In most common situations, “x = y” means that x and y are the same thing. However, I’m going to look at it from a new perspective: “x” and “y” are clearly different-looking symbols. Furthermore, statements are always said with a purpose. Therefore, there is some reason why I want x and y to be the same, or, that I want y to replace x, or vice-versa.

“x = y” means “x = y **and-or** x”, which means “x = y **or** x **and** *anti-*x.” (anti-x is used to represent “the former x.”) I know it's weird, but basically what this means is: If I say something like "x and y are the same thing" then I'm necessarily making a claim that goes against what our language uses to represent these two things anyway. However, if so, we don't have the "moral high-ground" to simply demand that we replace one with the other entirely in common parlance.

Therefore, x can either be *both *x and y, and called either x or y in different situations, perhaps, or, x and y together (where the fact that x is y is remembered and pointed out in the context of most if not all situations). However, we do need something to represent the counter-claim, too. If we are making a new symbol for x, and proposing its usage along with x, then we also need to pay for that with a new symbol for the counter-proposal, named “anti-x.”

Anti-x means both "against this proposal itself, x = y and-or x" as well as "opposite to x", i.e., x on the right does not refer to what's on the left, which is x, by default. So, everything to the right of x with the = sign as the pivot is the proposal, and x is the symbol used to represent the proposal. This is a self-referential proposal, which is not to be taken as the same thing as a self-referential equation.

So anti-x must mean "only whatever the symbol, literally the drawing 'x' contains within it or with it already, up until this point here, including all of the emotional baggage." It wouldn't contain any more than whatever the best-supported amount of emotional baggage was, which is usually, but not always, to the left of the equal sign (and no further).

But unfortunately, even this is not enough. It is the same as this except literally before the equal sign, **in time**, too. Now the weird part is this: The counter-proposal is the *status-quo*, meaning to keep things the way they are. We have to change the system while respecting it, too. So x cannot *just *be y. And therefore, the least anti-x can be is x, but earlie(r/st) in time.

This means we can't really write it on the same line, since it's not really happening at the exact same time, as "and" would normally mean. That being said, x = x and-or y is sort of the same proposal as x = y and-or x, with the former one being a lot closer to the anti-x than x is.

The **and-or** construct is necessary, because we typically are more justified in bringing both the new and the old with us, rather than just the new.

I want to point out too, cautiously, as this is a very grave matter: “x = y” in our system is not a function that evaluates to true or false. In fact, our system - though it recognizes such tokens - prefers to use 1 and Y. Y stands for anti-1. It can also stand for “anti-” by itself, when it is placed to the left of something else.

Where do we get **1** from? If you demand that we justify the introduction of the number 1, I can always retort that "respecting the system" would necessarily require using 1 somewhere in my argument. I would also say that x and anti-x (a number or a matrix and its inverse) have generally been said to be equivalent to **1** in most of the literature on the subject. Also, that **1** = **1** and anti-**1**. However, you will see that anti-**1** will have to be **Y**, if we follow my arguments henceforth and heretofore.

It is not enough to stop at "1 will be 1 in some contexts, and anti-1 in other ones." In fact, what we are really doing is saying that we are going to replace x not with y, but with something in-between x and y. Therefore, Y must stand for the actual new thing, the final product, as well as the proposal. But, Y is not the actual new thing yet, it just stands for it. Y by itself can also stand for Y[ ] or “anti-blank,” which sort of means the unknown thing before it has taken on a new value.

Why do we like to use the letter Y? Why? Well, because it is a downward facing chevron on top of a line. So that’s why it’s easy to recognize as anti-1 (since 1 is mostly an upwards arrow, or half of one, and also can dually be considered a blank line such as |).

We can't really say x = y. We *can *say x = x y (or x and y). Then anti-x represents the "well if that's too much for you even still, then what should "anti-x" stand for, if not your argument?”

In that case, anti-x = [] -> y. This means anti-x should be anywhere from nothing at all all the way up to y. That way, x anti-x acts as a slider, where you can have x scale from x (keeping it the same), to an x y (or whatever the merger of these two things are).

x anti-x = x [] → x y.

This means the same thing as:

x yx = x [] → x y. (because anti-x is the same as yx, given my definition of y).

You can now see why I’ve chosen y to mean “anti-”, which is primarily just because of my choice of x and y as my first two symbols.

Here are some intuitions in bullet-point form:

x = y is the aggressive assertion. (I want to fire you.)

x = x and-or y is the first restatement. (I am going to hire someone new, because you're under, but if you perform well enough and we can afford it, we'll keep both of you.)

x = y and-or x is the first challenge. (Wouldn't it be cheaper to just keep me, but hire the extra if we can?)

Also, if you’re paying attention, you may have deduced also that:

= = or (equals means or).

= or or (you could use or, or, =).

"=" implies they are the same, which they never are. "Or" implies that one could be better than the other in a similar context. Still rife with political implications, but arguably a little more honest. "And" is certainly much better, undeniably more benevolent, as well as sure to be more politically acceptable. If you ever showed anyone our true budget / coffers, which I do have on hand, you'll find it's quite affordable / in your best interests, I think.

I think I'll settle for and-or, though. I don't have to learn new skills unless they interest me personally. In fact, Y itself simply *means *and-or (as well as or by itself), and I just needed to learn that new concept.

I’ll leave it here for now.