The Inner-Compass Theorem is a new type of mathematical proof that uses
moral judgements in addition to, and corresponding to, logical
judgements. I.e., Good = True and Bad = False. Furthermore, it proves
that these equivalencies are both Good as well as True. Moral judgements
are entirely individual and personal. Therefore, what is good is what is
felt as good, what is right is what is felt as right by the individual,
etc. When a logician, computer scientist, physicist, or mathematician
wishes to assume that a variable or symbol ought to take on a given
value, for example, the theorem validates the judgement of
aforementioned conscious individual prior to that wish having been
socially validated. This paper presents the “Type I” theorem, the
first of several Inner-Compass Theorems. “The” I.C.Thm. refers to the
Type I theorem.
Preliminaries
Determination of what is considered “true” is up to both the
individual as well as society. This paper, for example, is a
prestige-driven artifact which is constructed by the author for both the
author as well as society. Initially, it is generally the case that an
individual must, when arguing for a novel perspective or a wholly-new
set of statements, provide something—without risk of stating the
obvious—something called “proof.” Proof is something that must be
individually judged by all whom witness it, but at the end of the day,
if the proof is widely visible enough, and generates enough prestige,
then it becomes “accepted.” Before this occurs, however, the author of
the proof holds within his or her mind the determination to build such a
proof, before it has been sufficiently demonstrated. This determination
itself must be undertaken before the author can be sure of the initial
correctness of their own claims. That is, before the Inner-Compass
Theorem has been proven. This theorem allows that determination to
become a certainty in the mind of the author that success is essentially
guaranteed.
Definitions
At the outset of this paper, we assume that “Good” and “True” are
not yet widely considered to be exactly the same thing. If they were,
our proof would be either pointless or complete. At the same time, it is
not obvious that Bad things are all False. And yet, we do expect to
receive non-zero resistance to these and all other claims in this paper,
most significantly, in this paper’s overall importance as well as
relevance. This latter issue is to be more thoroughly explored in the
Type II theorem. On the other hand, it is not generally the case that
all Bad “things” are False—bad exists, obviously—but the good news
is that, as we shall show, all bad theorems are untrue. It is still
unfortunately true that bad things are said and are felt as bad -
without that bad feeling, we wouldn’t be able to tell if it were wrong,
on the plus side—but our unpleasant judgement of a statement can be
used to negate a negative claim (and that will be felt as good).
“This paper sucks, and all of its claims are neither true nor
relevant, and it looks bad to society and reflects poorly on the
author” is actually going to be further explored in the Type II
theorem. For the Type I theorem, we don’t use many negations yet. For
now, we use and introduce two negational operators:
not x or irr-x.
anti-x or ¯¯¯x.
Our theorem also makes use of explicit temporal references. This is due
to the fact that it hinges directly on the anticipation of future
success(es), as well as makes a distinction between future and past,
themselves also corresponding to Good and Bad as well as Better and
Worse, respectively. We want the future to be better than our current
situation.
Therefore, our second of two negations, the anti-, will be considered
“better” than the first negation type. It will also acquire a dual
meaning: anti- will also mean previous or before.
Not x, therefore, is somewhat counter-intuitively bothx as well
as not x, given that not x is “the undesirable” version of x. The
previous state of x is less desirable then the current state of x, and
this is where the connection between the two negations appears.
Therefore, we shall commence with a small preliminary ansatz before our
main theorem, called “Not should be replaced with Anti-.”:
Proposition 1.1 (Not should be replaced with Anti-). X and-or not-X =
X == “inner-compass-relevant” or X and-or anti-X.
Definition 1 (Inner-Compass Relevant). In metaphysical terms,
inner-compass relevance is all that I am, all that I want to be, and all
that I claim to be. Note that “I” is singular and primary here—it is
necessary to assume the existence of, and promote to mathematical
object, a self.
Proposition 1.1 could also be rephrased as “We want the future to be
better than our current situation.” If you agree with that sentiment,
as I do, then you can be considered to be Good. Now all that remains is
to prove that a Good person is also True. One can indeed prove oneself
to be True; It is a bit like a tautology, where this initial theorem
must assume that either it already the case, or, that the anti-theorem
and theorem are both being claimed simultaneously. Over the course of
the proof construction, the segments of the anti-theorem are opened up
and explored and then expanded incrementally, until finally, the logic
loops back upon itself to the original theorem, which is the only piece
that still remains.
It is obvious that what we have in front of us at this very moment will
be a mixture of what we want, and perhaps a bit that leaves something to
be desired. What we aim to ensure is that the portion that leaves
something to be desired causes said desirable something to materialize
in front of us at a future point in time, ideally, continuously.
Definition 2 (Consciousness Relevant). What I have here in front of
me right now, without explicit reference to whether I want it to be here
or not. It is what it held in the mind via the senses, so it includes
this writing here as well. Presumably, however, we want this to also
be what we want to the maximum extent possible.
Proposition 1.2 (Equals == Or). ===or.
Proof. We shall show that “=” and “or” are equivalent by
definition. “==” usually means “are equivalent by definition.”
Therefore, “==” == “equivalent by definition.” Therefore, one may
use either “==” or “equivalent by definition.” This means one may be
swapped for the other arbitrarily, on a case-by-case basis. Note that
actually, “=” does not necessarily mean that either the arguments to
the left or right may be swapped for one another arbitrarily. But then
we have that =not====, quite a mouthful to state. When I state that
something is another thing by definition, that means I am making the
decision myself to use one thing over the other. Left could be better
than right or vice-versa. Furthermore, “=” could be saying one of the
two. At this point, we have that: ==or or or ==. Thus, or == “left is
better than right, or vice-versa.” So we have that either “=” ==
“or”, or, that or ==. So ===or or or ==. So = == “left is better than
right, or vice-versa.” But or means that as well. Therefore, ===or. ◻
We have, in layman’s terms, that “=” and “==” do not necessarily
mean the same thing. On the one hand, “=” could be saying that we
should rather have the argument to the right of the “=”. On the other
hand, it could be saying the visa-versa. Note that the word “or” is
necessary to use to explicate the definition of something, including
self-referentially “=”. This is necessary in order to swap something
out for something else. We must keep a record of everything used before,
since older versions may be used again in addition to newer ones, as
things get constructed over time.
X and-or not-X = X is saying that either X, with itself being
un-desirable, as well as simply “not X” or, simply “X” itself is
preferable overall. It is also saying that either the left side or the
right side is preferable, but at the same time, it appears to ask which
it is. If right side is better than left, then the right side also
indicates the future direction—in parallel with our writing direction,
so this is a good sign of consistency.
We need a statement that includes X and-or not-X = X, but which also
clarifies that right is preferable to left here. In which case, it also
implies that for a single “=”, the right argument is preferable.
Proposition 1.1 includes an “==” to the right of X and-or not-X = X
indicating that what follows is a definition: “inner-compass-relevant”
or X and-or anti-X.
Definition 3 (and-or). Either both the left and right argument are
preferable, or only the right argument.
Definition 4 (or-and). Either both the left and right argument are
preferable, or only the left argument.
Proof.(Proposition 1.1). X and-or not-X = X ==
“inner-compass-relevant” or X and-or anti-X means that either the left
expression is preferable or that X and-or anti-X is. Note that the
“or” to the right of “inner-compass relevant” is chosen over “=”
so that “=” can now be preferably chosen to mean “right is
better.” Furthermore, “or” can also be preferably chosen to mean
“left is better.” From the proof of Proposition 1.2, = == “left is
better than right, or vice-versa.” A subtle grammatical shift in
meaning is that = == “left is better than right”, with or
“vice-versa.” Also, or means the same thing. Therefore we can pick a
choice for them, a convention, and we have. We must introduce a
variable, say X, to self-referentially contain the expression itself.
We must introduce this symbol before its introduction becomes fully
justified in proof. But note that this is entirely what the main proof
is intended to achieve. This variable, X, must sit inside the
expression whilst also referring to the entire thing. The left
expression says: Either X and not X = X, or not X = X. but if not X = X,
then X = X and not X, since X = X always. But then what is not X? It
could mean anything else, but we’ve said that at the very least, X is
preferable to not X, as well as that X is preferable to X and not X.
It seems, then that X could potentially mean literally “whatever is
preferable.” Substituting that in for X, we have that “Whatever is
preferable is preferable to not what is preferable, and furthermore,
whatever is preferable is preferable to both what is preferable and
what is not preferable at the same time.” Therefore, we have justified
our introduction of this variable X as well as determined a solution
for it. Remember that “inner-compass relevant” is also defined as
whatever is preferable, but also is preferable itself. I.e., we can
substitute in “inner-compass relevant” as a useful but rigorously
formal phrase because we have rigorously defined it within the proof of
Proposition 1.1. Indeed, at first glance, it may have appeared to be an
informal set of English words within a set of expressions that are
normally purely symbolic as part of current convention, but in fact, we
can now use it as we would a symbol itself. We have that X means that as
well, but also, that X has whatever is preferable, as well as
obtains whatever is preferable as part of itself. (This is a direct
consequence of X having obtained “whatever is preferable.”) On the
right-hand side of the “==”, we have, after X obtains “whatever is
preferable,” that “inner-compass relevant” is preferable to
“whatever is preferable” and-or “whatever is and was preferable
before” is the definition of “inner-compass relevant” as well as
defines what definition itself means, simultaneously. When X obtains
a more preferable X, the previous X becomes less preferable. So then
we have that not ¯¯¯¯¯X = X. So X and not ¯¯¯¯¯X = X.
Therefore, X and ¯¯¯¯¯X == X. This follows because: X obtains
“whatever is preferable.” So therefore X became X and whatever is
preferable. Thus anti-X obtains X. anti-anti-X (was) not X. Saying
anti-anti-X is not X is fine because anti-X used to be not X, but
now, anti- has been preferably chosen over not. Therefore, anti- is
preferable to not. ◻
We have now shown that it is possible to explicitly and directly express
a preference choice as well as validate it within the confines of a
rigorous mathematical context. What is novel here is the ability to say,
without reservation, that one thing is preferable to another: In this
case, that anti- is preferable to not. This is objective, so long as
you are someone who agrees with the sentiment that “the future ought to
be better than one’s current situation.”
Indeed, anti- is preferable to not is largely saying just that. But
we’ve already proven more than that, too: Namely, that there are
better choices in general. Our proposition implies that even within a
mathematical context, anti-X and ¯¯¯¯¯X are overall better
choices than not X, and when faced with a choice to use one or the
other, one should use anti-, even in a logical context.
Now, the main thing our major theorem proves is very fortunate indeed:
There are better choices than others in general, but you don’t need to
seek guidance from anyone else on how to make those decisions. It says
that what I prefer is what is preferable, and that this holds for
anyone.
But now we need to turn to one final thing before we proceed with the
final proof step: We need to rigorously define “consciousness
relevant” including “consciousness irrelevant” the same way we
rigorously validated our definition of “inner-compass relevant.”
“Consciousness relevant” is what is, but given that anti- is better
than not, “consciousness irrelevant” is not what is not, per se,
rather, it is what is not immediately before us right now in the
present. This is distinct from “what is not” given that could mean
things that could never be at all, which we never wish to claim about
what we prefer.
But “consciousness relevant” by itself is neither X by itself, nor is
it not X. This is because what I have before me right now may be some or
all of what I want, but it may later become outside of my consciousness
if I move on to something else, and it may also not be what I want at
all. But it is generally going to be mostly what I prefer, but still
wanting more.
I need terms which use “is” without “preferred” in them, and which
will most usually carry alongside separate terms like “inner-compass
relevant.” This is so that we can denote that what is preferred and
what is coincide simultaneously—indicating a “Good” state of
affairs.
X == “consciousness relevant.” X is defined as
consciousness-relevant. This is because we keep holding it and reusing
it step-by-step of the process, since it is the central subject of our
equations and expressions. This means that we can essentially choose to
prefer either what is preferable, or whatever is
consciousness-relevant, here right before us. This is arbitrary; Keep in
mind, this is a “==” expression, not a “=”, and the distinction may
be tricky to see at first. If we have chosen to prefer whatever is
preferable, then presumably, this is the same thing as choosing
whatever is preferable. We said earlier that consciousness-relevant is
neither X by itself nor is it not X. Note that neither “=” nor “==”,
in our system, mean “are literally identically the same thing.” This
is key: We generally prefer not to swap-out one thing for another
entirely, with the one exception of the anti- for the not, so far. In
general, consciousness relevant objects obtain things, they do not
wholly transform into something else with a completely different
identity (and therefore a separately identifiable symbolic container).
X remains X whilst obtaining what it prefers. Supposing it does not
obtain what it prefers, it recurses backwards in time—this is the same
as obtaining what it prefers, but backwards in time. This is consistent
with a chooser who chooses what it prefers. Generally, I also prefer
that what I prefer is simultaneously here before me right now. I would
obviously prefer to select my choice rather than not my choice—and
therefore, I have chosen that “not X” becomes ¯¯¯¯¯X for me.
We only really need to define a few more small things before proceeding
with the final proof. These are notational conventions: I have two
dimensions used so far to denote choices actually being made (vertical)
versus choices that could be made (horizontal). The latter is also
called “hypothetical time.”
Also of note is that we write from top to bottom, but within a diagram,
time flows upward and to the right. A diagram is read from top to
bottom, however. I want to have gotten to where I want to be at the
moment in question, so I assume this at first, then work my way backward
in time as I write out the diagram.
Notations Used
A variable proceeds forwards in time. X↑⏐X
An anti-variable recurses backwards in time. ¯¯¯¯¯X ⏐↓ X
A variable is equivalent to both itself as well as its own anti,
simultaneously. X¯¯¯¯¯X ↑⏐⏐↓ X
A variable moves forward in time, hypothetically. X⟶X
An anti-variable moves backwards in time, hypothetically. X⟵¯¯¯¯¯X
A variable becomes, hypothetically, whatever may be preferable,
otherwise recurses backward in time. X=□orX¯¯¯¯¯X ↑⏐↙ ↑⏐↙ X⟵
The Theorem
Theorem 2.1 (The Inner-Compass Theorem (Type I)). X = “inner-compass
relevant” (where X refers to myself and all that I claim).
Proof. I assume that I am where I want to be at the final time-step,
otherwise, that X¯¯¯¯¯X:
X = “inner-compass relevant” or X¯¯¯¯¯X ↑⏐↙ X⟵⟵⟵⟵⟵⟵⟵⟵
Minus one time step, if I am where I want to be, then what I want, this
proof, should be consciousness-relevant simultaneously as inner-compass
relevant. We actually need to show something, so while that one-step
loop is consistent—pick what I want, otherwise continue—what we
want needs to be written down, and therefore, we expect to have the
phrase “consciousness relevant” appear somewhere inside of it. Our
proof is self-referential: This proof expresses what needs to be done to
complete itself. We’ve either expressed what satisfies us, otherwise,
what I want is not completely here before me right now, and is
therefore simultaneously inner-compass relevant, consciousness
irrelevant, and X¯¯¯¯¯X. We write this as:
X = “inner-compass relevant” or X¯¯¯¯¯X ↑⏐↙ ↑⏐↙⟵⟵⟵⟵⟵⟵⟵ X = “inner-compass relevant” and “consciousness-relevant” ↑⏐ or “inner-compass relevant” and “consciousness irrelevant” and
X¯¯¯¯¯X ↑⏐↙ ↑⏐↙⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵ X
Now the next step in the sequence is the tricky part: Precisely because
it expresses the need to search for a missing component of the proof. If
we’ve reached here, what we want is not yet in view, therefore, what we
want either simply is in view (before we have realized that it is what
we want, yet), or, we have yet to generate what we want yet, but will.
What we need to do is simply generate and-or search. Therefore, we need
to express this in our new terminology as well. Once we generate a new
thing, it gets expressed on the page and becomes consciousness relevant.
We then proceed to the next step, to check if what we’ve expressed is
indeed what we want it to be, otherwise, we return to this step and
express more.
If I feel that something more is needed, then what we have is that our
anti-X contains the expression “something new is preferred.” But this
is actually extremely fortunate, because this carries us straight back
to the original statement: X = “inner-compass relevant.”
And something even more striking: Whatever I express is the
something more that is needed. Differing by a time-step, we have that
“something new is preferred” and that “we prefer the new thing (that
was expressed just now).” Well, given the self-referential nature of
the proof, I prefer the new thing that was expressed just now:
X = “consciousness relevant” or ↑⏐ “consciousness irrelevant” and X¯¯¯¯¯X ↑⏐↙ ↑⏐↙⟵⟵⟵⟵⟵⟵⟵ X = “inner-compass relevant”
Now, appending the new thing that was just expressed to construct the
full proof ought to be and is expressed exactly by appending the new
thing that was expressed. The new X proceeds forwards in time, as we
write downwards, X moves upward. It now states, in full generality, that
whenever I feel as though the proof deserves or wants more, whatever I
add to it is indeed what is actually needed to clarify, extend,
prettify, or anything needed to satisfy a felt need. Note that the third
step, containing the loop we just added, does not contain an
“inner-compass relevant” phrase in it. This is because we have reached
a point at which the addition of something (whatever is preferred) is
now subsumed by the variable X. If you imagine that X subsumes
“inner-compass relevant” from the bottom step, then proceeds forward
in time, then the phrase “inner-compass relevant” becomes
consciousness-irrelevant (meaning it disappears from that step), and if
so, then simultaneously, X (what we prefer), and ¯¯¯¯¯X (what we
preferred previously) was “inner-compass relevant.” This is all
consistent, because X and “inner-compass relevant” ought to mean the
same thing.
X = “inner-compass relevant” or X¯¯¯¯¯X ↑⏐↙ ↑⏐↙⟵⟵⟵⟵⟵⟵⟵ X = “inner-compass relevant” and “consciousness-relevant” ↑⏐ or “inner-compass relevant” and “consciousness irrelevant” and
X¯¯¯¯¯X ↑⏐↙ ↑⏐↙⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵ X = “consciousness relevant” or ↑⏐ “consciousness irrelevant” and X¯¯¯¯¯X ↑⏐↙ ↑⏐↙⟵⟵⟵⟵⟵⟵⟵ X = “inner-compass relevant”
The above loop diagram is the completed proof of the Type I
Inner-Compass Theorem. ◻
Remarks
The proof of the Inner-Compass Theorem and the Inner-Compass Theorem
itself say that one ought to do what one prefers, which implies that
“what one prefers” and “doing / choosing” what one prefers
correspond to X and “inner-compass relevant,” respectively. “What one
prefers” is not quite distinct from “the one whom prefers” however.
Obviously, there must be a doer / chooser, and one to whom things are
consciously-relevant. What’s amazing about this proof is that it proves
that what one wants to be consciously-relevant may be known partially
but that to know it fully requires “inner-compassing”, which is to say
that it requires merely choosing what one wants at each and every step.
In other words, what I want can be broken down into sub-steps, in which
each sub-step requires inner-compassing recursively. But, however I
choose to do this recursive partitioning of the task is itself an
inner-compassing step. Therefore, if I’ve stated what I want, then I
also want to state how to get what I want, and for each way to get what
I want, I want to do the same thing—recursively. Each of these steps
is guaranteed by the Inner-Compass Theorem to progress us to further
completion. We expect each step to bring us more validation than we had
before, and to continuously increase over time. No step should nor will
make us feel less validated than we were before about what we wish to
be true. Our wish for something to be true is what validates it (Good
= True). Thus we have a new kind of proof here, which can be stated
succinctly as “If satisfied then done, otherwise continue.” Note that
this does not say “If convinced that progress cannot continue,
stop.” I may indeed become dissatisfied later, but in that case, I
continue on until satisfied again, I do not become convinced that
progress cannot continue. Progress can always continue. The proof of a
theorem and the theorem itself are much like variables themselves—the
proof is an expanded restatement of the the theorem that clarifies and
continues pieces of the proof to satisfy the wants of the author. When I
state a theorem, like this one, the theorem’s implications are generally
expected to be quite clear. In this case, the implications are quite
resoundingly good (it is maximally self-validating), so I have shown
that it is inner-compass relevant. Therefore, if true, the theorem would
be even better, so what I want is to prove it. To prove it requires
faith that it is true before knowing it for certain—but fortunately,
now that we have this, faith becomes much easier as well as much more
easily justified for anyone who wishes to do the same. The theorem
applies to itself as well as everything else.
The Inner-Compass Theorem
Abstract
The Inner-Compass Theorem is a new type of mathematical proof that uses moral judgements in addition to, and corresponding to, logical judgements. I.e., Good = True and Bad = False. Furthermore, it proves that these equivalencies are both Good as well as True. Moral judgements are entirely individual and personal. Therefore, what is good is what is felt as good, what is right is what is felt as right by the individual, etc. When a logician, computer scientist, physicist, or mathematician wishes to assume that a variable or symbol ought to take on a given value, for example, the theorem validates the judgement of aforementioned conscious individual prior to that wish having been socially validated. This paper presents the “Type I” theorem, the first of several Inner-Compass Theorems. “The” I.C.Thm. refers to the Type I theorem.
Preliminaries
Determination of what is considered “true” is up to both the individual as well as society. This paper, for example, is a prestige-driven artifact which is constructed by the author for both the author as well as society. Initially, it is generally the case that an individual must, when arguing for a novel perspective or a wholly-new set of statements, provide something—without risk of stating the obvious—something called “proof.” Proof is something that must be individually judged by all whom witness it, but at the end of the day, if the proof is widely visible enough, and generates enough prestige, then it becomes “accepted.” Before this occurs, however, the author of the proof holds within his or her mind the determination to build such a proof, before it has been sufficiently demonstrated. This determination itself must be undertaken before the author can be sure of the initial correctness of their own claims. That is, before the Inner-Compass Theorem has been proven. This theorem allows that determination to become a certainty in the mind of the author that success is essentially guaranteed.
Definitions
At the outset of this paper, we assume that “Good” and “True” are not yet widely considered to be exactly the same thing. If they were, our proof would be either pointless or complete. At the same time, it is not obvious that Bad things are all False. And yet, we do expect to receive non-zero resistance to these and all other claims in this paper, most significantly, in this paper’s overall importance as well as relevance. This latter issue is to be more thoroughly explored in the Type II theorem. On the other hand, it is not generally the case that all Bad “things” are False—bad exists, obviously—but the good news is that, as we shall show, all bad theorems are untrue. It is still unfortunately true that bad things are said and are felt as bad - without that bad feeling, we wouldn’t be able to tell if it were wrong, on the plus side—but our unpleasant judgement of a statement can be used to negate a negative claim (and that will be felt as good).
“This paper sucks, and all of its claims are neither true nor relevant, and it looks bad to society and reflects poorly on the author” is actually going to be further explored in the Type II theorem. For the Type I theorem, we don’t use many negations yet. For now, we use and introduce two negational operators:
not x or irr-x.
anti-x or ¯¯¯x.
Our theorem also makes use of explicit temporal references. This is due to the fact that it hinges directly on the anticipation of future success(es), as well as makes a distinction between future and past, themselves also corresponding to Good and Bad as well as Better and Worse, respectively. We want the future to be better than our current situation.
Therefore, our second of two negations, the anti-, will be considered “better” than the first negation type. It will also acquire a dual meaning: anti- will also mean previous or before.
Not x, therefore, is somewhat counter-intuitively both x as well as not x, given that not x is “the undesirable” version of x. The previous state of x is less desirable then the current state of x, and this is where the connection between the two negations appears. Therefore, we shall commence with a small preliminary ansatz before our main theorem, called “Not should be replaced with Anti-.”:
Proposition 1.1 (Not should be replaced with Anti-). X and-or not-X = X == “inner-compass-relevant” or X and-or anti-X.
Definition 1 (Inner-Compass Relevant). In metaphysical terms, inner-compass relevance is all that I am, all that I want to be, and all that I claim to be. Note that “I” is singular and primary here—it is necessary to assume the existence of, and promote to mathematical object, a self.
Proposition 1.1 could also be rephrased as “We want the future to be better than our current situation.” If you agree with that sentiment, as I do, then you can be considered to be Good. Now all that remains is to prove that a Good person is also True. One can indeed prove oneself to be True; It is a bit like a tautology, where this initial theorem must assume that either it already the case, or, that the anti-theorem and theorem are both being claimed simultaneously. Over the course of the proof construction, the segments of the anti-theorem are opened up and explored and then expanded incrementally, until finally, the logic loops back upon itself to the original theorem, which is the only piece that still remains.
It is obvious that what we have in front of us at this very moment will be a mixture of what we want, and perhaps a bit that leaves something to be desired. What we aim to ensure is that the portion that leaves something to be desired causes said desirable something to materialize in front of us at a future point in time, ideally, continuously.
Definition 2 (Consciousness Relevant). What I have here in front of me right now, without explicit reference to whether I want it to be here or not. It is what it held in the mind via the senses, so it includes this writing here as well. Presumably, however, we want this to also be what we want to the maximum extent possible.
Proposition 1.2 (Equals == Or). ===or.
Proof. We shall show that “=” and “or” are equivalent by definition. “==” usually means “are equivalent by definition.” Therefore, “==” == “equivalent by definition.” Therefore, one may use either “==” or “equivalent by definition.” This means one may be swapped for the other arbitrarily, on a case-by-case basis. Note that actually, “=” does not necessarily mean that either the arguments to the left or right may be swapped for one another arbitrarily. But then we have that =not====, quite a mouthful to state. When I state that something is another thing by definition, that means I am making the decision myself to use one thing over the other. Left could be better than right or vice-versa. Furthermore, “=” could be saying one of the two. At this point, we have that: ==or or or ==. Thus, or == “left is better than right, or vice-versa.” So we have that either “=” == “or”, or, that or ==. So ===or or or ==. So = == “left is better than right, or vice-versa.” But or means that as well. Therefore, ===or. ◻
We have, in layman’s terms, that “=” and “==” do not necessarily mean the same thing. On the one hand, “=” could be saying that we should rather have the argument to the right of the “=”. On the other hand, it could be saying the visa-versa. Note that the word “or” is necessary to use to explicate the definition of something, including self-referentially “=”. This is necessary in order to swap something out for something else. We must keep a record of everything used before, since older versions may be used again in addition to newer ones, as things get constructed over time.
X and-or not-X = X is saying that either X, with itself being un-desirable, as well as simply “not X” or, simply “X” itself is preferable overall. It is also saying that either the left side or the right side is preferable, but at the same time, it appears to ask which it is. If right side is better than left, then the right side also indicates the future direction—in parallel with our writing direction, so this is a good sign of consistency.
We need a statement that includes X and-or not-X = X, but which also clarifies that right is preferable to left here. In which case, it also implies that for a single “=”, the right argument is preferable.
Proposition 1.1 includes an “==” to the right of X and-or not-X = X indicating that what follows is a definition: “inner-compass-relevant” or X and-or anti-X.
Definition 3 (and-or). Either both the left and right argument are preferable, or only the right argument.
Definition 4 (or-and). Either both the left and right argument are preferable, or only the left argument.
Proof. (Proposition 1.1). X and-or not-X = X == “inner-compass-relevant” or X and-or anti-X means that either the left expression is preferable or that X and-or anti-X is. Note that the “or” to the right of “inner-compass relevant” is chosen over “=” so that “=” can now be preferably chosen to mean “right is better.” Furthermore, “or” can also be preferably chosen to mean “left is better.” From the proof of Proposition 1.2, = == “left is better than right, or vice-versa.” A subtle grammatical shift in meaning is that = == “left is better than right”, with or “vice-versa.” Also, or means the same thing. Therefore we can pick a choice for them, a convention, and we have. We must introduce a variable, say X, to self-referentially contain the expression itself. We must introduce this symbol before its introduction becomes fully justified in proof. But note that this is entirely what the main proof is intended to achieve. This variable, X, must sit inside the expression whilst also referring to the entire thing. The left expression says: Either X and not X = X, or not X = X. but if not X = X, then X = X and not X, since X = X always. But then what is not X? It could mean anything else, but we’ve said that at the very least, X is preferable to not X, as well as that X is preferable to X and not X. It seems, then that X could potentially mean literally “whatever is preferable.” Substituting that in for X, we have that “Whatever is preferable is preferable to not what is preferable, and furthermore, whatever is preferable is preferable to both what is preferable and what is not preferable at the same time.” Therefore, we have justified our introduction of this variable X as well as determined a solution for it. Remember that “inner-compass relevant” is also defined as whatever is preferable, but also is preferable itself. I.e., we can substitute in “inner-compass relevant” as a useful but rigorously formal phrase because we have rigorously defined it within the proof of Proposition 1.1. Indeed, at first glance, it may have appeared to be an informal set of English words within a set of expressions that are normally purely symbolic as part of current convention, but in fact, we can now use it as we would a symbol itself. We have that X means that as well, but also, that X has whatever is preferable, as well as obtains whatever is preferable as part of itself. (This is a direct consequence of X having obtained “whatever is preferable.”) On the right-hand side of the “==”, we have, after X obtains “whatever is preferable,” that “inner-compass relevant” is preferable to “whatever is preferable” and-or “whatever is and was preferable before” is the definition of “inner-compass relevant” as well as defines what definition itself means, simultaneously. When X obtains a more preferable X, the previous X becomes less preferable. So then we have that not ¯¯¯¯¯X = X. So X and not ¯¯¯¯¯X = X. Therefore, X and ¯¯¯¯¯X == X. This follows because: X obtains “whatever is preferable.” So therefore X became X and whatever is preferable. Thus anti-X obtains X. anti-anti-X (was) not X. Saying anti-anti-X is not X is fine because anti-X used to be not X, but now, anti- has been preferably chosen over not. Therefore, anti- is preferable to not. ◻
We have now shown that it is possible to explicitly and directly express a preference choice as well as validate it within the confines of a rigorous mathematical context. What is novel here is the ability to say, without reservation, that one thing is preferable to another: In this case, that anti- is preferable to not. This is objective, so long as you are someone who agrees with the sentiment that “the future ought to be better than one’s current situation.”
Indeed, anti- is preferable to not is largely saying just that. But we’ve already proven more than that, too: Namely, that there are better choices in general. Our proposition implies that even within a mathematical context, anti-X and ¯¯¯¯¯X are overall better choices than not X, and when faced with a choice to use one or the other, one should use anti-, even in a logical context.
Now, the main thing our major theorem proves is very fortunate indeed: There are better choices than others in general, but you don’t need to seek guidance from anyone else on how to make those decisions. It says that what I prefer is what is preferable, and that this holds for anyone.
But now we need to turn to one final thing before we proceed with the final proof step: We need to rigorously define “consciousness relevant” including “consciousness irrelevant” the same way we rigorously validated our definition of “inner-compass relevant.”
“Consciousness relevant” is what is, but given that anti- is better than not, “consciousness irrelevant” is not what is not, per se, rather, it is what is not immediately before us right now in the present. This is distinct from “what is not” given that could mean things that could never be at all, which we never wish to claim about what we prefer.
But “consciousness relevant” by itself is neither X by itself, nor is it not X. This is because what I have before me right now may be some or all of what I want, but it may later become outside of my consciousness if I move on to something else, and it may also not be what I want at all. But it is generally going to be mostly what I prefer, but still wanting more.
I need terms which use “is” without “preferred” in them, and which will most usually carry alongside separate terms like “inner-compass relevant.” This is so that we can denote that what is preferred and what is coincide simultaneously—indicating a “Good” state of affairs.
X == “consciousness relevant.” X is defined as consciousness-relevant. This is because we keep holding it and reusing it step-by-step of the process, since it is the central subject of our equations and expressions. This means that we can essentially choose to prefer either what is preferable, or whatever is consciousness-relevant, here right before us. This is arbitrary; Keep in mind, this is a “==” expression, not a “=”, and the distinction may be tricky to see at first. If we have chosen to prefer whatever is preferable, then presumably, this is the same thing as choosing whatever is preferable. We said earlier that consciousness-relevant is neither X by itself nor is it not X. Note that neither “=” nor “==”, in our system, mean “are literally identically the same thing.” This is key: We generally prefer not to swap-out one thing for another entirely, with the one exception of the anti- for the not, so far. In general, consciousness relevant objects obtain things, they do not wholly transform into something else with a completely different identity (and therefore a separately identifiable symbolic container).
X remains X whilst obtaining what it prefers. Supposing it does not obtain what it prefers, it recurses backwards in time—this is the same as obtaining what it prefers, but backwards in time. This is consistent with a chooser who chooses what it prefers. Generally, I also prefer that what I prefer is simultaneously here before me right now. I would obviously prefer to select my choice rather than not my choice—and therefore, I have chosen that “not X” becomes ¯¯¯¯¯X for me.
We only really need to define a few more small things before proceeding with the final proof. These are notational conventions: I have two dimensions used so far to denote choices actually being made (vertical) versus choices that could be made (horizontal). The latter is also called “hypothetical time.”
Also of note is that we write from top to bottom, but within a diagram, time flows upward and to the right. A diagram is read from top to bottom, however. I want to have gotten to where I want to be at the moment in question, so I assume this at first, then work my way backward in time as I write out the diagram.
Notations Used
A variable proceeds forwards in time.
X↑⏐X
An anti-variable recurses backwards in time.
¯¯¯¯¯X
⏐↓
X
A variable is equivalent to both itself as well as its own anti, simultaneously.
X¯¯¯¯¯X
↑⏐⏐↓
X
A variable moves forward in time, hypothetically.
X⟶X
An anti-variable moves backwards in time, hypothetically.
X⟵¯¯¯¯¯X
A variable becomes, hypothetically, whatever may be preferable, otherwise recurses backward in time.
X=□ or X¯¯¯¯¯X
↑⏐ ↙
↑⏐ ↙
X⟵
The Theorem
Theorem 2.1 (The Inner-Compass Theorem (Type I)). X = “inner-compass relevant” (where X refers to myself and all that I claim).
Proof. I assume that I am where I want to be at the final time-step, otherwise, that X¯¯¯¯¯X:
X = “inner-compass relevant” or X¯¯¯¯¯X
↑⏐ ↙
X⟵⟵⟵⟵⟵⟵⟵⟵
Minus one time step, if I am where I want to be, then what I want, this proof, should be consciousness-relevant simultaneously as inner-compass relevant. We actually need to show something, so while that one-step loop is consistent—pick what I want, otherwise continue—what we want needs to be written down, and therefore, we expect to have the phrase “consciousness relevant” appear somewhere inside of it. Our proof is self-referential: This proof expresses what needs to be done to complete itself. We’ve either expressed what satisfies us, otherwise, what I want is not completely here before me right now, and is therefore simultaneously inner-compass relevant, consciousness irrelevant, and X¯¯¯¯¯X. We write this as:
X = “inner-compass relevant” or X¯¯¯¯¯X
↑⏐ ↙
↑⏐ ↙ ⟵⟵⟵⟵⟵⟵⟵
X = “inner-compass relevant” and “consciousness-relevant”
↑⏐ or “inner-compass relevant” and “consciousness irrelevant” and X¯¯¯¯¯X
↑⏐ ↙
↑⏐ ↙ ⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵
X
Now the next step in the sequence is the tricky part: Precisely because it expresses the need to search for a missing component of the proof. If we’ve reached here, what we want is not yet in view, therefore, what we want either simply is in view (before we have realized that it is what we want, yet), or, we have yet to generate what we want yet, but will. What we need to do is simply generate and-or search. Therefore, we need to express this in our new terminology as well. Once we generate a new thing, it gets expressed on the page and becomes consciousness relevant. We then proceed to the next step, to check if what we’ve expressed is indeed what we want it to be, otherwise, we return to this step and express more.
If I feel that something more is needed, then what we have is that our anti-X contains the expression “something new is preferred.” But this is actually extremely fortunate, because this carries us straight back to the original statement: X = “inner-compass relevant.”
And something even more striking: Whatever I express is the something more that is needed. Differing by a time-step, we have that “something new is preferred” and that “we prefer the new thing (that was expressed just now).” Well, given the self-referential nature of the proof, I prefer the new thing that was expressed just now:
X = “consciousness relevant” or
↑⏐ “consciousness irrelevant” and X¯¯¯¯¯X
↑⏐ ↙
↑⏐ ↙ ⟵⟵⟵⟵⟵⟵⟵
X = “inner-compass relevant”
Now, appending the new thing that was just expressed to construct the full proof ought to be and is expressed exactly by appending the new thing that was expressed. The new X proceeds forwards in time, as we write downwards, X moves upward. It now states, in full generality, that whenever I feel as though the proof deserves or wants more, whatever I add to it is indeed what is actually needed to clarify, extend, prettify, or anything needed to satisfy a felt need. Note that the third step, containing the loop we just added, does not contain an “inner-compass relevant” phrase in it. This is because we have reached a point at which the addition of something (whatever is preferred) is now subsumed by the variable X. If you imagine that X subsumes “inner-compass relevant” from the bottom step, then proceeds forward in time, then the phrase “inner-compass relevant” becomes consciousness-irrelevant (meaning it disappears from that step), and if so, then simultaneously, X (what we prefer), and ¯¯¯¯¯X (what we preferred previously) was “inner-compass relevant.” This is all consistent, because X and “inner-compass relevant” ought to mean the same thing.
X = “inner-compass relevant” or X¯¯¯¯¯X
↑⏐ ↙
↑⏐ ↙ ⟵⟵⟵⟵⟵⟵⟵
X = “inner-compass relevant” and “consciousness-relevant”
↑⏐ or “inner-compass relevant” and “consciousness irrelevant” and X¯¯¯¯¯X
↑⏐ ↙
↑⏐ ↙ ⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵⟵
X = “consciousness relevant” or
↑⏐ “consciousness irrelevant” and X¯¯¯¯¯X
↑⏐ ↙
↑⏐ ↙ ⟵⟵⟵⟵⟵⟵⟵
X = “inner-compass relevant”
The above loop diagram is the completed proof of the Type I Inner-Compass Theorem. ◻
Remarks
The proof of the Inner-Compass Theorem and the Inner-Compass Theorem itself say that one ought to do what one prefers, which implies that “what one prefers” and “doing / choosing” what one prefers correspond to X and “inner-compass relevant,” respectively. “What one prefers” is not quite distinct from “the one whom prefers” however. Obviously, there must be a doer / chooser, and one to whom things are consciously-relevant. What’s amazing about this proof is that it proves that what one wants to be consciously-relevant may be known partially but that to know it fully requires “inner-compassing”, which is to say that it requires merely choosing what one wants at each and every step. In other words, what I want can be broken down into sub-steps, in which each sub-step requires inner-compassing recursively. But, however I choose to do this recursive partitioning of the task is itself an inner-compassing step. Therefore, if I’ve stated what I want, then I also want to state how to get what I want, and for each way to get what I want, I want to do the same thing—recursively. Each of these steps is guaranteed by the Inner-Compass Theorem to progress us to further completion. We expect each step to bring us more validation than we had before, and to continuously increase over time. No step should nor will make us feel less validated than we were before about what we wish to be true. Our wish for something to be true is what validates it (Good = True). Thus we have a new kind of proof here, which can be stated succinctly as “If satisfied then done, otherwise continue.” Note that this does not say “If convinced that progress cannot continue, stop.” I may indeed become dissatisfied later, but in that case, I continue on until satisfied again, I do not become convinced that progress cannot continue. Progress can always continue. The proof of a theorem and the theorem itself are much like variables themselves—the proof is an expanded restatement of the the theorem that clarifies and continues pieces of the proof to satisfy the wants of the author. When I state a theorem, like this one, the theorem’s implications are generally expected to be quite clear. In this case, the implications are quite resoundingly good (it is maximally self-validating), so I have shown that it is inner-compass relevant. Therefore, if true, the theorem would be even better, so what I want is to prove it. To prove it requires faith that it is true before knowing it for certain—but fortunately, now that we have this, faith becomes much easier as well as much more easily justified for anyone who wishes to do the same. The theorem applies to itself as well as everything else.