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  1. I know that the logical statement $p \leftrightarrow q$ is defined using its truth table, shown below. It basically just asserts that if $p$ is true, so is $q$, and vice versa.

    enter image description here

  2. And it's my understanding that $p \Leftrightarrow q$ is the same as $p \leftrightarrow q$, except it also comes with 100% guarantee that it's truth value is always T. I.e., $p \Leftrightarrow q$ cannot be an untrue statement, otherwise we shouldn't be using $\Leftrightarrow$. Is my understanding correct?

  3. So, if I wanted to define the limit of a function, could I write $$\left\{ L=\lim_{x \to a} f(x)\right\} \Longleftrightarrow \left\{(\forall \varepsilon \in \mathbb{R}^+) (\exists\, \delta \in \mathbb{R}^+) \text{ such that} \\ 0<|x-a|<\delta \implies |f(x) - L| < \varepsilon\right\}$$ or should I use $\equiv$ instead of $\Longleftrightarrow$? (There doesn't seem to be any difference between the two symbols to me—except that $\equiv$ also carries some connotation of definition.)

ryang
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Anis Manuchehri-Ramirez
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    For relevant commentary: https://math.stackexchange.com/questions/500644 – C. Caruvana Aug 25 '24 at 12:35
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    The real question for each of symbols such as $\leftrightarrow,,\iff,,=,,\equiv$ is whether they intend "accidental" or "necessary" equality of truth values. There are so many symbols each meaning is represented in multiple ways, which hopefully don't overlap. – J.G. Aug 25 '24 at 12:37
  • Regarding Question 3. $\quad$Regarding Question 2; in mathematics, ≡ is seldom used to connect statements whereas P⇔Q (which could well be a false claim) just means that P and Q are mathematically equivalent statements (equivalent under the agreed mathematical definitions). – ryang Aug 25 '24 at 15:34
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    Unfortunately, the $\leftrightarrow$, $\Leftrightarrow$ (and also the $\equiv$) can be used different between different texts. But the important distinction to keep in mind is between material equivalence and logical equivalence. Material equivalence expresses that two statements have the same truth-value given some world or domain. Many (most?) texts use $\leftrightarrow$ for this, but some will use any of the other two. Logical equivalence expresses that two statements have the same truth-value in any world. For this, most texts use $\Leftrightarrow$, but again, not all. – Bram28 Aug 26 '24 at 15:05
  • The way you use them in your post is compatible with what I just said: $\leftrightarrow$ is a logical symbol expressing material equivalence, and the truth-table you gave is showing how $\leftrightarrow$ is a truth-functional operator, meaning that the truth-value of $p \leftrightarrow q$ is a function of the truth-values of $p$ and $q$: once you know the truth-value of $p$ and $q$, you know the value of $p \leftrightarrow q$. So in that sense $\leftrightarrow$ is like $\land$, $\neg$, etc. – Bram28 Aug 26 '24 at 15:11
  • The $\Leftrightarrow$, however, is quite different. It is a meta-logic symbol, that says something about the truth-conditions about two logic statements: namely that these two statement always have the same truth-value, no matter what domain you evaluate them in. Also, $\Leftrightarrow$ is not truth-functional: for example, just because I know that $p$ and $q$ are both true (as evaluated in some specific domain, e.g. mathematics), does not tell me that $p \Leftrightarrow q$ is true: maybe in some other world $p$ is true but $q$ is false. – Bram28 Aug 26 '24 at 15:14
  • As far as $\equiv$ goes: that one is actually the one that is used most inconsistently (and thus confusingly) between different texts. Some use it to capture material equivalence, while others use it to capture logical equivalence. – Bram28 Aug 26 '24 at 15:19
  • Now, in Answer to your specific questions. 1: yes. But again, with the understanding that the truth-values of $p$ and $q$, and $p \leftrightarrow q$ are all relative to some domain. So, $p \leftrightarrow q$ does not express that *whenever $p$ is true, $q$ is also true (and vice versa), but simply that that is true for some specific domain .. like mathematics. – Bram28 Aug 26 '24 at 15:27
  • 2: It is certainly not that $p \Leftrightarrow q$ is always true. It can certainly be false. Indeed, if $p$ and $q$ are simply atomic variables of our logic language, then $p \Leftrightarrow q$ is false, because it is possible for $p$ to be true and $q$ to be false. In fact, not being a logic statement (but rather a meta-logic statement), I would avoid saying that $p \Leftrightarrow q$ has a truth-value like $T$ (or $F$) at all. It is simply true (or false). – Bram28 Aug 26 '24 at 15:33
  • So what does it mean that $p \Leftrightarrow q$ is true? It means that $p$ and $q$ always have the same truth-value. For example, $p \land q$ and $q \land p$ have the same truth-value, no matter what world or domain you evaluate those statements in. So, we can say that $p \land q \Leftrightarrow q \land p$. Interestingly, it is also the case that $p \leftrightarrow q \Leftrightarrow q \leftrightarrow p$. If you understand that, you understand the difference between $\rightarrow$ and $\Leftrightarrow$. – Bram28 Aug 26 '24 at 15:35
  • 3: So given that you are dealing with 2 statements that are to be interpreted as being about the specific world of mathematics, I would use material equivalence ($\leftrightarrow$), rather than logical equivalence ($\leftrightarrow$). But, many texts do use $\Leftrightarrow$, and say that the $\Leftrightarrow$ expresses mathematical equivalence (which is really just material equivalence given the domain of mathematics). – Bram28 Aug 26 '24 at 15:39

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