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Where universality breaks: about Kant's triads and the dual to Laplace's demon

Hi. Let’s wait for more people to show up… Or it’s just us? OK. So listen up, I wrote this very cool post that gathers almost all my mathematical and philosophical interests in one place…

OK, whaterver, if you are not interested I will stop.

OK, let’s try again * clears throat * “The age-old mystery of Kant’s triads has baffled academics for centuries: what is the significance of the third element that finishes each of his triad. And is it possible that the categories were given to Kant by aliens?”

What, now it’s too dramatic? No, impossible, Kant never used any drugs! OK, whatever, I give up, no more introductions for these articles, let’s just get on with it!

The mystery of Kant’s triads

In “The Critique of Pure Reason”, Immanuel Kant introduces his famous list of the pure concepts of the understanding, also known as the categories, which, according to him, are the basic building blocks from which all other concepts are derived. They are split into four groups called quantity, quality, relation and modality. Each of these groups contains three categories, and for this reason the groups are known as triads. If you’ve read Kant, the fact that each group contains three categories instead of two is very weird, as all other Kantian taxonomies are based on dichotomies, and it gets more baffling when you realize that the third category in every triad are not at all as obviously derived as the other two, e.g. in the category of quality, the first two categories basically correspond to true and false, so what does the third one correspond? According to Kant it corresponds to some combination of the two. Hmm…

For context, here is the list of categories (descriptions are mine.)


  • Unity (measure) - recognition of a thing as one
  • Plurality (plurality) - recognition of there being several things
  • Totality (whole) - unification of all things (of a given group) as one.


  • Reality - recognition of presence
  • Negation - recognition of absence
  • Limitation - recognition of a thing as both present and absent (e.g. in different time periods).


  • Inherence and subsistence - representation of a thing as being inherent, everpresent
  • Causality and dependence - representation of causal effects (things being dependent upon other things)
  • Community - representation of a relation where one thing causes the other and the other way around (causality of a substance as reciprocally determining another substance)


  • Possibility - Impossibility - speculation for the possibility/impossibility of a given thing
  • Existence - Non-existence - perception of a thing as existing/non-existing
  • Necessity - Contingency - acceptance of a thing as inherently true (existence given through possibility itself)

Combination, but not entirely

Here is what Kant has to say about the third categories.

II. The number of the categories in each class is always the same, namely, three — a fact which also demands some consideration, because in all other cases division à priori through conceptions is necessarily dichotomy. It is to be added, that the third category in each triad always arises from the combination of the second with the first.

But wait, if the third member of each triad is just a combination of the other two, then doesn’t this mean that it’s not really “pure”/atomic/first-order and should therefore be removed from the list? Kant claims that it is not the case, but his explanation why is vague:

Thus totality is nothing else but plurality contemplated as unity; limitation is merely reality conjoined with negation; community is the causality of a substance, reciprocally determining, and determined by other substances; and finally, necessity is nothing but existence, which is given through the possibility itself. Let it not be supposed, however, that the third category is merely a deduced, and not a primitive conception of the pure understanding. For the conjunction of the first and second, in order to produce the third conception, requires a particular function of the understanding, which is by no means identical with those which are exercised in the first and second. Thus, the conception of a number (which belongs to the category of totality) is not always possible, where the conceptions of multitude and unity exist (for example, in the representation of the infinite). Or, if I conjoin the conception of a cause with that of a substance, it does not follow that the conception of influence, that is, how one substance can be the cause of something in another substance, will be understood from that. Thus it is evident that a particular act of the understanding is here necessary; and so in the other instances.

What is this particular acts of the understanding for which Kant talks about and which enables the creation of each third category from the other two. For me it has to do with a peculiar process which converts empirical knowledge to universal knowledge, using the concept of the (understandable) universe (the world) for which I want to talk about in this article. This process is based on faith, not knowledge, or more precisely, it is based on faith of knowledge.

If empirical knowledge is about creating models, universal knowledge is about accepting the models that we have created as universally true. This this is precisely what the third pure concepts of understanding enable us to do.

Universal knowledge and the concept of the universe

Here I will try to prove Kant wrong by putting forward the following thesis: the third member of each triad of categories is not a pure concept of understanding. It rather represents a way to bridge the other two concepts of understanding to reason. That is, the third member allows us to form a mental image of the understandable universe, or of *substance * as philosophers call it, and to apply the other two concepts to it.

To elucidate my point, consider the problem of induction which I talk about in my text about Gettier and almost everywhere else, which, incidentally, is also the problem which inspired Kant to write the Critique in the first place. You can formulate this problem in the following way: We all know that our knowledge becomes obsolete all the time and that what is valid in one time/place/context isn’t generally valid for another. So how can we justify all the generalizations that we make all the time (and on which our whole thinking is based on?) The simple answer is that we just cannot justify them - that has been known for millenia. The answer is so simple that I am inclined to ask a follow-up question: How and why are we even able to make such claims in the first place? From where do we have the logical apparatus to make them them (which other animals don’t seem to have) and what does this logical apparatus looks like?

The answer lies in the concept of the understandable universe, or of substance (I will use the term “universe” as nobody is sure what substance means). The universe is a concept that, although it sounds obvious, is a bit weird if you think about it. Like, why do we even have such a clear idea of something that we cannot ever possibly observe in full?

It is the concept of the universe that allows us to make general claims, and to act as if we know it all. And general claims about the universe shape our understanding in a way that is wrong but useful. It allows us to make predictions about the world we are living in (which are also wrong but useful.)

Digging a bit into Kant’s epistemology

The section of the critique where Kant puts forward his theory of knowledge (and which occupies most of the book) is split into two parts - transcendental analytics and transcendental dialectics which correspond to the two main types of knowledge which are, according to him, understanding and reason, the main distinction between those two being that the concepts of understanding (which are based on the categories) are related to possible experience, while concepts of reason (such as the concept of the soul, of God and the world) are not. And for that reason they are, according to Kant, kinda illusory.

A key question: to which of the two faculties does the concept of the universe belong? Kant explicitly puts it under reason (the universe and the world being pretty similar.) But also implicitly puts it under the understanding (by including it in the categories). For me, it’s place is somewhere between the two. It is not a concept of experience, because, while we all experience the universe all the time, nobody can ever know anything that is valid for all of it, in neither in practice, nor even in theory. At the same time, the universe is not a concept of reason, simply because it is a precondition for the existence of reason, (as causality is a precondition for the existence of experience.)

But we are getting ahead of ourselves, so let’s start from the beginning - by examining the reincarnations of the idea of the universe in each third category of the four triads.


Unity Plurality Totality

We will examine the categories of quantity, or of number. Categories are based on concepts from logic and the categories from these triad are based on the logical predicates singular, particular and universal, which are often articulated as “one/unique”, “some” and “all”.

Note that the difference between singular and particular predicates, and therefore between unity and plurality, is obvious, it is, for example, the difference between perceiving a flock of 20 birds as 1 flock, or as 20 birds. But the difference between those two and totality is massive - they are different types of knowledge - the first two represent knowledge with limited scope, while, the third one represents knowledge that is universally valid.

To elucidate that point, let’s go back to the predicates, which come from the classical Aristotelian syllogisms:

  • Plurality - One A is B / Some A-s are B (limited scope)
  • Totality - All A-s are B (universal scope)

Presenting things that way makes my point more obvious - statements that have only limited scope can be justified only by observation - if I observe two or three objects that I categorize as As (e.g. “apples”) and I find that they possess the property B (e.g. “tasty”), I can conclude, based on those observations alone, that Some A-s are B or that One A is B.

Statements with universal scope, on the other hand, are axiomatic by their nature. Note that although, All A-s are B and Some A-s are B are very different statements, the basis/evidence from which I conclude that All A-s are B is not at all different from my basis for saying Some A-s are B.

If I enjoy apples and I have never eaten a bad one, I might say that “All apples are tasty” and someone that has the same experience as me, but who is in a more sceptic mood might say that “All apples are tasty”. Our experiences are the same, what is different is my decision to assume that this piece of knowledge is universal.

In other words, despite the fact that they contain just two variables (A and B), statements of the form All A-s are B, reference a “secret third thing” that is not mentioned explicitly - a thing that we called the universe i.e. you can rewrite All A-s are B to All A-s in this universe are B.

Here lies a paradox - the concept of the universe is used to construct empirical statements, but at the same time it is not empirical by itself, as we cannot make empirical observations about the universe (unless we are Laplace’s demon, but more on that later.) One way to escape this paradox is to assume that universal statements are not really empirical - that they create reality as much as they describe it, e.g by saying that all A-s are B-s you are not saying something about the world we are living in, but instead you are defining what A is (something that has the quality B.)

For more info about this, see the second chapter of my time notes.


Reality Negation Limitation

The category of Limitation is a very misunderstood one, as this triad is based on the concept of truth and classical logic is inherently bivalent - its propositions have just two possible truth values - true and false, and the triad of quality has three categories. So what is the third one?

Well, a path for answering this question is examining intuitionistic logic, where a proposition does not have to be either true or false (this connection is particularly interesting because intuitionistic logic is related to a mathematical theory called “category theory” on which I wrote a book about which as the word suggest is inspired by Kant’s categories (or maybe by Aristotle’s, no way to be sure.))

Anyways, classical logic is based on Platonic epistemology, according to which we humans, can have universal knowledge - his view is that knowledge is given by God and that we are not investigating, but actually recollecting the ideas that are given to us. To postulate this universality, classical logic uses the category of limitation e.g. the predicate B is universal as everything is either B or non-B i.e. the B/non-B distinction is an inherent quality of each object in the universe.

Intuitionistic logic, represents the opposite view - the view that knowledge is subjective and it has a limited scope. For this reason, it contrasts reality not with limitation, but with negation i.e. a thing can be B or not B, but it doesn’t have to be one or the other, because, since the predicate B is made up, there is no reason to think that it should apply to every object in the universe.

Again, using reality and negation, I can only make a statement about a given object, e.g. saying that this object is B or saying that it is not B. While when using limitation, I am making a statement about the whole universe - postulating that it is divided into B’s and non-B (or perhaps un-B).

This is how Kant puts it in the chapter on “The transcendental clue to the discovery of all pure concepts of understanding”:

For example, if I say of the soul, “It is not mortal”—by this negative judgment I should at least ward off error. Now, by the proposition, “The soul is non-mortal,” I have, in respect of the logical form, really affirmed, inasmuch as I thereby place the soul in the unlimited sphere of immortal beings. Now, because of the whole sphere of possible existences, the mortal occupies one part, and the immortal the other, neither more nor less is affirmed by the proposition than that the soul is one among the infinite multitude of things which remain over, when I take away the whole mortal part.

There isn’t much more to say after this quote. Here the term “the sphere of possible existences” is used in the sense in which I use “the universe”.

For more info, check the chapter on logic in my Category Theory book.


Inherence Causality Community

We are continuing with reviewing the so called dynamic categories which, unlike quantity and quantity, don’t concern the phenomena themselves, but the way the phenomena are perceived. In these triads, the relations between the third categories and the rest is a bit different from the first two, but it still elucidates the idea of the universe.

The first such triad is the triad of the categories of relation, whose schema are the relationships between different events in time.

The first category which (as all other first categories) is something like the base of the triad, is the category of inherence which determines the relations of appearances to time itself. The second category, the most famous one, is the relation of events based on cause and effect. And the third one concerns another relation - that of community or of interaction as Kant also calls it.

What is community? Let’s first see the explanation for the disjunctive logical statement on which it is based on, in “The transcendental clue…”:

Finally, the disjunctive judgment contains a relation of two or more propositions to each other—a relation not of consequence, but of logical opposition, in so far as the sphere of the one proposition excludes that of the other. But it contains at the same time a relation of community, in so far as all the propositions taken together fill up the sphere of the cognition.

like the Here if we replace “the sphere of the cognition” with “the universe” we wouldn’t be too far from my initial thesis. The disjunctive relation reminds us of limitation and of the relation between A and non-A that we reviewed earlier, the difference being that in this case we aren’t talking about predicates, but about things that change over time.

But there is something else we should note - this quote makes it seem like community, (which is based on disjunctions) is different from causality (which is based on consequence), but later Kant says that “nothing determines the position of anything else in time except that which is it’s cause?”

What is this community then? Later in the book he defines it as a reciprocal cause and effect. He says that that two substances exist simultaneously in community when each of them “contains within itself the causality of certain determinations in the other substance and, at the same time, the effects of the causality of that other substance.”

And where do the two approaches meet? Although the way Kant frames it is a little different, I view community as the idea of the universe as a system in which everything is in a causal relationship to everything else, including the things that are simultaneous (on which Kant stresses on the most.) It is the main “causal chain” which does not allow for the existence of other causal chain and demands that everything be connected to it in order to be considered real.

We don’t talk about this causal chain but intuitively we are always aware of it. For example, we see a phenomena that is at the same time unexplained and very real (it affects our senses very much), we often say that there must be some explanation for it. And I believe that the reason why we say that is because we are operating withing the bounds of the category of community.

Conclude with one more quote, this time from “Systematic representation of all synthetic principles of pure understanding”:

These are the three analogies of experience. They are nothing but the principles for determining the existence of appearances in time, according to all its three modes, namely:

  • the relation to time itself, as a magnitude (magnitude of existence, that is, duration);
  • the relation in time as a series (successively);
  • and finally also the relation in time as a sum total of all existence (that is, as simultaneous).

Here the term used to mean “the universe” is “the sum total of all existence”.


possibility/impossibility existence/non-existence necessity/contingency

The last triad, that of modality, is very interesting too. Unlike the other three, it is not dealing with knowledge itself, but with the relation to knowledge and the knower i.e. you. Again the first two categories form a clear dichotomy - the dichotomy between the possibility and actuality i.e. between things that may theoretically happen and ones that have actually happened. It is based on the two types of logical proposition which Kant calls problematic i.e. ones that lack relation to reality and assertoric propositions i.e. ones that you assert as true or false.

But out of those two types of propositions a third one emerges, namely apodeictical propositions, or ones that are necessarily true. Let’s read “The transcendental clue…”:

The apodeictical proposition cogitates the assertorical as determined by these very laws of the understanding, consequently as affirming à priori, and in this manner it expresses logical necessity. Now because all is here gradually incorporated with the understanding—inasmuch as in the first place we judge problematically; then accept assertorically our judgment as true; lastly, affirm it as inseparably united with the understanding, that is, as necessary and apodeictical.

For me, this quote clearly describes the three stages of evolution of our understanding which finally gives birth to the concept of the universe.

  1. We are making up explanations based on what we see, and hear without confronting these explanations with reality (we don’t expect for things to make sense)
  2. We see that some of these theories are good and start basing some of our decisions on them (we think it is possible for things to make sense)
  3. We accept these theories as true, thereby making them part of ourselves, we stop perceiving any pieces of knowledge that confronts to them (we insist that things should necessarily make sense)

We can see that the differences between stages 2 and 3 are qualitative. Actually they are very similar to the differences between the categories of negation and limitation - possibility implies that something may be true some of the time, for some cases, while necessity implies that it is true for all cases.

Let’s highlight other differences between the two stages that are interesting.

  1. At stage 2 we are operating at phenomenological level - that is, we only deal with things that we directly perceive, while at stage 3 we start imagining things that are based on what we perceive, but which cannot be perceived directly, by themselves.

  2. At stage 2, our observations are scattered, while at stage 3 they form a cohesive whole (which is based on the presumption that the universe itself forms a cohesive whole.)

  3. At stage 2, we can make a random observation at any moment and also drop any observation that are no longer true. At stage 3, every observation must be connected with every other one (it is this connection which is necessary.) So we cannot just observe a new thing, nor stop observing anything without making changes to our whole worldview.

In short, at stage 3 the observer thinks that they understand the fabric of reality - that they are able, through collection of some scattered observations to grasp facts that are valid at all places and times i.e. again, we are talking about universal knowledge, as Kant says in the “Systematic representation…”

That the connection of which with the actual is determined according to universal conditions of experience is (exists as) necessary.

The concept of necessity, is the idea that we are able to do that. The idea that the universe is understandable and, furthermore, that we are able to understand it.

Where universality breaks

Consider this quote about negation and limitation:

“the whole sphere of possible existences, the mortal occupies one part, and the immortal the other.”

This sentence contains the assumption that all objects in the universe are either mortal or immortal, which is questionable e.g. what about the things that are not alive in the first place? e.g. is a chair mortal or immortal?

This issue is not present in the dichotomy between reality and negation e.g. the dichotomy between “mortal” and “not mortal” - here it is easy to sort all possible objects to mortal, e.g. a chair is clearly not mortal. In my view this is the difference between reality and negation.

This is one of the many examples where the principle of universality breaks i.e. cases where it leads us ashtray.

Universality and logical biases

Whether or not we are capable of having understanding without the categories of totality, limitation, community and necessity depends on how do you define understanding, but what we can be sure of is that if we are to remove these categories from our brains, we would still be able to do quite a lot: we would be able to identify different kinds of phenomena (unity/plurality), we would still be able to classify some of these phenomena are as present and absent from our views (reality/negation) and we still be able to identify relations between those phenomena that allow us to perceive them as objects (inherence/causality). Finally, we would be able to make up different possible interpretations of what we see and then classify some of them as actual (possibility/existence.)

Simple living organisms that most certainly are able to get by without the concepts of totality, limitation, community and necessity. And even for us, turning them off can actually be humbling and, you might argue that it would make our thinking more objective, albeit less capable.

  • Totality represents the bias to make claims that are valid for all objects in the universe. Without totality, we would remember that nothing we say is valid about all things or all people and at all times.

  • Limitation represents the bias of applying systems of predicates to objects that don’t confront to this system (AKA boolean thinking). The loss of limitation, would prevent us from, seeing everything as being either or, according to some subjective criteria, when in reality many things can be both or neither.

  • Community represent the bias of thinking that all events must be causally connected with all other events and excludes everything what doesn’t fit in this picture as unreal. So without it, we can have a richer understanding of the events around us.

  • Necessity is the bias of establishing inexistent hierarchical connection between different bits of knowledge. By abandoning the necessity for this hierarchy, we can put our preconceptions aside and see things more objectively.

So, if you take understanding to mean having an unbiased view of what we perceive, one can even make the point that not only it is possible without the third categories, but that these categories cloud our understanding.

Universality and entropy

So if those principles (which are based on universality) are wrong, then why have we adopted them? Well, as we said at the beginning, the existence of each third category follows from that of the other two. At first sight, that might look like a good-enough reason for them to “exist”. But, if you think about it, this line of thought is just an application to the same principle of universal knowledge that is embedded in all of those categories - considering something as necessarily true just because it follows from other things that we consider true i.e. disregarding the fact that we might be wrong and that every piece of knowledge has scope of validate (so it’s not universal.) So, just because we have the first category and we have the second one, does not mean that we have one that is a combination of the two. This is a problem that Kant himself acknowledges and talks about a lot in the transcendental dialectic - the problem that each piece of knowledge has a scope of validity and if you go outside of that scope, then the piece of knowledge becomes irrelevant.

But if the scope of our knowledge is not universal, then how do we know where does it end?

That’s the root of all issues - we cannot know that.

Perhaps the only satisfiable (albeit wrong) answer to this question is that it somehow doesn’t end, and that we must be capable of somehow perceiving the whole world, the whole universe. This is the answer given by Plato, who says that there is this divine world of forms which we are somehow capable of “recollecting”, and this is also the answer that Kant gives - to justify it, he resorting to a separation between things in themselves and things as they appear to us and talking about the second category. We can adopt this as some kind of axiom about the world, the axiom of causality as I will call it (I know that this name coincides with one of the categories, but still I will use it because it’s very appropriate.) If we do that, then the concept of the universe and each third category would just follow from the other two and of this principle.

Or alternatively, if we don’t insist for our answer to be satisfiable (or we want it to be 100% true) we can go on the skeptic’s road and say that it ends as soon as it begins, that we don’t really know anything. That would mean that everything that we observe as a law is not permanent and our knowledge would slowly disintegrate with time - a principle that I dub with the often-misused word entropy (which I am also probably misusing here.)

Laplace’s demon and its archnemesis

The two different answers not only paint two different pictures of the universe that we are living in, but they describe two ways of seeing our role in this universe and our modus operandi. To end the article with a twist, I will write something about them here.

But first, let’s talk about entropy. We can say that the entropy of a given system corresponds to how easy it is to describe it’s state. e.g. an ordered deck of cards has a low entropy because it is easy to describe, while the only way to describe a shuffled one is to just make a list of all cards in the order in which they appear in the deck. The second law of thermodynamics follows directly from this definition - there are billions of ways a deck of cards can be unordered, but there are a just a few ways that it can be ordered. So for this reason the entropy increases with time (provided that we fiddle with the deck) - that is, the more we shuffle it, the less accurate would be our description of its state. An amazing fact that is, I think, underappreciated, is that is valid regardles of the description that we use - no matter if we pick a precise one that describes the state completely (e.g. the it’s 1, 2, 3, 4… etc) or a more imprecise one (e.g. “the first half of the deck is only red, the second one, only black”), no matter if the description is really simple, or massively complex, it will gradually become less and less valid when time passes. This is entropy. And similar things happens when we move in space as well.

In one of his essays, the philosopher and mathematician Pierre Simon Laplace, describes a godlike creature known henceforth as Laplace’s Demon.

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

If we presume that our knowledge is (kinda) universal, this means we humans are some crippled versions of Laplace’s demon that we are theoretically capable of predicting everything and knowing everything about the world we are living in, and are unable to do so only because of constraints in terms of perceptions and computing power. And (this is an extension of this idea that we see all the time) even if a person by themselves is not able to be Laplace’s demon, the humanity as a whole can collectively function as such. This thesis might sound weird, but it is the dream and promise of the human civilization and what many people are subconsciously subscribed to it.

But let’s now look into the unnamed archnemesis of Laplace’s demon that is hidden in a dark dual universe that is governed by entropy other than causality (I am not saying that it has to be called “Marinov’s Demon”, I am just putting the name out there and letting you make a decision). This demon might look inferior to the all-knowing demon of Laplace. Unlike Laplace’s demon, who seems to know everything, this demon does not comprehend any general principle for the way the universe works (except for the lack of such principles.) Unlike Laplace’s demon, which exist beyond time, this demon is always stuck in the present moment and it cannot really make any decision for the future, as it is not sure whether the knowledge it has in a given moment won’t become obsolete in the next one.

You might pity this incapable demon, but in fact it’s situation is no worse than that of it’s more powerful counterpart: like Laplace’s demon, it cannot be disappointed by anything, albeit for opposite reasons: Laplace’s demon cannot be disappointed because it just knows everything and our other demon cannot be disappointed because it knows nothing. The principle of entropy teaches it that everything it knows and everything that it is slowly disappearing. In fact at every instant, this demon sees everything disappearing before its eyes.

And, as it knows that nothing is universal, this demon does not feel attachment to any principle that it discovers, while Laplace’s demon certainly is intimately tied with the principles of the universe. So, for Laplace’s demon life is boring, and being the opposite demon is interesting.

You might call this demon an agent of chaos, a villain (as its personality resembles that of many movie villains), but it actually is very harmless, as it has no motivation for destroying anything or anyone. As a matter of fact, it has no motivation of doing anything - if he is reborn in a human form, he would probably become a really good Buddhist monk.

The only evil thing about it, according to today’s standards, is that it sees no inherent benefit to progress, and to action that is based on a given doctrine. But this is not and evil view. As a matter of fact, it would be good for some of us to adopt this view too.

Not sure how to finish this, as I have a lot of other things that I can write, but thanks for reading my article and stick around for more updates.


Written on February 5, 2022

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