Distinction between equality, logical equivalence and biconditionality.

Question

This is my first post here so I'm not sure about the proper etiquette, but I have quite a few questions all pertaining to one underlying concept, and I hope it's okay that I've grouped them all into one post.

I'm completely new to logic, or at the least the academic study of logic (we all make use of some logical concepts in every day life, but it gets much more complicated and abstract than that). I'm taking a Discrete Math course in first year university, and because it's only a one term course I feel like we're rushing through things and not fully explaining each concept. Just enough to get by. We rushed through truth tables and logical connectives in about a week (or 3 hours of lectures and a handful of practice exercises) and then went straight into set theory and set operations and proving relationships between sets. But the properties of set operations require a solid understanding of the properties of logical connectives, which I just don't have because we barely spent any time on them. My professor just tries to get through the material as fast as possible, even if it means cutting corners here and there. That's what it feels like, anyway, maybe I'm just slow on the uptake and not fully connecting things in my mind.

Anyway, what I'm curious about right now is exactly what the following symbols mean and when to use them:

$=$, $\equiv$, $\leftrightarrow$, and $\Leftrightarrow$

Question 1

Up until university (now), $=$ is the only such symbol that I have ever used. My understanding is that $=$ is simply the statement that the two mathematical expressions on either side of $=$ have the same numerical value.

$2=2$ (true)

$4+5=3+6$ (true)

$5 = 3$ (false)

$4\neq7$ (true)

$4x-y=6x+3y$ (only true or false depending on the values assignments of x and y)

(That last one is not a statement, right? It's an open sentence? But the equal sign is still asserting the equality of values on either side)

So my question is can the $=$ sign be used in any other context, perhaps between two statements, rather than inside of a statement, in order to make a new compound statement? In general, for two statements P and Q, am I ever allowed to say:

$P=Q$

After all, P and Q do have a value of true or false, and, for instance, if P was true and Q is true, then the above compound statement would be

$T=T$

Which would make the compound statement $P=Q$ true (and if the truth values differed, the compound statement would be false).

Is this allowed?

(EDIT: You know what, I was just proofreading this post before sending it, and I realized that $P=Q$, as I've just described it, makes the same statement as $P\wedge Q$. So I assume that since there is already an operator $\wedge$ that denotes such a relationship, the $=$ sign is not to be used in this way)

Question 2

My professor essentially told us that $\leftrightarrow$ and $\Leftrightarrow$ are interchangeable, but based on the bit of research I've done outside of class, I get the feeling that's not entirely true. Maybe they can be interchanged for the purposes of our very condensed course, but I just don't feel like I'm being told the entire story. That, or I just misunderstood her and she didn't actually say they are interchangeable. Do both symbols represent biconditionality between two statements? Perhaps $\Leftrightarrow$ is reserved for use between two compound statements, whereas $\leftrightarrow$ is reserved for use between two component statements within a compound statement?

Question 3

While my professor didn't come out and say that $\equiv$ and $\Leftrightarrow$ represent the same thing, she seems to use them in very similar situations and I don't know why. My understanding of $\equiv$ is that it represents logical equivalence between two compound statements (ie. the two compound statements being compared will have matching truth values for each possible assignment of truth values to the component statements they consist of). Obviously if two compound statements are logically equivalent, then one being true will result in the truth of the other, and one being false will result in the other being false as well. But is it correct to say that they are biconditional? Do the two compound statements actually depend on each other or do their truth values just happen to coincidence due to logical equivalence?

It seems to me that biconditionality between two compound statements is a different kind of relationship than logical equivalence, even if they happen to be closely related (oh god, I'm talking about relationships between relationships, my head hurts). Am I right in stating that $\Leftrightarrow$ assesses the dependency of statements on each other, and $\equiv$ assesses the equivalence of their truth values? If so, is this distinction important to consider, or can I interchangeably use the two different symbols whenever I want?

Assuming that $\equiv$ and $\Leftrightarrow$ are really distinct and not interchangeable in the way that my professor seems to think they are (I'm still allowing the possibility that I might have misunderstood her), it seems to me that, given two compound statements R and S, the logical equivalence of R and S implies the biconditionality of R and S, and vice versa. Or in other words, the logical equivalence of R and S is biconditional with the biconditionality of R and S. Even if the two relationships are not the same kind of relationship, you can't have one without the other. Is this correct? It almost feels like I'm dealing with the logic behind logic itself. Am I wasting my time trying to understand this?

Question 4

I like to think of compound statements as the logic version of the functions I deal with in Calculus (f(x) and all that jazz), where you input certain values of x (or whatever), and the function spits out a resulting value y (or whatever) based on the operations that take place inside the function. So, in my head, $\equiv$ is the logic version of the mathematical $=$; it means that two different "logic functions" (or compound statements) that take the same inputs will give the same outputs, even though the operations being performed on the inputs in order to achieve the output may be different. In other words, compound statements are like functions that deal with truth values rather than numerical values. Is this a good way to think about it? I always worry that the analogies I create in my mind will only work until I find some example later on that breaks my analogy, and then at that point I'll be so accustomed to my analogy that it will be difficult to think contrary to it.

Recap

1) Is the $=$ sign reserved for use inside of the component statements that might make up a compound statement, or is it acceptable to relate two statements using the $=$ sign?

2) What is the difference between $\leftrightarrow$ and $\Leftrightarrow$, if there is a difference? My current theory (which I've made up entirely on my own) is that $\leftrightarrow$ is used to connect basic statements into higher compound statements, and that $\Leftrightarrow$ is used to make a kind of "metastatement" about two compound statements.

3) What is the difference between $\equiv$ and $\Leftrightarrow$? It seems to me that they assert different relationships between compound statements, but that if the relationship expressed by one is true, then the other has to be true as well. Is it proper to assign truth values to these "metastatements", as I call them, and then compare those metastatements by saying that they, too, are biconditional (thus creating a new, even higher statement asserting a relationship between two compound statements, which themselves consist of relationships between component statements)?

4) Are compound statements analogous to mathematical functions in the way I think they are?

I tried asking these questions to a few other students in my class but they weren't quite sure themselves.

Answer 1

1. The identity sign is standardly used for a two-place relation, and to get a well formed sentence, a two-place relation needs to be combined with two terms (referring expressions denoting objects), not two propositions. So $P = Q$ is ill-formed in a standard syntax where $P$, $Q$ are propositional variables.

2. Conventions differ. One common convention is that $\leftrightarrow$ belongs to the formal language of propositional or predicate logic (to express "if and only if"), while $\Leftrightarrow$ is shorthand added to mathematician's English, and means "entails and is entailed by" or "is logically equivalent to".

3. Conventions differ. But perhaps most common is that $\equiv$ belongs to the formal language of propositional or predicate logic (to express "if and only if") — so is just a stylistic variant of $\leftrightarrow$.

4. "Compound statements are like functions that deal with truth values rather than numerical values". Applying the square of function to a number gives us a number. Applying the logical operator $\neg$ to the proposition $P$ (grass is blue) gives us another proposition $\neg P$ (grass is not blue). True, the compound proposition has a truth-value, but it isn't happy to say that it deals with a truth-value (it deals with the colour of grass!). But to be sure, the logical operator is such that the truth-value of $P$ fixes the truth-value of $\neg P$ — so, as they say, it is truth-functional.