The Axiomatic Law of Universal Congruity

I am afraid this post will be a particularly difficult read for some audiences, but I do believe that most people should be able to get something out of it if they try hard enough.  However, if any of my readers should happen to have a degree in philosophy—for whatever strange reason—I should expect that he or she will find this particularly interesting.

If there’s something you don’t understand, please comment; ask questions.  I don’t have an editor (such is the nature of blogging) and so it is likely that the piece could use some revision, and questions from readers could help with that process.  No question is too shallow; even if you don’t understand this at all, readers and I could benefit from any question you might come up with.

In order to accommodate for this article’s richness in footnotes and such, I have implemented a new format: Whenever you see a *, click on it to open the footnote in a new tab, and whenever you see a word highlighted, click on it to open a note that has been included for increased accessibility—also in a new tab.  When you hear the chime, turn the page.  For a printer friendly version of this post, click here.

Please read scrupulously; it should make sense.


I would like to propose an argument for the necessity of the fractal structure that I have hitherto used for modelling Reality.  In my post, “Fractal Reality,” I have begun to describe the practicality of understanding reality as an infinite structure of concrete truths; however, by my assessment, I have not adequately addressed the necessity nor the consequences of such a model.  I intend to undertake the former of those tasks here and complete the latter in a later post, but I suspect I might end up using more posts than that for a more complete investigation of this subject.

The nonphysical

In order to model reality, we must begin by considering what reality is.  It seems the most obvious place to begin such an inquiry is with the debate between materialism, idealism, and dualism.  However, as delightfully cliché as such a method of argument would be, I find it infeasible.  For it doesn’t seem reasonable for me to use logical argument, something from the purely “idealistic” realm, to ponder the validity of the materialistic realm.  Just as it doesn’t make sense to debate relativism using absolutism, so is the materialist required to hold his or her beliefs without theoretical reason, for the existence of nonphysical reasons for a set of beliefs seems to imply a belief in the nonphysical.  However, by the same thinking, we might also suppose that holding a set of beliefs at all constitutes the act of investing faith in the idealistic realm.  Therefore, within a reasonable scope of thinking, one may be either an idealist or a dualist, but not a materialist.  Whether one happens to be a dualist or an idealist is immaterial to this argument at present so long as it is agreed that there is at least some component of reality that is nonphysical.

With that in place, let us examine this nonphysical component.  We might consider this component to be something like Plato’s world of forms; that is, the nonphysical is a sort of normative understanding of reality.  Things in the nonphysical behave in accordance with our cognition.  For example, whenever one imagines a circle, it exists in the nonphysical, because all that is required for the spawning of an object in the nonphysical is the decision that it exists.  If I decide that there is a circle of radius R, then there is.

Let us further explicate this nonphysical realm by using the physical realm as its analogue.  If we presume that the physical realm is governed by the laws of physics, we might similarly regard the nonphysical as being governed by the laws of logic.  Therefore, while I can decide that a circle exists in the nonphysical, I cannot decide that a square circle exists, as that defies the laws of logic.  We may also understand the physical realm as being perceived by us via our five senses, but the nonphysical realm must be perceived through a nonphysical sense: our reason.  All this seems a quite necessary part of any scope in which logical argument can have significance.

Definition of logic

Continuing with our analogy, let us define logic.  We are able to use the word “physics” in two primary senses: (1) physics is a field of study, a branch of science, and (2) physics is something that belongs, in some sense, to a physical system (e.g. the physics of airplanes).  Likewise there are two common uses of the word “logic”: (1) logic is a field of study, a branch of mathematics, and (2) logic can belong to a nonphysical system, an argument.  We often speak of “the logic of an argument,” or “the logic behind an argument.”  This is the thing that I wish to define.  Logic in this sense is a chain of reasoning, or to be broader, a normative construct, that adheres to the laws which govern the nonphysical.  As has already been said, the laws that govern the nonphysical realm are the laws of logic, in the first sense of the word.  So logic in the second sense is a normative construct that adheres to the laws of logic in the first sense.  Therefore, to examine logic in this second sense, we must understand it in the first sense; hence I propose the question: What are the laws of logic?

In order to arrive at the laws of logic, it seems prudent to make a distinction between laws and methods.  On the surface, it appears that the laws of logic must be very complex and there must be many of them.  We could list all of the logical operators, explain how they work, and use them to derive what we would call the “rules of inference,” but I would categorise all such work as the derivation and identification of functional methods of logic.  The methods used to solve physics problems involve complex mathematical equations, but the actual laws of physics are the reasons that physical systems behave in a way that can be modelled by such methods.  For example, it is a law of physics that matter is subject to gravity, but it is a method of physics to use a parabolic function to model gravity.  Gravity itself is the way physical systems behave, and all formulas and explanations about gravity merely constitute a methodology for understanding that behaviour.  Indeed, the laws of physics are the very things that make physics what it is; all the rest can be viewed differently by different people and still function.  That is, I can write the equation for gravity differently, and I can use different words to define gravity, but I can’t change what gravity is.

The same is true of the methods and laws of logic.  The laws of logic are what make logic what it is.  On a fundamental level, I would argue that there exists only one law of logic, in this sense, and it is the law of noncontradiction.  (Ironically, the law of noncontradiction is considered the “second law” of aristotelian logic, but I regard the other two as “methods” under the linguistic framework I have set up.)  Noncontradiction is the only law of logic because it alone is what the methods of logic are intended to accommodate for.  A logician can execute an intricate and extensive proof with all sorts of complexities, but when he or she gets to the QED at the bottom, noncontradiction alone is what makes it all logical.

This seems an easy premise to object to.  Surely, if noncontradiction were the only requirement, logical argument could include all sorts of absurdities.  For example, one might argue, “All chickens are green; Hamlet is a chicken; therefore, Hamlet is green” .  And by this understanding of logic, that argument is logical; it doesn’t contradict with itself.  What’s wrong with the argument is not that it is illogical but that its premisses are false.  Therefore, it must be understood that an argument may be logical and still not accurately reflect the empirical facts of reality.  However, a logical argument which accurately reflects reality in its premisses will also accurately reflect reality in its conclusion.

Therefore, logic is that which is noncontradictory with itself.

A qualification of reality

And now I will indirectly return to the discussion from earlier regarding materialism and idealism.  The definition of logic which we have just arrived at tells us what logic is, but not how it functions.  Logic is designed to accommodate for its function: that of discovering truth.  Logic describes reality to us via the derivation of conclusions from premisses.  Hence, we suppose that if we are given accurate premisses which describe reality, we can manipulate them using any number of methods, and so long as we adhere to the law of logic, the law of noncontradiction, we will arrive at an equally accurate conclusion about reality.*

This tells us something of the nature of reality.  As it turns out, all reality must adhere to the law of logic, not just the nonphysical.  For the very reason that logic has the law it does is that we presume reality to have that same law.  That is, reality is naturally subject to the law of noncontradiction, and the nonphysical is thus modelled after such a stipulation.


And now I should like to explain some calculus to make this argument more clear:

In calculus, infinity is assumed.  That is, if there exists any system that increases without bound, then it is assumed that the system approaches infinity.  Hence, we can determine what a system will approach, i.e. its limit, based on its rate of change.  If a system increases at a decreasing rate, it will have a finite limit, but if it increases at a constant or increasing rate, the system will approach infinity.

The second mathematical concept that must be understood before the argument may continue is orders of infinity:

This is something of a paradox that we live with in calculus.  It is supposed that, while one system might have a limit of infinity, another might have a limit of infinity squared, and though both are equal to infinity, the one is infinite times less than the other.  Hence the limit of y = x as x approaches infinity is infinity, but the limit of y = x ^ 2 as x approaches infinity is infinite times greater than the former infinity.  In fact, the application of any operation of higher power than addition/subtraction to infinity will affect the order of infinity (i.e. infinity times, to the power of, etc. any finite number is a different order of infinity).

Why the nonphysical is infinite

Let us suppose that the nonphysical realm, which is subject to the law of logic, is a subset of some “conceptual realm.”  This conceptual realm is not subject to the law of logic but is made up of everything that can be conceptualised.  In fact, such could be its analogous law: the law of conceivability.  By this I mean that all things in the conceptual realm are governed by the law of conceivability, which dictates that all its subjects must be conceivable.  Let us call each of these “things” in the realm “declaratives,” meaning statements in the indicative mood.

I would argue that this conceivable realm is infinite—that there is no limit to what can be conceived.  By this, I do not necessarily mean that there is no limit to what the human mind can conceived, but that there is no limit on conceivability in itself (I do not wish to make any comment on the former of those claims).  This is because there is no limiting factor on the system of conceivability; the law to which it is bound excludes nothing from its domain, and therefore, if we imagine the realm as some universe that expands as an omniscient being continues to conceive of more and more things, there is no reason we should expect its expansion to ever slow down.  It is a system which increases at a constant rate, which means that it approaches infinity because infinity is assumed.  However, the nonphysical is a subset of this conceptual realm in that it is possible to conceive of illogical things, but by definition, such things cannot spawn in the nonphysical (e.g. a square circle).

This poses a problem.  If we look at the nonphysical by itself, we may very well expect it to be a finite realm, for the more things which are spawned in the nonphysical, the harder it is to come up with things that don’t contradict any of them.*  One might relate the expansion of the nonphysical to the covering of an elaborate lie.  As a suspicious other asks the liar for more and more information about the subject, the liar’s task becomes more and more difficult as he tries to avoid contradicting himself through creativity and strategy.  The difficulty lies in the fact that each thing he says is required to be in noncontradiction with the growing construct of falsehood that has come before it.  For this reason, it seems the nonphysical must have a limiting factor; it appears to be decreasingly increasing, which, in calculus, means that it has a finite limit.

However, appearances are often deceiving, and a statistical approach to the problem proves such to be the case here:  As has already been said, the nonphysical is a subset of the conceptual, where the conceptual is an infinite set of declaratives.  For every declarative, there exists a negation.  For example, if there exists a declarative, A, which states, “the pen sits on the table in the room,” then there also exists a negation, ¬A, which states, “the pen does not sit on the table in the room”.  Both A and ¬A are, in this case, dependent on other implied declaratives, the most obvious one being a declarative, B, which might state, “the room has a table in it”.  Therefore, spawning ¬B in the nonphysical excludes the possibility not only of spawning B, but also of spawning A or ¬A, and therefore one might at first suppose that this reduces the number of possible inclusions by a greater quantity than that which has been included; i.e. we have included only one declarative, ¬B, but in so doing have excluded two: B and (A or ¬A).  However, we have also opened up the possibility of including other declaratives which are dependent on ¬B.  For example, declarative C might state, “the absence of furniture makes the room feel bland”.  Both C and ¬C would have been excluded by spawning B; therefore, while by spawning ¬B, we exclude the possibility of spawning two other declaratives, we do the same, in quantitative terms, by spawning B.*  In fact, within an infinite set of declaratives, there will exist an equal amount of declaratives which become includable as which become no longer includable upon the affirmation or negation of any given declarative.  This is because within a finite set of declaratives, X of them might be excluded upon the inclusion of declarative A and Y of them upon the inclusion of ¬A, but we have no statistical reason to suppose that either X should be greater than Y or Y greater than X (for in fact, A could be reassigned the value of ¬A, in which case, X and Y would also switch values), and therefore, on average, X is equal to Y, which means that, in the case of an infinite set of declaratives, X always equals Y.*

What this tells us then is that with every expansion of the nonphysical, an equal number of declaratives become includable in the nonphysical as become no longer includable, and therefore, the percentage of declaratives in the conceptual which may be added to the nonphysical remains constant.  Thus, the nonphysical is a fractional subset of the conceptual, and is therefore infinite (though by a lower order of infinity than that which describes the magnitude of the conceptual).

By this model, we should indeed expect the system to behave the way it did in the case of the liar.  For the liar is only capable of thinking of a finite quantity of declaratives quickly enough to use them (I’m still making no comment on the full capacity of the human mind).  Therefore, while each addition to his lie is opening up an equal number of possible additions as it is taking away, some of the new possibilities are not present in his finite selection of declaratives, and he is therefore only affected by any of the exclusions which happen to be in said selection.

A nonphysical construct can now be defined as “any infinite construct which is noncontradictory with itself”.  We should expect there to be multiple such constructs based on the calculus.  Theoretically, there are infinite declaratives that exist in the conceptual which were excluded from the original nonphysical construct, but any one of those can serve as the starting point for an entirely unique, infinite, nonphysical construct.  Thus, there are at least two possible nonphysical constructs, but only one reality, and for this reason, it must be possible to conceive things that are not real.*

A definition of reality

From two sections ago (“A qualification of reality”), we have found that it is possible to put anything which exists in reality into the nonphysical, and anything that exists in the nonphysical might exist in reality.  One must then ask, does everything in the nonphysical exist in reality?

I do not so much wish to answer that question directly, but rather propose a model of reality that relates very specifically to the nonphysical.  Elsewhere on this blog, I have discussed The Necessity of the Omnipotent.  In that post, I wrote that due to the nature of causality—causality being an inescapable facet of reality under the logical scope—there must exist something in reality that is somehow “omnipotent,” or as the word came to be used in the jargon of the piece, “uncaused.”  This primal cause argument is often referred to as the “cosmological” argument by people even more esoteric than myself.  Simply put, there must either be a primal cause which exists without cause and which caused all the rest of reality or else there must be an uncaused, infinite chain of causality that makes up reality.  As I have elsewhere observed, the two of these possibilities seem very much to be merely two different ways of expressing the same thing: the omnipotent, or uncaused, thing is both the cause of reality and the essence of reality.  All this means that reality is necessarily infinite.  Everything is real.

That being the case, reality is an infinite construct that adheres to the law of logic; in other words, reality might be defined as “that which is noncontradictory with itself.”  I say “that which,” and not “a subset of that which” because reality includes all existing things that are noncontradictory with themselves.  By definition, nothing exists outside of the domain of reality.  I do not mean that nothing can be imagined that does not exist, but rather, everything that exists is a part of reality, and all those things are noncontradictory.

Because reality is infinite, we know that it is made up of infinite declaratives, for the phrase “reality is infinite” could be reworded “there exist infinite truths”.  Therefore, reality has the exact same form as a nonphysical construct: it is an infinite construct of noncontradictory declaratives.  The law of logic rules both the realm of the nonphysical and that of reality and insists that their respective systems be defined by their noncontradiction with themselves.  In other words, they are defined in terms of themselves.

The singularity of reality

Saying that reality is defined in terms of itself may seem prima facie objectionable to some.  It is not immediately evident that reality is defined in terms of itself, but rather that each of its parts are associated with certain qualifications that relate them to each other part (by “part” I mean “declarative”).  But as it turns out, these qualifications do in fact serve as definitions as well.  A definition is a description for which only one thing is qualified to match.  This is the nature of the noncontradiction qualifier.  We understand reality as being entirely causal, even in the realms beyond the natural (see “The Necessity of Causality in the Logical Scope”), and as such, reality must exist in the only possible state which is logically permissible.  For each set of causes has but one set of effects; it is not possible for some part of reality to be different than it is unless its cause is also made to be different than it is, and then that cause’s cause would have to be modified as well, and one would need to trace the whole thread all the way back along the infinite chain of causality until he reached the Omnipotent, who would also need to be changed, which is an Omnipotent impossibility (see “Absolute Nonsense”).  Hence, if we change any single declarative that makes up reality, it will be in contradiction with the whole, and for this reason, the system of reality is defined by noncontradiction.  Noncontradiction describes each part of the system such that only one thing is qualified to match the description.  And because the system is defined by noncontradiction with itself, it may be said to be recursively defined.

The structure of reality

To better understand what sort of structure this forms, we must subscribe for a moment to a scalar model of reality.  It is generally presumed that any individual is capable of perceiving some portion of reality, but not the whole; i.e. everyone knows something, but no one knows everything.  However, I would like to propose that the “something” which everyone knows is a particular scalar view of reality.  What each individual knows about reality is not just some random subset of the whole, but some finite-scaled scope, however incomplete, of reality.  By this I mean that a person may know or be capable of learning all sorts of things on a given level, but there will be some nuances of reality that are, in a sense, too “small” or “detailed” for anyone to understand, as well as some truths that are too large.  We can’t comprehend the entire universe, and neither can we understand why protons and electrons attract and repel.  We might think of this scalar construct as something that is explored via inquiry.  That is, we might be within one scalar scope when we know A, but when we ask how A works, we move to a finer scope, and when we ask what A does in the context of systems outside itself, we move to a coarser scope.  However, though not perceivable all at once, each of these scopes are contained within one another, and there are infinite of them.

It is this concept of unperceivable scopes which troubles many a modern thinker into some form of relativism.  It is supposed that if there exist infinite scopes which we cannot perceive, then all our knowledge is useless.  However,  the recursive nature of reality at which we have already arrived would suggest that such a conclusion does not follow.  For in fact, every level of reality is defined directly in terms of every other level.  As I have said, it is not the case that the parts of reality merely relate to each other according to noncontradiction, but that, under this scalar model, each level defines each other level via noncontradiction.  This creates what appears to be a paradox on the surface.  Each level is defined as the only thing that is noncontradictory with each other level.  In other words, if a level of reality A contains a finer level B and B contains C, then A is the only thing that is noncontradictory with B, but C is also defined as the only thing that is noncontradictory with B.

Most will think I’ve simply made a slight oversight in inventing this paradox.  One solution might be as follows:  (1) It is not that A is the only thing which is noncontradictory with B, but that it is the only thing which is noncontradictory with B and C, and hence, each part is allowed to be the only thing noncontradictory with the remaining structure outside of itself.  As compelling as such a solution to the paradox is, it is not entirely sufficient.  For we do expect A to also be noncontradictory with itself, and so it must be the only thing which is noncontradictory with A, B, and C, but in that case, B is also the only thing noncontradictory with A, B, and C.  However, there is a second possible solution to the paradox that deals with the structure of the levels:  (2) Perhaps B is the only thing which is noncontradictory with existing inside of A, and C the only thing that may exist inside of B.  But even this does not solve the issue all together.

Each of those parts—A, B, and C—is the only thing which is noncontradictory with the whole in its particular structural context.  B is the only thing which does not contradict A, B, and C when it is structurally related to those parts in the particular way that it is.  Think of it like a car engine.   In a car engine, the only thing that may function in the particular place where the cylinder is located is the cylinder itself.  If we put the gas tank where the cylinder is, it would contradict the function of the machine, but the gas tank is noncontradictory with the function of the car when it is located in the place it is supposed to be.  In other words, each of the parts of the car are noncontradictory to its function when they are structurally related in but one particular manner.  However, in the case of the levels of reality, structure is redundant.  We presume there to be infinite levels of reality all of which contain each other.  Therefore, while C is structurally related to B in one way, B is also structurally related to A in the same way, and there are infinite other levels in which A is contained as well as infinite other levels which are contained in C.

C is the only thing that can structurally relate to B in the way it does, but B relates to A in the same way.  With this being the case, we can suppose that A, B, and C are different from each other, but they cannot be unsimilar.  “Contains” is a transitive relationship; that is, if A contains B and B contains C, then A contains C.  Clearly, this does not mean C is structurally related to A in the exact same way that B is, but the relationship is similar—congruent, if you will.  And because each of the levels of reality are what they are in accordance with their structural relationship to the rest, the levels themselves are also congruent.  This gives rise to The Axiomatic Law of Universal Congruity: “Every understanding and misunderstanding of a given scope of reality is congruent to that of the whole.”  Some readers might find it humorous to call this a “Categorical Declarative”.*

Therefore, reality is self-similar.  On every level from which we observe reality, we see something that resembles the whole.

There are many consequences of The Axiomatic Law of Universal Congruity which I am very excited to tell you all about, but I imagine that if you have bothered to read this far, you are already far too kind, and I cordially thank you for your interest.  In light of that, I will refrain from subjecting you to any further mind numbing activity.

If I imagine people more esoteric than myself, then they exist.

9 thoughts on “The Axiomatic Law of Universal Congruity

  1. Hi …read your entire article and got the gist …… I also understood that you are a true believer in the infiniteness of knowledge. God lives through man and when you combine all the knowledge in all the brains … you come face to face with the infiniteness of The Singularity and the layers that keep on showing up forever … revealing true infinity . Each nuance of each thought and concept has so many levels that its mind boggling. You are a student of philosophy so you wrote your article in depth and yet come to any village or small town in India from where I come and the people who have never been to any fancy school or college will share so much raw knowledge of our world with you that it is humbling. Who taught them … my conclusion … we share the same mind … we are the same … we will always be. I am constantly testing my power to influence other minds unbeknownst to them … simply to test this theory … and I am always proved right. I always receive that which I seek … i.e. when I want someone to be nice to me .. I think good thoughts about them … and lo and behold … even if they were daggers drawn earlier … they are like lambs now. We do control our destinies in the nicest way possible … we are the creators of our worlds. We hold within ourselves the power of our own God.

      • You were analyzing philosophy so I jumped to that conclusion ….. ok I get it O over O means nothing. Pray what was so disagreeing about my perspective … correct me so that I am understand it better. I don’t mind learning from another’s point of view and if it makes sense to my mind I shall know you’re right .. deal ?

        • Very well.

          I agree with you about the universality of human understanding. Indeed, I was just recently discussing this matter with a doctor of sociolinguistics. She and I agreed that it is often shocking just how similar the philosophies of different cultures around the world are, in spite of the fact that they have had no known historical influence on each other. For example, in India, as you know, it is largely believed that time works in cycles: people are born, they die, and they are reborn in a different from based on karma from their previous life. However, in ancient Greece, a very similar philosophy was held. Zeno, a philosopher from a school in ancient Athens, is credited as being one of the founders of stoicism. The stoics, like the modern indians, believed in a cycle of life, death, and rebirth from which one could only escape through complete stoicism–meaning purity and detachment.

          So that is very interesting, and a powerful source of consolation for the modern absolutist.

          Also, I completely agree that one does not have to go to any sort of fancy school to acquire wisdom. Indeed, if it is wisdom a person is after, the last place he or she should go is the university. But schools are rich sources of knowledge. They are a place for people to come together and discuss ideas–not wisdom. One can acquire knowledge at the university, but the wisdom to use that knowledge well must be found elsewhere.

          Therefore, going to a “fancy school” doesn’t give a person more merit; it is up to the individual to effectively use the blessings that he or she has been given.

          I believe we are in agreement with respect to all of these things. Here is where we disagree:

          I don’t see reason to believe in mind control, nor do I see value in such an ability. Perhaps, when you think good thoughts about others, you start treating them better without realising it, merely because you have made positive suggestions which are influencing your actions.

          I also don’t believe that we are gods, or that we hold the power of God, or that, as you put it, “we are the creators of our worlds. We hold within ourselves the power of our own God”.

          I believe in a single, absolute, and universal Truth, and likewise, a Single, Absolute, and Universal God. He rules everything. He is Omnipotent, Omnipresent, Omniscient, and Omnibenevolent. Jesus Christ Himself, being equal with God did not think equality with Him was something to be sought, but became a man and died for us. That’s what heaven is: the complete, altruistic surrender of all that you have been given, for it isn’t yours anyway. Heaven is about self-sacrifice which springs from love.

          • Wow! I loved everything about your response. Concise and clear, truth based and detailed, and most of all organized and loving (gentle). Truly a very nice writing style.

          • Carl Jung has a very interesting theory to explain the similarities between cultures. He calls it “the universal unconscious.”

  2. Hi … forgot to add another experience I have been having lately. It is to do with reality …… I was having this series of discussions with God yesterday morning as is my want every morning just before I get out of bed and while doing this …. as one subject ended and before I could go onto the next my body would shudder (like a brake being applied ) and as it concerned on going subjects for a while now … it seemed to me that reality was being folded in layers … like I was going in and out of time zones and different things I was talking of was being filed in different layers of my reality. I understood that it has to do with time …. for I am always in a hurry being in my late 50s. I actually reacted like a piece of machine would.
    Couldn’t find your profile so I don’t know who I am talking to …. but I’m generally comfortable for I do believe that others are intrinsically as good as I am.

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