Abstract

The Ogdoad is a process that occurs between two objects or individuals, and returns eight possible perspectives between both objects / individuals. The original individuals have the ability to freely choose to obtain traits / attributes from one another, and doing so will never threaten to cloud nor corrupt their judgement. In other words, whenever there are two parties which need to resolve a disagreement about what is better, both parties will naturally Ogdoad, as long as at least one of them is open to seeing both perspectives. The Ogdoad will provably result in both parties becoming better with respect to each of them simultaneously.

Introduction

Whenever there are disagreements about what is right and wrong, good or bad, it usually happens that disagreeing parties maintain some level of safeguarding against the risk of obtaining what might be considered an "opposing" viewpoint. Generally, there is always a tendency to desire to keep one's preferences whenever there seems to be a possibility of having them altered to resemble someone else's. For example, I might see my preferences as being good (and do at the outset, always), and come across someone else's insistences that we all ought to be having what they're having from now on - and I see their choices as significantly sub-par. I could assume that there is some unseen reason why it would actually be better to prefer their choices, or, it might just be necessary to be them to feel the same way they do. That being said, they might be being extra pushy about this to me, and that indicates that they presumably believe that it is possible for others to adopt their choices.

On the other hand, I could decide to push back - and this would also feel warranted, given that they also presumably feel justified in being pushy about it initially. If they do not push back after that, then I can assume some resolution has occurred. Perhaps they decided to adopt my preferences instead. If they did, and seem happier after that, perhaps my preferences were actually better. I may remember times earlier in my own life when I adopted a new preference - those situations did not exactly mirror this one. I may have learned through experience that when my preferences are actually better, actually trying both preferences freely over multiple occurrences and situations does not result in the initially worse choice to usurp the initially better one. Whenever I did adopt a new preference, the only time there was ever doubt about this being a good idea was whenever I believed that my original preferences would suddenly seem bad if I adopted a new preference. This never occurred - my original preferences stayed where they were, and in the most extreme case, the best preference only becomes second place to a new best preference.

We often assume that the worse-case scenarios are the best choices for analysis, given that they present the largest possibility of uncertainty. Let's refer to this as the Principle Of Maximum Disagreement (POMD).

Definition 1 (Principle of Maximum Disagreement). The assumption at the outset of a scenario involving two or more parties that the situation most ripe for insights to be gained from analysis is the situation in which the parties disagree with one another to the maximum extent possible. The resolution of this situation provides the greatest opportunity for parties to converge on agreement of other principles and shared axioms / beliefs.

The Principle of Maximum Disagreement serves as an initial offering for a starting point, given that if parties choose to disagree with that starting point (or any), they would be fulfilling the POMD. The maximum disagreement presumes that if we are able to solve the most difficult case, or rather solve the most difficult situation to the next-most difficult situation, then the problem becomes simultaneously more-solved as a whole as well as less difficult at each step. Each step requires getting parties to at least partially agree with one-another on something. Therefore, we can all at least agree that we disagree. In the worse case, some or all of us believe that this disagreement must be resolved, we cannot agree to disagree, in other words.

The Ogdoad does not explicitly assume a position of full-subjectivity - objectively superior preferences are possible - but it allows for this to be mutually determined by a superior and an inferior party, without the inferior party "losing" anything. Both the inferior and superior party can assume both of their perspectives simultaneously, allowing them both to witness each other's preference choices, as well as how the other party feels about the ongoing process as well. Each party can witness the other party witnessing their perspective, and vice-versa, which results in an infinitely recurring chain of nested perspectives, up to any desired depth.

The POMD allows us to resolve the situation, "But what if it isn't even possible to adopt another person's perspective even in the slightest?" In that case, we must agree to disagree. But if POMD is assumed, then the other parties have presumably ruled out agreeing-to-disagree. They require parties to adopt some preference to become uniform. Agreeing to disagree is an incrementally better-resolved situation than full POMD: We can either live separately, or engage in combat, to determine the preference resolution state. However, if living separately is not possible (the world isn't big enough for us both), and combat has resolved in a direction that favors the most open-perspectived party, then the other party might request that we return to the full POMD situation again (since combat has resolved in the direction of one party getting their preference as the uniform preference). If the least open-perspectived party wins combat, then either their preference choice becomes uniform, or it does not even in the case where the other party (the open-perspectived party) attempts to adopt the other other preference choice. In either case, we have returned to POMD: In one case, because one party has refused to adopt the other's preference choice, in the other because one party cannot successfully adopt the other's preference choice (the latter's being determined as the worse of the two).

Combat need not be limited to actual physical violence - combat could consist of any demonstration of capabilities from one party to another. That being said, it is likely to be a process that attempts to most directly compare two capabilities on the most equal footing possible, to most accurately judge the "better" of the two, from the perspective of both parties simultaneously. Combat could consist of, e.g., but not be limited to, debate, sports, games, simulated combat, dancing, and fashion. For example, if we've mutually agreed on fashion as our arena, then we have presumably agreed that not only do we wish to impress ourselves, but perhaps more importantly, other people as well. Because there is bound to be disagreement, we may have decided on "overall number of notices" as our score, rather than actually-expressed approvals in the affirmative.

If an open-perspectived party first attempts to adopt an inferior perspective at the insistence of a closed-perspectived party, then the former party will not be able to successfully do so, and will inevitably emit negative responses towards the closed-perspective party, whom may be also attempting to win combat in the arena of social approval. Therefore, they may, if their preferences are inferior enough, lose in the social-approval arena, even if they previously won in the combat arena. The closed-perspectived party will have (by assumption) presumptuously insisted that preference adoption is equivalent to losing combat. They therefore may have also prematurely declared victory over the combat arena, due to the presence of an open-perspectived party. Open-perspectivity is not the same thing as throwing out the old preferences in favor of new ones. As mentioned earlier, the "old" preferences remain exactly the way they were, and the new preferences are inserted in-between the old ones where appropriate, depending on their relative ranking.

Furthermore, it may also be the case that old preferences and new ones occurring simultaneously is an even higher preference than either the previous or new best by themselves. Even if newly acquired preferences are determined to be inferior than the previous set of preferences, it can still be a preference in and of itself to have knowledge of the higher, enclosing set of preferences (union of the two). Therefore, we will assert that it is preferable to be open-perspectived, in general.
Thus, the first new preference that a closed-perspectived individual can obtain is open-perspectivity. This occurs when the closed-perspectived individual realizes that they do not need to sacrifice any of their deeply cherished preferences, nor run the risk of accidentally or mistakenly doing so. The Ogdoad is a naturally occurring process, and occurs even without explicit knowledge of the process or its mechanics. It does not result in either party losing or changing their preferences entirely; None will come to see something they liked before suddenly becoming "bad." On the contrary, there is ample opportunity for them to understand their own preferences better - to obtain additional justification for them.

It is actually more likely that one is likely to see a previously self-doubted or self-restrained preference as more acceptable to have. In other words, one might have held back or restricted themselves from having or utilizing a preference that they would have otherwise allowed themselves to adopt had it not been for external pressures which they had up to that point, chosen to allow to override their own internal judgement. The Ogdoad allowed them to adopt the perspective of someone else who had allowed themselves to adopt that same preference (or a similar one), realizing that the external pressures were not actually legitimate. These external pressures - we assume to be illegitimate - were likely coercive in nature, emanating from the direction of other closed-perspectived individuals, themselves likely still carrying the fear that open-perspectivity could result in the loss of cherished beliefs.

Open-perspectived individuals are more likely than closed-perspectived individuals to view thoughts and beliefs as having inherently more "believability" the more actually-true it is, regardless of the manner in which they are expressed. Closed-perspectived individuals, on the other hand, are more likely than open-perspectived individuals to see thoughts and beliefs as having more "believability" only at the subjective viewpoint of the person it is being expressed to, which closed-perspectived individuals are also more likely to believe have far more variation as well as overall influence over the probability that a given thought will be accepted by someone. Thus, closed-perspectived individuals tend to, contrary to open-perspectived ones, believe that actually-true beliefs can be experienced as worse than actually-false beliefs. This leads to preference uncertainty: One may, like presumably everyone does, wish to only believe true things. Therefore, a belief that is judged as true will feel better to have than one judged as false. But if actually true beliefs can be inherently less preferable to hold, even despite their veracity, then such a belief will feel some combination of good and bad to hold. Therefore, if we still presume the assumptions given in the last few sentences, there would be far more difficulty in arriving at the truth, given that a priori, it is uncertain how "good" a true belief will need to feel via simply being true to override any negative feeling it may by assumption hold.

Open-perspectived individuals may have high self-evaluations of their own ability to determine how true things are - this is why they would feel that more believability would be felt as such, and therefore would be a stronger indicator for truth. It would be better if true things felt better to believe too, and this is also a potentially-true thing, which would be better if true, therefore I believe it for consistency's sake. This is the self-reinforcement principle for open-perspectivity. Closed-perspectived individuals therefore face an extremely up-hill battle when attempting to convince someone to adopt a low self-evaluation of their own ability to determine how true things are. In principle, this actually is not really possible, given the self-reinforcement principle.

Definition 2 (Self-Reinforcement Principle). It would be better if more true beliefs felt better to believe as well. That, too, is a belief which (as it states) would be better if true. Therefore, it is consistent to believe it, and reinforces itself.

Ironically, closed-perspectived individuals - adding to the already steep incline they face - are forced to argue for "closed-perspectivity" in general to an open-perspectived individual, which would require the open-perspectived individual to either be open-minded in order to consider the argument presented to them, or closed-minded in order to not consider the argument presented to them. In other words, it would require the open-minded person to obtain closed-mindedness - and therefore remain the way they are - in order to consider both perspectives. Closed-perspectived individuals would need to adopt open-perspectivity in order to consider the arguments presented to them as well, but they may see this as succumbing to the argument already. That being said, they need to obtain it to some degree to be able to engage at all. Therefore, they are faced with adoption of open-perspectivity to some extent, or to disengage completely. Thus open-perspectivity possesses the advantage in general.

It is difficult to argue to someone that, "You should switch your belief to this one, which I also claim is incompatible with yours (so can't be held simultaneously), and then become closed to new perspectives." Yet, this is indeed what closed-perspectived individuals are inherently poised to argue. Since they believe that beliefs in general can feel bad to believe even when true, holding two beliefs - which might-or-might-not contradict each other to some extent - simultaneously would prefer the better-feeling of the two over time anyway. Presumably, the closed-perspectived believe that holding a belief always carries with it the risk that it could cause someone to believe it by feeling better to believe. Therefore they must insist upon not even considering some beliefs at all. But it is definitely quite a hard ask to insist on switching to the position that they ought not only to adopt this new set of preferences, but also actively exclude all other sets of preferences including the open-perspectivity preference.

Closed-perspectived individuals may also believe in preventing the open-perspectived person from seeing their perspective, too. This may result from choosing to argue for closed-perspectivity in general. But we also offer a scenario that adds an interesting dynamic: That the closed-perspective individual is being deceptive. They may be aware that open-perspectivity would cause their deception to be detected. In this case, the closed-perspective individual must attempt to cause the open-perspectived person to adopt an intentionally deceptive belief. If the open-perspectived person adopts both perspectives, they would both agree that the closed-perspective person's view is worse. So the closed-perspectived person tries to prevent this - but this makes it also harder for them to successfully deceive. If the deception fails, the closed-perspective person may simply retreat to a position of "you may not know why I keep choosing to deceive you nor can you convince me to stop."

The open-perspectived person may then realize that adoption of the other's preferences is not, strictly speaking, possible by design. If the other is keen on antagonizing them, merely for the sake of being antagonistic, then there is nothing to adopt besides switching to a combative stance. Agreeing-to-disagree, in this case, amounts to "okay, I won't convince you to stop. Nor do I care why you keep doing what you do." The closed-perspective person wishes for the other to attempt to convince them to stop, which is why this stance amounts to combat. Given that the closed-perspectived persons beliefs are mutually agreed upon to feel worse, the open-perspectived person's already-held beliefs continue to hold. The closed-perspective person will then be forced to accept this, and the open-perspective position wins out, inevitably.

The closed-perspective preference cannot really be enforced - it can only be held as a belief that would be "better" if true. But the person holding the belief that open-perspectivity either isn't possible or would be worse if true can only hope that this is the case to some degree, weakly. Since it is not strictly speaking, better for closed-perspectivity to be true, nor do closed-perspectived people tend to believe that things that feel better to be true are in fact true in general, they cannot hold to this view very strongly. Therefore, we find that open-perspectivity, and with it the Self-Reinforcement Principle, wins out in general.

The Ogdoad

The Ogdoad begins, of course, with at least two arbitrary objects, put together for comparison. It is not necessary that these objects differ. However, if they do not differ, than we gain a new bit of knowledge from this already: There can be more than one of the same type. Let's say we have two X's: X and X'. We decide to name one of them "X'" to allow us to notice which one is which, although at first, they may have been both "X." Now we have both objects are of the type "X" and that one of them is "X and not X at the same time." We could also say that "X" is the original X (so we keep it without markings) and that X' is an additional X. It could also be said that we are taking the perspective of X and not X', at the outset. So X can be considered to be "us." On the other hand, there is no injunction against taking X' as our perspective, and we will remember this fact throughout this paper.

Keep in mind, as the last point says, that we can always swap our perspective with the other one, and also could have done so at any point in time, all the way from the very beginning. This automatically gives us twice as many possibilities to whatever we have without doing do. Thus, we have X and X' or X' and X. We said that X' was "X and not X at the same time" which is a way of saying that it is an X but not literally which is, in this scenario, The X. We could get away with just saying "not X" instead of saying "X and not X at the same time" because "not X" implies that whatever we are referring to could be assumed to be X (if it looked very similar, let's say), but we wanted to denote that it was "different." So, we can say that X' = X and not X or that X' = not X, and that both or either of these are okay to use. So we could also "substitute" X' for not X and get that X' = X and X'. We like this because X' is clearly a compound symbol that contains an X as the "main" part of it and an optional extra symbol along with it. We could also say that X' = X and '. Since an and is optional (simply placing two things next to each other is equivalent to an and), we can say that ' is a symbol in and of itself, also not an X (but this is obvious visually in this case), although is supposed to appear alongside of one, it is implied.

With X' = X and ', we have introduced the concept of a merger of two objects into one. And with X' = X and X', we have also introduced the concept of a merger with overlap, in which (at least in this case) the overlapping regions are not duplicated. We can have and $X = X$ . We have also implicitly introduced the concept of an anti-merge (a split) and that a merge is an anti-split. But what if we said X' = X and ? This statement could be posed as a question or as an assertion. If posed as a question, the box symbol seems to imply that the answer to the question ought to be placed inside the box, or swapped for the box. If stated as an assertion, the statement seems to insist upon providing it with an explanation - it is not as clear-cut as when the symbols already seem to fit together. In the question form, the answer seems to be $'$ or = '. We can pose a new question with the options we've been given so far: What is X = and '? We are essentially asking what symbol, when merged with a ', becomes an X. Obviously different (and more difficult) than if we had used X'. Given the symbols I have available on the keyboard, I will propose that "Y" is the symbol we are looking for, or at least closest to it. Y looks like an X with one of the legs broken off. This broken off leg can become our tick mark. Therefore, if we merge this tick back with Y, we can rebuild our X.

This operation is slightly different than either a merge or a split. It is asking us to un-merge two symbols from one symbol that has not been previously produced by a merge, by giving us one symbol of the two, and asking us to return the other. So Y is a new symbol, one that behaves a bit like an X (given that it is roughly the same size), and so also follows the relation: Y' = Y and Y'. But also, Y' = X, so Y' = X and Y and Y'. So we can also say that X = X and Y, and that this corresponds to the same operation that says that X' = X and X'. This operation is essentially asking us to take a single symbol as input, find the most significant sub-component of that symbol, and return both the input symbol and the input symbol with the sub-component removed as output. This process is necessary for the Ogdoad in the following way: Suppose we return to our situation in the Introduction, in which we were discussing two parties attempting to "try out" the preferences of one another. A "preference" is a bit like a sub-component of a larger symbol. You can consider yourself a large symbol composed of many preferences, traits and attributes (which themselves determine your preferences, we presume), of which some of these traits and attributes you might be able to imagine are possible to swap and-or replace for different ones. The "larger" of the two sub-symbols becomes your "core" (self you choose not to alter for the time being) and the smaller of the two sub-symbols becomes the "exterior" (something you consider to be not yourself or potentially not yourself for the moment).

"Core" components and "exterior" components do, we expect, have noticeably different characteristics which we will hopefully be able to agree upon, if we are two individuals engaged in this process. An "exterior" component, we might be able to agree, is more easily swappable between us than a "core" component. That being said, we may also disagree upon that, or perhaps disagree on what constitutes a core vs. an exterior component. Note though that is more obviously an "exterior" than a core, however. An enclosed shape or one forming a loop behaves more like an exterior. But also, pieces that can be broken off or attached behave like an exterior as well. Imagine a loop formed around an X and then squeezed in from all around, such that it was intended to be tightened around all the edges. Such a loop, if tightened in as many places as possible, would have one leg around the X which was untightened, and thus from the outside, would appear to have an extra length or spoke attached to it.

An X is wrapped with a loop around it which is tightened such that it forms a tag on the top right corner: $X ⟶ X^{'}$

A wrapped X is unwrapped without untightening the wrapper first, then it is untightened: $X^{'} ⟶ X X^{'} ⟶ X$

Now X and X' have been shown to be different to the extent that the former is a core and the latter is an exterior. We will introduce the term "dual" which is pervasive throughout mathematics to describe the relationship between X and X', and this term will be used in precisely its typical, widely established meaning. Our symbols, typeset in latex, are intended to mean what they look like and vice-versa, so when we take shortcuts to represent what would likely be an animated process, we also have to use symbols to represent that we've made an incremental time-step (such as the arrow). Our justifications for our assumptions are expected to continuously bear themselves out through time. Our major, as yet implicit assumption so far, has been that we expect symbols to mean (and behave) the way they look, otherwise they would not function as well as they do. Fortunately, most of our already available symbols already satisfy this, and we have the opportunity to show why this is formally (since it needn't have already been done, as they would be expected to be more used the better they serve their purpose).

A Y is wrapped with a loop around it which is tightened such that it forms an extra leg: $Y ⟶ X$

A wrapped Y is unwrapped without untightening the wrapper first, then it is untightened: $X ⟶ Y X ⟶ Y$

Using the above, it is not (yet) clear if the original "X" shape had been formed from a tightened loop or not, but we can see that copies made from it only need to wrap around it - if they want to mimic the X as much as possible, they can first be tightened (loosely) around the X, which is then removed, and then the loop is tightened until it is fully closed. Loops that are intended to be removed might be slightly larger, which is why they possess the tag (it may make for easier removal). This introduces the concept of dual-type: There is more than one kind of X, as well as more than one kind of anything, as well as difference inside one type of X as well:

X or Y.
X or X'.
Y or Y'
X or X.
X or XY.
X' or XX'.

Let's ask the question: What would X' look like if unwrapped / split into an XY, rather than an XX'? X', looked at from the perspective of an X, would think: "If I'm an X, then I should be able to split into an XY too." X' is considered a type of X.

$X' ⟶ X X^{'} ⟶ Y X X^{'} ⟶ Y X'$

An X' unwraps first into an XX', then realizes that the first X is a wrapped Y, removes the Y, then re-wraps the inner wrapper to form a YX'.

Similarly, we can ask the dual-question from the X: "If I was actually composed of two different X shapes, what would those look like when separated?" Well, it could be asking two different of the same shape, which in this case would be YY'. However, it is not necessary that all X's actually consist of an inner Y with an outer wrapper which is long enough to produce a full extra leg. It is also possible that the inner X is solid, the the wrapper is fully tightened around it, such that separating them forms two X's which are exactly the same. It is also possible that sections can be broken off and re-attached. If this is allowable as well (and we allow everything, given open-perspectivity), then a thick X (X) can be unwrapped in (at least) two ways, depending on how it was formed:

$X ⟶ X X .$
$X ⟶ Y X^{'} ⟶ Y^{'} X ⟶ X X .$

These two X's are drawn to look the same when separated, but in reality, could differ slightly. For example, the legs may not be the same angles from one another. But these two X's are both considered "X" whilst an X and an X' are "different types of X's." Both of the above situations require one to imagine that there a subtle variations present within an object that are not distinguishable from a distance. In the first case, we have that a wrapped Y and a wrapped X are hard to distinguish from an X alone, since a wrapped Y will look bold over part but not all of it. In the second case, we have that an X and a wrapper formed into the shape of an X are hard to distinguish because this is intended: The wrapper is formed into the shape of an X as closely as it is possible to do so.

We have done a lot of work so far manipulating symbols into different shapes and calling them different types and versions of one another. We are doing this because we are beginning from the context of having different frames of reference. We assume that the way we experience the world is based on who we are. Therefore, things outside of us might appear differently to us because of the way those things are transformed via the exterior layers as they are viewed by our core. To obtain a new preference is to pick a new frame of reference by placing someone else's exterior over ourselves. This exterior, too, is picked via our preferences. One must prefer to have either their own exterior, someone else's, their exterior underneath someone else's, or someone else's exterior underneath their own. This is four possibilities from one core perspective. Adding the other person's corresponding four gives a total of eight possibilities.

The Ogdoad

Me from my perspective.
You from my perspective.
Me from your perspective.
You from your perspective.
You looking at (Me from my perspective).
You looking at (You from my perspective).
Me looking at (Me from your perspective).
Me looking at (You from your perspective).

We use the term "dual" to describe the relationship between "me" and "you" here, and then also to describe the dual-perspectives that arise as a result of it. One can also be self-dual, which is when one is self-reflective. In this case, we use the term "anti-" (introduced in our previous papers). If I am X, my anti-self is $¯ ¯¯¯ ¯ X$ . My anti-self is my previous self, or one time-increment before my current self. When I am self-reflecting, I am considering my self at one time-step before my current self, because my current self is engaged in the self-reflective process (that is not to say one cannot analyze their self-reflective self). One time-increment is whatever the smallest unit of time one chooses to analyze (a decision or action, for example). These may also be called a "moment."

When we consider ourselves an anti-self, then presumably we are open to change - we consider our older versions of ourselves to be "different" or at least potentially different. That being said, we may not be as incentivized to do so unless there was a different anti-self present there to incentivize us (the other self, in this case "you", or X'). The other self, we know, is always going to be another self, whereas our anti-self is going to become us, changed or not. Therefore, we are presented with the opportunity to incentivize X' to be subjected to the same pressures they are subjecting us to. If I already believe in the Self-Reinforcement Principle, for example, I am probably not going to abandon it, even under the greatest pressure to. If X' demands that we abandon the Self-Reinforcement Principle, I see no reason not to demand that they adopt it. If they are pressured to adopt it, they will then need to provide a reason or set of reasons it would be good to not use it - but these reasons will be, by assumption, difficult to adopt due to being a mixture of good or bad. I will also notice that I cannot be entirely sure that they do not believe in the SRP, only that they continue to insist that I abandon it, which is different from what I am doing in a significant way, still. In other words, I can be sure that there is something substantial that we differ in terms of that could be enumerated or circumscribed, but that we may have yet to do so. But I can be hopeful that if we both agree on mutually coming to a shared enumeration or circumscription of some set of describable beliefs, traits or preferences, then the only step remaining after we do that is to test both traits on ourselves (there will be one from each of us, each a member of a type or set of traits which we give a name to).

For example, I could break each of us down to being an "Inner-Compasser" or an "Anti-Inner-Compasser" which are members of the "Inner-Compasser" trait. Inner-Compasser is what I choose to use to define myself, and I believe is already enumerated and circumscribed, and by assumption, you have a problem with this. I can do my best to help you enumerate and circumscribe your disagreement with me, but if you seem to be using my help more than I am using yours, then I note that my preferences seem superior (to us both) because we are using more of them. You may have approached our interaction with the initial set of prejudices that consist of the claim that my beliefs require or demand the "burden of proof" to be taken upon by me, as opposed to you. I may have agreed to undertake the task of justification even if our initial states were based on completely opposing expectations: You expected that demanding proof was a sign to us both that I would find it difficult or impossible to do so, but would still enjoy the process of watching me attempt (and presumably fail) to achieve it, since this would end with me adopting your perspective. I expected that I had already done the work of justifying it fully, I only needed to demonstrate it personally (or provide even more impressive justifications). Furthermore, even if this occurred prior to my own state of deciding that I had enough justification or proof fully developed yet, I could not imagine that adopting the perspective of being anti-self (convinced that my objectives could not be accomplished) would ever seem or feel like the "better" of two preferences. From experience, one learns that even the worst type of perceived failure must always lead to more advanced techniques and methods being used and developed, as well as the vast majority of all failures being followed by success.

A
not B
not A
B
A and not A
B and not B
A and not A
B and not B

The above is filling in "The Ogdoad" with two individuals, A and B, who are assumed to maximally disagree. Here we write "not A" or "not B" to describe the state of seeing A or B with dislike, respectively. When either one of them chooses to see themselves from the other's perspective, they will see A and not A or B and not B. (A and not A) will be perceived as better than not A by itself; Likewise, B and not B will be perceived as better than not B. If A chooses to adopt A and not A for themselves instead of A, then not A becomes not A and not not A. Call this new not A, A'. Then likewise B' for the new not B. So A can choose to become AA' and B to become BB' if they so desire. In expanded form, A = A and not A and not not A, which translates to: "Take the part of A that is agreed upon as good, remove the part that is agreed upon as bad, and create room for a piece of A that is desired but does not yet exist." When A looks at themselves via their own view as well as B's simultaneously, they will not see that they are wholly disliked or bad (and thus experience an information-less blend of pure neutral across any of their attributes), rather, that some pieces of them offend B more than others. If they agree with B's offense, they will add this trait to not A. If they disagree with B's offense, they will add it to not not A. If they disagree with B's offense, they will also switch to looking at B through both perspectives. Then B possesses the burden of proof to justify that chosen piece's being labeled not A. B now seems to have an attribute that makes them offensive (they become dislikable in the presence of traits or attributes possessed by someone else).

Actually, in our scenario, A and B both initiate the situation from seeing each other as only not A or not B, respectively, and therefore, add no extra information about themselves or the other for why they feel this way, until they see each other from the other's perspective. If A decides that not A is empty (there is nothing about themselves they dislike), then B shifts to become (from A's perspective), B and not A, so that now B has a trait called "not A." Then not B, from A's perspective, becomes B and not not A (if there is nothing else to dislike about B besides the fact that B dislikes A, apparently). A and B are not assumed to inherently contain negations, unless that makes itself clear via their view of the other (or themselves) potentially. Negations must be assumed to exist at the outset, but there will not be unknown negations. Negations are presumed to be the source of disagreement, and therefore the lack of such would be an end to the solution of the problem.

If A agrees that there is something to change about themselves, then this viewpoint, presumably, will be looked upon favorably by B. Otherwise, if they like everything about themselves and wish to change nothing, then B could only obtain a better perspective (one more pleasant to live in) by sharing A's perspective, since A's only adds favorability to B's apparently negative view of A. The only thing that could deter the situation from improving, at this point, is if B was determined to remain closed-perspectived and yet continually engaging with A. "Continually engaging" could mean employing a cold-shoulder, actively-looking-like-not-engaging kind of behavior, intended to cause A to feel negated continuously while attempting to avoid perspective-sharing. This situation is inherently unstable, as more distance between A and B will make A feel less negated, but less distance will cause more perspective-sharing. Furthermore, A possesses the advantage by preferring both more freedom from negation as well as operating more strongly in the perspective-sharing domain.

In the above examples, we have commonly employed symbols such as X and X' and A and B to denote a person which we choose to identify with, who is usually listed first, and a different person (who could be anyone) who we choose not to identify with, who is naturally assigned the second symbol. X' or B commonly act to oppose the person we identify with, but we do not typically imagine ourselves as doing the same, although in theory, one could choose to do that. We postulate that "single-negation" behavior exists but is provably difficult to justify via open-perspectivity. This is behavior that, via the above reasoning, would prove to be a non-desirable trait to have as well as be subjected to. Furthermore, the SRP is a sticky-belief, one that becomes harder to dislodge over time. We speculate that those who obtain the SRP initially become targets for single-negationary behavior, until those who target them face enough double-negation to cease doing so.

One this initial "worse-case" Ogdoad iterates one step, it can proceed to one in which the negations have been replaced with more useful "anti-" and "dual" formulations, which results in more interesting sequences of updates (and expands the number of possibilities). I can begin to reject attempts to initiate perspective-sharing with those who have, in the past, proven to be opposed to perspective-sharing in general, as this will proceed us to the "agree-to-disagree" stage. Furthermore, if I detect resistance to move toward this stage from the other party, I feel more confident that during this stage, the other feels somewhat more strongly that I will be successful than they will. In this stage, our perspective will automatically become shared because we will both be inside of and experience a shared space of some kind in which traits can be visibly tested between us.

Mixture and Re-Combination

We start from the position of two individuals who wish to remain wholly separate as well as inherently "different" on some aspect of themselves, while also having the ability to consider each other members of the same "type" or class of things. We have chosen to use the symbol "X" to denote what this class / type is or looks like. While both individuals are X's, the X's can also visually differ in several ways, e.g., X vs. X, X vs. X, X vs. X', and also combinations of these variations, X', etc. Furthermore, we have introduced the stipulation that while Xs can potentially differ along continuous axes, between two individuals, there will only be two variants available having a specific value along those axes.

We use the symbol "Y" and the presence of it inside of the X or not as the marker for the desired "inherent separator" between the two individuals - a type within a type, in this case, binary. Recall:

A Y is wrapped with a loop around it which is tightened such that it forms an extra leg: $Y ⟶ X$

A wrapped Y is unwrapped without untightening the wrapper first, then it is untightened: $X ⟶ Y X ⟶ Y$

Y is inherently a sub-symbol of X, so this can be shown to be the case via the wrapping / unwrapping of an elastic X "rubber band" around it.

We can start with our two individuals, constructed as so:

$Y ⟶ Y ⟶ X$

$⟶ X ⟶ X ⟶ X'$

Upon realizing that the "rubber bands" are common between them, in the sense that they each can be pinched inwards to create an X shape, as well as stretched, these two individuals can be deconstructed like so:

$X ⟶ Y ⟶ X Y$

$X' ⟶ X ⟶ X X^{'}$

Note that the "Y" individual has a slightly different X (X vs. X). Now, we have some implicit rules defining what constitutes an "individual" here:

The presence or absence of the Y.
Each individual is composed of two "objects."
Each individual is an overall "X" shape.
Rubber bands are squeezed / pinched into X shapes, which can be done around another object or not.
Xs can be merged into a single X.

The last rule is one that we have come up with ourselves in order to "break" the rules somewhat. We obtain justification for this rule in the following way: The rubber band around the Y in the first individual could be swapped for one of the rubber bands from the second individual. If this is done, then we have that a "1/2" rubber-band becomes a "1" rubber-band and vice-versa. If we ask what the "dual" of this transformation is, we get that it is when two rubber-bands are made to form one rubber-band, by placing one inside the other before pinching them into an X shape. This is essentially just the way that the second type of individual is constructed.

XY
XX'
XY
XX'
X'Y
XXX' $⟶$ XX'
XXY $⟶$ XY
XX $⟶$ X

Actually, there are more than eight ways (an infinite number of ways) to continue to construct new individuals. These eight are simply the shortest list of possibilities to define how to continue past these indefinitely. Each of these individuals can be used as input to the same process by placing them in one of the first two positions. We would expect, intuitively, eight possibilities to emerge from:

2x from the presence of the Y or not.
4x from number of ways to choose two halves of one object starting from an initial pair.

Without positing the "rubber-band" construction as I have above, then generalistically, if we imagine that Y is simply an extra possibility on top of all of the rest of the combinations, then we just need a generalistic way to merge two objects in some compatible way. If I have any two objects which I've judged to be the same overall shape but different internally, e.g. color, then if I arbitrarily choose to split both of them in half in the same position for both (it doesn't have to be the exact middle), then the two can be recombined by selecting the left (or top) half from one, then the right (or bottom) half from the other, to give four total possibilities. Each of those halves can be recursively recombined in the same manner.

We've made explicit correspondences between these eight and our original Ogdoad, but what we still need to do is show how these correspondences are meaningful. In other words, how do these mixtures and re-combinations correspond to different perspectives as well as perspectives "through" other perspectives?

We can first make a correspondence between "perspective" and the rule that defines how an X is constructed - in this case, there are two definitions of an X: One with a Y in the center and an X on the outside, and one with an X in the center and an X on the outside. For example, earlier in this section, we asked how an X' could choose to view itself as an X with a Y in the center, and also its dual-question, which is how an X could view itself as two Xs. From this, we obtain that either an X or an X' could choose to view itself as inherently whatever its "core" happens to be. The merger-rule (that two Xs can become one X or one can split to become two) allows for an individual composed of three (or more) objects to view themselves in terms of either type of individual. E.g., a "Y"-type individual could reason that the closest it could be to becoming the other type of individual is to wrap the entire other individual around itself (being then composed of three objects) with the two outer layers becoming its exterior. Then there would be choice about whether or not it wished to keep, remove, or swap its own exterior for the core of the X' individual. Likewise, the X' individual could reason that the closest it could become to being the X individual would be to swap its own core for the exterior of the X individual. This introduces the question of how each individual determines what to consider "themselves" - either the core itself, or instead the rule about which type of individual they are.

It could be both - consider that from X's perspective, when looking at how X' views themselves, X' would be defined by the core X, and so if that X were swapped with another, X' would no longer be X'. If that X wrapped around Y, it would not be X', it would be considered Y, now armed with both Y's and X"s perspectives. From X"s perspective, if looking outward from X, X' may see the Y as offering no more than a fixed location of reference - a signifier for whom is doing the looking, but providing no different information in how it feels to be them. Y makes the point-of-view feel like a unique, separate identity, but not necessarily a different kind of experience altogether. The X's and their combinations contribute to the overall variation in experience, from X"s perspective.

We Can Agree On How To Divide Ourselves

For two conscious individuals who disagree with each other about what is good, they will likely both start with uncertainty about what precisely they disagree on and why, and therefore will both feel better if they can at least agree on how to describe the difference. Any given point at which a line could be drawn to imagine swapping out a piece of themselves for the piece from the other individual is not necessarily assumed to be the one that will switch their preferences completely over to the other set of preferences. But any point which seems like a good point to try will have the characteristics of being a) easy to describe and b) feasible to implement. A feature can be described such that any individual can optionally be described as that feature and the anti-feature simultaneously. Then, four possible individuals can be produced based on the combination of: Feature from A or B and Anti-Feature from A or B. The feature that A and B are separated across have to be chosen in the shared perspective, meaning that it will naturally be something that both A and B feel is relevant. Relevant means that the adoption of that feature is likely to be preferred strongly by either one of the parties at least. A party could prefer to obtain it from the other, or prefer that the other obtained it from themselves. From the perspective of the party that does not wish to replace their version of the feature with the other (but this is preferred by the other party) they will at least prefer to use that feature to factor down the differences even further - it will still contain highly relevant information, and guide in the acquisition of a new feature to split across.

We are predisposed to rating ourselves highly at the outset. If we do not rate ourselves highly, then presumably that implies that we have observed others which we rate more highly than ourselves. Upon performing the Ogdoad with someone whom we regard highly, if they see us more highly than we do ourselves, we will probably choose to adopt their frame of reference as well. Otherwise, we may try to adopt their traits as ours until they see us more highly. We would expect that if we view them highly, and they view themselves highly as well, then adopting their traits would cause at least them to see us highly now as well. When one of us dislikes a feature from the other, then that feature becomes the new axis around which we attempt to discover more common ground. This is done via the overlapping perspectives, overlaid on top of the feature of interest. It is in both of our best interests to uncover the boundary between that feature and a new feature, where this boundary best delineates the section of the feature most liked by one / disliked by the other, and vice-versa for the feature across the boundary.

For example, imagine one of the individuals is wearing a long coat, bright red, which is a style enjoyed by the one wearing it, but thought to be obnoxious by the other. Sharing perspectives, they realize the obnoxiousness results from the bright red color. The wearer of the coat actually prefers long coats in general, and could tolerate wearing one with less loud / bright colors, or a more subdued red. The initial feature here is "bright red, long coat" and then is further delineated into "long coat" and "bright colored / bright red" (brightly colored may potentially be an even further delineation). The coat-wearer finds a better coat, still long, and still having some red along the lapels, which seems to satisfy both parties.

When features appear to have mixed good and bad in them, the overall "loudness" of the feature is going to be both the indicator of relevance as well as the source of more jarring disagreement. That loudness could emanate from either the feature or the other person with respect to one of them (or both, given that the feature is likely to be obtained from one of them).

We now have a full set-up that can be described to show how two individuals, with two assumed-to-maximally-disagree sets of preferences, will result in a shared-perspective situation in which the objects of preference contain the preferences themselves as well. In other words, these individuals can choose not only what to prefer, but better or worse preferences overall. We do not assume, from the metaphysical standpoint, complete subjectivity or moral relativism. On the contrary, we do believe that there are objectively better preferences, but we may initially begin either from the standpoint of not knowing this at all, or at minimum, from not being able to tell what those objectively-better preferences are, until and unless we meet a person who has them.

If we assume that the two individuals are not identical, and therefore that one of them (initially unknown as to which one) will have an overall better set of preferences, it can be shown that the one with the overall better set of preferences will have a specific set of identifiable characteristics that can be described. For one thing, it will become clear, over the course of their interaction with each other, that the one with inferior preferences will eventually wish to adopt more from the other than the vice-versa. The one with inferior preferences will also, as a rule, have a tendency to more often disfavor things or find more preferences disagreeable, rather than simply finding alternatives to those more preferable. E.g., a person with this tendency could consider "cold weather" to be "bad" in general, or at least inferior to warm weather in all circumstances.

Their dual, within the context of the Ogdoad, could suggest the usage of a removable exterior that could be adopted in cold weather and removed in warm weather environments. Their preferences will be notably better in more distinct circumstances than the other's. For example, "present in a cold or warm environment" is one feature, and "wears a coat or not" is another.

    Person          Coat Wearer           Season           Preferred
      A              Wears Coat         In Winter             Yes
      B            No Outer Wear        In Summer             Yes

      A              Wears Coat         In Winter             Yes
      A              Wears Coat         In Summer          Sometimes
      B            No Outer Wear        In Winter              No
      B            No Outer Wear        In Summer             Yes

      A              Wears Coat         In Winter             Yes
      A            No Outer Wear        In Winter              No
      A              Wears Coat         In Summer          Sometimes
      A            No Outer Wear        In Summer             Yes
      B              Wears Coat         In Winter             Yes
      B            No Outer Wear        In Winter              No
      B              Wears Coat         In Summer              No
      B            No Outer Wear        In Summer             Yes

Above, three tables are shown which depict Persons A and B split across two traits. Initially, it is observed that person A lives in a place with cold winters, and person B lives in a place with warm summers. A person's habitat can be considered a feature of their exterior, something that can be traded for one another. Person B initially cannot tolerate living in a place with cold winters. Person A can tolerate living in a place with warm summers, but it is not as ideal as their usual circumstances. If "environment" is considered to be more exterior than "clothing", then it will seem possibly more plausible for each person to imagine themselves (or even actually testing) being / living in the opposite environment. Person A and Person B both consider that sharing the same exterior (environment) would be better than each having a different one. This is equivalent to the desire to solve the problem of preference disagreement in general (living in a shared space or not has a greater effect on the degree of magnitude of the difference). But both would have to sacrifice their comfort to some degree if this happens. One point of contention is that it is not clear if person A has the better preference set because they are more amenable to living in both situations, or if person B is more correct about their choice in particular (warmer weather) because the disagreement about it has a lesser felt magnitude.

    Person             Season           Preferred      
      A              In Winter             Yes         
      B              In Summer             Yes

      A              In Winter             Yes         
      A              In Summer          Sometimes      
      B              In Winter              No         
      B              In Summer             Yes

Fortunately, it is not required to settle that matter. In reality, the difference in preferences has to do with the features that each person has available to make them more compatible with that environment, and obtaining the ability to transform their exteriors at will obviates the disagreement. As the above table shows, if we split on the first feature only (weather / environment), then we obtain a disagreement that only seems insurmountable without the presence of the other feature. The other feature (clothing) is a more-inner exterior also sharable by both. Neither party has to sacrifice their initial set of preferences: Even when person B obtains a coat, they still dislike living in cold weather without one, just as before. When person A removes theirs, they still feel good having it on when it's cold, and good having it off when it's warm. Their initial state (being incompatible with the opposite weather) hadn't referenced the values of other variables which existed, but were not used or changed when the referenced variable did. When those variables take on the values they had at the outset, the initial preferences return to exactly the same ones they had before.

Person B needn't worry that they would have to somehow "train" themselves to tolerate cold weather, or that this would alter them permanently such that they did not enjoy their usual warm weather as they did before. No training is needed at all. Person B will still hate cold weather without a coat, just as before. Person A will still enjoy cold weather, and now get to enjoy both. The goal here, for both of them, is to both enjoy more possibilities than they had been able to before.

Conclusion

If people's preference disagreements were unsolvable, i.e., that one could not place themselves in another's situation and feel the same way they do about it, without permanently altering their own preferences (against their current preferences), this would seem much worse than if the reverse were true. The SRP states that what feels better if true is true. It would be much better if I could find a way to still like the things I like, feel less discomfort at the things I find discomfortable (that is, never intolerable discomfort), find better resolutions to things I prefer to still find some discomfort in, and find things to like even better than the best things I liked before. Thus, if we believe the SRP, we believe that the previous sentence is possible, too.

Belief in discapability cannot resolve discomfort, since capability is discomfort resolution, by definition. It would be discomfortable if the SRP were not true, since that would mean what feels better if true may not always be true. But if the anti-SRP were true, along with the natural desire to arrive at truth in general, that would imply discapability in general. It would mean that truth feels good to believe in, and especially when truth is verified, but that our natural desire to find truth ends everywhere except at the final moment that a belief is chosen to be considered "verified." A belief in discapability cannot cause (or increase) capability. It would be better if only those who believed in discapability were discapable, because all they would need to do is willfully accept the SRP to attain capability again, whereas, if I accepted their belief in discapability, I would have to believe that some instances of discapability were insurmountable, and therefore not even attempt to overcome them. Thus I choose to believe in the SRP, because the worst-case scenario imaginable for that is only that for some things, continued attempts to arrive at the truth (the correct way to do something, for example), might be met with difficulty. There may even require more than several attempts before arriving at the truth. But this worst-case scenario requires that continued attempts at success being met with "failure" have to also be arbitrarily labeled as instances of discapability, which is an instance of circular reasoning.

The above paragraphs are an example of a generalistic Ogdoad, in which I place myself in the position of someone who does not believe that the Ogdoad is possible. Firstly, it might be believed that the Ogdoad is possible, but undesirable, which can be shown not to be the case (it is desirable) because it perfectly preserves initial preferences for any two individuals. What allows one to be this confident in the absolute assertion that the Ogdoad is desirable? Because to believe the Ogdoad - defined as a direct consequence of the SRP, as a process that preserves the preferences of any two individuals, such that both obtain better preferences afterwards consistent with the preferences they had before - is undesirable, would be the same as having a preference that believes the SRP would be worse if true, a contradiction. A person with such a preference would obtain that, at the very least, the SRP would indeed feel better if true. Next, they may not believe that the Ogdoad is possible, albeit desirable. This is equivalent to belief in discapability; That it is better to believe that one should prefer maintaining disagreement rather than attempting to find a resolution.

But this is only the same as the belief that one could unintentionally acquire a preference they did not want to have in the first place, something we have shown not to be possible with the Ogdoad. How resistant they are to acquiring the SRP depends entirely on how much insistence they place on maintaining closed-perspectivity. The most basic thing that could be "worse/better if true" that extends from the SRP as a natural next-question would be: Are people's preferences inherently incompatible with one-another? Because the SRP does not naturally dissuade one from believing it (which is why we call it self-reinforcing), it is not obvious that it isn't true right off the bat. Therefore, one could potentially try using both beliefs throughout life, testing to see if one works better than the other. If the SRP were false, we would expect that although it would also be expected to reappear as an at least somewhat commonly held belief-pattern among individuals, those individuals who held it would find themselves encountering more difficulty because of it, and eventually lose it, to pave way for a superior alternative to it. That being said, a superior alternative to the SRP would need to be far more complicated; Given that beliefs would in general not always feel better if true, one could not be certain based on their known knowledge, feelings, or intuition, whether or not they were correct about anything. Even if one had acquired much knowledge and experience throughout life, which would normally mean that new information would more frequently "jive" with said knowledge and experience (and thus feel better to hold), if the SRP was not true, even this could not be guaranteed. It feels better to be correct, we can assume is also true for anyone.

Luckily, the SRP, if held long enough, would "jive" with the experience that new knowledge "jives" with past experience. Thus, someone holding the SRP would find themselves adept at acquiring new skills, since it would be better-if-true that obstacles could be surmounted, in general. It would also be a belief that sat nicely in the "firmware" of someone's mind, something that didn't oft need to be repeated verbally in one's head in order to succeed at new tasks. What use for an intuition is there besides the ability to learn something by one's natural "feeling" of right/wrong, correct and incorrect? The Ogdoad extends from the reasoning that if what feels better to be true is true, then we can move on to solve the next most basic questions after that, and so on and so forth. One's intuition will, at the outset, feel most strongly on things it already has verification for, and then more weakly on things it wants to learn about (but feel strongly about the benefits of learning to clarify and strengthen itself). When one acquires more knowledge and experience, it will feel better to have acquired it, and also feel better that one's intuition responds more correctly to more instances. With that, we will close with the statement that "It feels better to be correct" is equivalent to "It feels better if true" which is equivalent to "It feels better if true that this feels better if true" which is the same as the SRP - thus the first statement in quotation marks, which is more likely to be agreed upon by everyone, immediately implies the SRP as well.