if they have the same truth value in every model." – Mauro ALLEGRANZA Feb 14 '24 at 06:50
  • But for Tao, see also page 150: "Then the following two statements are logically equivalent (each one implies the other)." But it is implicit: in the context of the tehory of sequnces. Thus, at most we may say that $1+1=2$ and $99+1=100$ imply each other in arithmetic (not in general). – Mauro ALLEGRANZA Feb 14 '24 at 06:56
  • Thus, we say that in the context of a specific theory, say $\Gamma$, we have that two statments imply each other is $\Gamma \vDash p \leftrightarrow q$. This "sounds" very similar to "logical equiv": if $p,q$ are theorems of theory $\Gamma$, we have that $\Gamma \vDash p$ and $\Gamma \vDash q$ and thus there is no way to find an interpretation satisfying the axioms in $\Gamma$ such that $p$ is false and $q$ is true (or viceversa). – Mauro ALLEGRANZA Feb 14 '24 at 08:19
  • I would say they are arithmetically equivalent ... or maybe even mathematically equivalent, but certainly not logically equivalent. – Bram28 Feb 14 '24 at 14:24

1 Answers1

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Whether or not two statements are logically equivalent depends on what they mean. So "the real number $x$ is equivalent to the real number $3$" is the same statement as "two times the real number $x$ is equivalent to the real number six". There are other ways to interpret both those statements where they are not equivalent, but Tao is talking about real number arithmetic where that is in fact true.

Gary
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Jonathan Beer
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  • So, in real number arithmetic, $1+1=2$ is logically equivalent to $99+1=100$? – user182601 Feb 14 '24 at 05:30
  • I'm not certain, but it sounds reasonable. You'd just have to prove that 1+1=2 iff 99+1=100 which appears to be correct. – Jonathan Beer Feb 14 '24 at 05:31
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    They are materially equivalent, and logically equivalent if you live in a world where both statements are true. It's really only worth further nuance if OP actually wants to study formal logic and such. But I feel like it's worth telling people that you can happily have an entire math career without worrying about these things if desired. – pancini Feb 14 '24 at 05:43