Straightening out the cladistic confusion

Cladists have obviously confused harnessing horses to the cart, putting the cart in front of instead of behind the horses. They obviously (see, for example, Mikael Härlin) think (and claim) that classification is posterior rather than prior to hypotheses.The fact that every hypothesis requires a prior classification has obviously escaped them. How on earth could a hypotheses originate without a prior classification. What on earth would it in that case specify?

This confusion has obviously led cladists into a search for a single True Classification of biological organisms, which they call the "natural" classification or The Tree of Life, USING prior classifications of biological organisms and logical reasoning, just as if there is a True Classification lurking beyond all possible prior classifications.

This irrational idea is easily falsified by considering the fact that every logical reasoning is orthogonal (ie, diametrically opposed) in a classificatory sense between its premises and conclusion. If, for example, the premises are that:

1. All men are mortal
2. Socrates is a man

(thus allocating Socrates to the class "men"), then the conclusion

3. Therefore, Socrates is mortal

allocates Socrates to the class "mortals", which is orthogonal to "men", because although all men are mortals, not all mortals are necessarily men. The two classes are orthogonal, ie, acting on orthogonal classificatory levels. Every logical conclusion is simply orthogonal in a classificatory sense to its premises. Logical reasoning can thus not retrieve its initial classification in conclusion, and if the number of possible logical reasonings are infinite, then the number of cladistic hypothesized True Classifications is either infinite or zero. There is thus no single "natural" classification, or True Tree of Life, lurking beyond all possible prior classifications.

This finding of, and investigation of, the cladistic confusion ought to be unnecessary. No knowledgeable researcher ought to fall into it. The question why cladists (beginning with Willi Hennig) obviously fell into it anyway remains to be answered. The reasons may be many, but only they themselves can answer it. A confusion is it in any case, actually a conceptual melt-down. And, a specific solution to the confusion is not to be found. The number of solutions are either infinite or zero, meaning, in practice, infinite.             

     

Is "clade" a class or an entity?

The approach called cladistics in biological systematics rests on the notion of "clades". This kind of "thing" is, however, confusing. The fundamental problem is that if "clade" (in indefinite form), analogous to "human", means both the class clade, analogous to the class human, and a single clade, analogous to a single human, then every single clade must consist of ONLY clades, and there must also be NEITHER anything else than clades NOR any clades at all, contrary to "human". "Clades" does thus, obviously, exclude everything, including itself. Accepting it does in practice mean denying everything, including itself.

What, then, is the reason for this confusion? Well, the reason is that "clade" is the opposite to "object". "Clade" is thus not a kind of "object", but actually the opposite to "object". It means that we have to think up-side-down and in-side-out when we discuss "clades". Thus,

- whereas there are different kinds of objects, there are different clades of kinds.
- whereas "object" is ambiguous between pattern and process, "clade" is instead contradictory between pattern and process.
- whereas objects are classified, classes are cladified.
- whereas objects are different from classes, clades are classes.
- whereas objects exclude clades, clades exclude objects.

"Clade" is simply using the ultimate conclusion if one starts with objects as axiom, that is, Russell's paradox, as axiom, thus landing in the opposite to object as ultimate conclusion (wherein the conclusion equals the axiom). It is a play with words landing in a paradox (i.e., Russell's paradox) as both axiom and conclusion.

The confusing property of "clade" does thus reside in that it turns assumption (actually axiom) and conclusion up-side-down and in-side-out. However, it is not the turning up-side-down and in-side-out itself that is confusing - it might just as well have been a normal development in science, a so-called "paradigm shift - but instead the fact that "clade" is a paradox, since it means that the axiom can't be real. Not even if all biological systematists would agree on this axiom could it be real, because it is contradictory, i.e., a paradox. "Clades" simply can't be real because they are contradictory, i.e., a paradox. The confusing property of "clade" is thus that it is contradictory (i.e., a paradox) and thus can't be used as an axiom because they are far from self-evident (as axioms ought to be). A contradiction may, of course, be self-evident to some of us, but not without contradiction.

The answer to the question "Is "clade" a class or an entity?" is thus "both". It is both a class and an entity. The problem is that class and entity are contradictory. The only difference between cladists and non-cladists is thus that cladists interpret this "both" erroneously as a confirmation (i.e., it is both at the same time), whereas non-cladists interpret it correctly as a contradiction (i.e., it is neither). This difference sends cladists on a vain search of a contradiction they call "clade", which currently seems to occupy a large part of the "research" in biological systematics. This search is thus correctly diagnosed as a mental desease - paranoia.
      

Den biologiska klassificeringens historia

Klassificering (egentligen begreppsbildning) är internt självmotsägande genom att splittra varje klass i (minst) två klasser. Varje klass är alltså både en klass och två eller flera klasser på samma gång. Denna inneboende självmotsägelse formulerades i logiska termer av Bertrand Russell 1901 och kallas idag Russels paradox.

Problemet med denna paradox kringrände (på engelska "bypassed") dock Aristoteles redan för 2350 år sedan genom att klassificera på två diametralt motsatta (kallade "ortogonala") nivåer, vilka han kallade "specificiteter" (dvs enskildheter) och "generaliteter" (dvs allmänheter), genom att särskilja det han kallade "specifika skillnader", dvs skillnader mellan specificiteter (enskildheter) av ett genus (dvs en allmänhet). Hans system "bröt" alltså upp verkligheten in i detta ortogonala system, dvs specificiter av olika generaliteter, istället för att helt platt klassificera objekt.

Aristoteles hade också en djup förståelse av vad han gjorde och varför, och förklarade det utförligt i många skrifter (i vilka han också skapade logiken som disciplin). Han förklarade bland annat att definitioner av begrepp är "en uppsättning ord som är ett, inte genom konjunktion [som logisk operator], såsom i Iliaden [det troligen äldsta bevarade Europeiska eposet], utan genom att referera till ett enskilt objekt". (Han avfärdade alltså uttryckligen den logiska konjunktionen som definitionsmetod, vilken kladistiken idag använder konsekvent (t ex "en klad är en monofyletisk grupp", dvs i en allmän mening: "en si är en så")).

Aristoteles löste alltså den "paradox" (dvs problemet med klassificeringens inneboende självmotsägelse) vi kan "känna" långt innan Russell kunde förklara den logiskt. Utvecklingen av detta problem tog dock ytterligare ett stort steg innan Russell kunde förklara det, i och med att Carl von Linné i början av 1700-talet konstruerade den hierarkiska sammansättningen av Aristoteles grundbult. Linné "flätade" alltså ihop Aristoteles specificiteter och generaliteter i mer omfattande specificiteter och generaliteter, i mer omfattande specificiteter och generaliteter, och så vidare... Detta system är alltså "Den enda konsekventa (dvs icke självmotsägande) klassificeringen" i en allmän mening, vilket är allt vi kan uppnå med Aristoteles system. Det utgör alltså en slutpunkt för denna startpunkt (vilken också är startpunkten för dagens traditionella vetenskap, dvs objektiv vetenskap, eller empirisk deduktion). (Linné är alltså VÄLDIGT stor, de facto oomkullrunkelig, faktiskt upphovet till dagens "appar".)

Linnés system (liksom Einsteins teoretiska upptäckt att tiden är relativ till rummet, vilken senare bekräftades empiriskt) skapade naturligtvis förstämning hos företrädare för den motsatta uppfattningen, dvs att klasser är reella och att definitioner således ska vara konjuktioner, kallad realism (åsyftande dess axiom att klasser är reella). På 1960-talet gick de dock till frontalangrepp på objektiviteten ledda av den Tyske entomologen Willi Hennig genom att helt frankt hävda att en klass, vilken han kallade "monofyletisk grupp" men vilken senare kom att kallas "klad", är vad han kallade "naturliga grupper". Angeppet återupplivade den för-Aristotelianska (och således för-vetenskapliga) uppfattningen som företräddes av Parmenides och representeras i Iliaden, att klasser är reella (med den ultimata logiska slutsatsen att världen "egentligen" är oföränderlig, dvs att förändring endast äger rum i våra huvuden). Hur Hennig kunde komma på idén att återuppliva denna förkastade uppfattning trots de brister som ledde till dess förkastande är obegripligt om man inte antar att Hennig var okunnig om såväl begreppsbildningens, vetenskapens, och filosofins utveckling de senaste 2500 åren. Den går endast att förstå genom att anta att Hennig (och därmed också hans efterföljare) helt enkelt är okunniga.

Problemet med realism är, naturligtvis, detsamma idag som det var för 2500 år sedan, dvs att den är inneboende självmotsägande (i.e., paradoxal), dvs praktiskt orealistisk, av den enkla anledningen att klasser är internt självmotsägande genom att splittra varje klass i (minst) två klasser. Om man, likt kladister, letar efter den så söker man alltså förgäves. Bättre vore att att de tänkte om själva från botten och upp, isället för att lura in unga studenter i denna fälla genom konsekvent begreppsförvirring.


       

Realism vs nominalism (or cladistics vs Linnean systematics)

Classification can only be contradictory or ambiguous: internally contradictory or ambiguous with regard to reality. It is internally contradictory when applied on a single level (like cladistics), and ambiguous with regard to reality when applied on two levels (then necessarily orthogonal, like Linnean systematics). Classification thus can't reach consistent unambiguity, for the simple reason it IS consistently ambiguous, that is, splits every class into two classes.

Now, there are fundamentally two opposite approaches to conceptualization of reality: realism (i.e., subjectivity) assuming as an axiom that classes are real (and thus that objects are abstract), and nominalism (i.e., objectivity) assuming as an axiom that objects are real (and thus that classes are abstract). Since realism assumes that classes are real, it tends towards classification on a single level, that is, towards internal contradiction, whereas since nominalism instead assumes that objects are real, it instead tends towards classification on two levels, that is, towards ambiguity with regard to reality.

In summary: there are fundamentally two opposite approaches to conceptualization: realism and nominalism, whereof realism is internally contradictory and nominalism is ambiguous with regard to reality. Between them resides the idea of a consistent and unambiguous classification (e.g., the True Tree of Life), thus totally impossible to reach.

The idea of a consistent and unambiguous classification is thus a running point, which realism tries to catch and nominalism tries to encircle. Neither of them will, of course, succeed to catch the impossible idea, but whereas nominalism (i.e., objectivity) ought to close up on it, realism instead drifts without a rudder. The realistic (cladistic) idea that parsimony might do the trick is a return alley, since it does not offer any criteria to distinguish between just as parsimonious hypotheses, which, thus, will always remain. A worse problem with realism is, however, that it is both consistently inconsistent and at odds with facts.

The sooner we get rid of realism in the form of cladism in biological systematics the better thus for biological systematics. How can anyone believe that biological systematics can defend consistent inconsistency and incongruence with facts in the long run?
     

What is a rabbit, really?

Human discussion about reality has forever been split between the two approaches nominalism (i.e., comprehending classes as artificial constructs, that is, resting on the axiom that only objects are real) and realism (i.e., resting on the axiom that classes are real). The two approaches can't be distinguished in discussions except by that when a nominalist for example says: "this is a rabbit", a realist asks: "how do you know?". The difference between them is that the nominalist views the classification as a provisional agreement to discuss common properties of objects of this kind, whereas the realist views it as a statement of a fact.

Neither of these two approaches is thus completely satisfactory concerning the question "what something really is", or "how the universe is constructed"; nominalism not considering the question and realism just asking the question (i.e., not providing the answer). Considering the great interest in this question, it is, however, important to straighten out if there possibly can be an answer to this question and what it, in that case, possibly can be.

The first question (i.e., super-question) is thus: can the question (i.e., sub-question) "what is a rabbit, really?" have an answer? It implies (i.e., assumes as an axiom) that there is an existential "True" class for every object, i.e., that every object basically IS a single something, that is, belongs to a single fundamental class. The super-question is thus if this axiom is sensible, that is, if there is there an existential "True" class for every object, if each object is a single something and if each object belongs to a single fundamental class? The answer to this question can be found in the fact that the axiom requires that every single object also is a single class, since every class that includes more than one object has to turn at least one object ambiguous between at least two classes. The answer to the super-question whether the sub-question  "what is a rabbit, really?" can have an answer is thus that it can, given that every object also is a single class.

This answer also gives the answer to the second question, that is, what this answer can possibly be. The answer is that the answer must be that every object is its own class. The sub-question is thus sensible ONLY IF each object is its own class.

The simple logical reasoning above thus discloses that realism is sensible ONLY IF each object is its own class. It means that the question "what is a rabbit, really" does not have an answer per definition, since it doesn't specify which rabbit, and that the more general question "what something really is" has the answer "what it is". The latter is thus a feedback-answer to the question. (For a more thorough understanding of this conclusion, please read Gödel, Escher and Bach by Douglas Hofstadter.)

The generic conclusion of this logical reasoning is thus that realism is a feedback-loop from answers to questions, that is, paranoia. It ports wherever the question leads it to end up, thus drifting with the question it poses, but always contradictory.

On clades and (the) Higgs boson - chasing paradoxes

The two facts that (1) logic is rational and has two entrances: the two axioms that objects respectively classes are real, and (2) that reality is irrational means that rationality meets reality as an orthogonality. Such meeting point means that reality is a passage (i.e., a process) in a rational sense - the passage between the two entrances to rationality. Reality thus can't be caught by rationality, because it actually IS the passage between the two entrances to rationality. This fact is expressed by Russell's paradox - neither of the two entrances to rationality can reach reality. This fact is more simply illustrated by Magritte's pipe - an illustration of a pipe is not, of course, a real pipe (can't be smoked).

The reasoning above explains and illustrates how we can get lost among words, but, more importantly, points at the illusion that resides within words called Russell's paradox. This paradox is actually the orthogonal opposite to object, and is thus just as real as objects are in a rational sense (i.e. not distinguishing the two entrances to rationality: objectivity and subjectivity). It is thus both a paradox and a reality at the same time (although paradoxes are not real per definition) in a generic sense in both of the  rational approaches. It means that rational reasoning may acknowledge this kind of phenomenon independently of which entrance one takes into rationality, although it can't be real (i.e., can't exist). This kind of phenomenon is thus a lure for rationality. It displays all necessary properties of being real in a rational sense, although it isn't real per definition. Its existential problems is further emphasized by that it denies the existence of objects, which it rests on.

Two examples of this kind of phenomenon is the class clade and (the) Higgs boson. Much can thus be said about this kind of phenomenon, but most important is that it is contradictory (i.e., paradoxes). Although it thus is real in a rational sense, it can't be found in reality. If this fact means that paradoxes are real and empirical realities (like objects) are not real, or not, is a matter of words, but the fact that paradoxes are contradictory is a fact. It means that acknowledging such paradoxes leads into contradiction, independently of whether they are considered real. Contradiction comes with the acknowledgement. Ironically, the "acknowledgement" is actually a confession to (rational) belief instead of rationality. Generic agreement between the two entrances to rationality - subjectivity and objectivity - is thus actually an entrance to their contradiction, that is, belief. The passage that rationality tries to catch does in practice lead to belief.

A much simpler explanation of the problem complex described above is that conceptual contradiction, what we normally call contradiction, is contradictory. It may be comprehended as The Truth by some rationalists, but it is actually an elevator to belief. It ought to be a no-no for all rationalists, but in the battle of the formulation privilege, some of them (us) just can't resist the temptation of diving into it, although it is their own end. Fact is that neither clades nor (the) Higgs boson can be found in reality. Both of them are just varieties of Russell's paradox.

 

  

Why (the) Higgs boson will never be found

(The) Higgs boson will, of course, never be found, for the simple reason that kinds do not exist (i.e., are not real), as theoretically shown by Russell's paradox and empirically evidenced by the fact that time is relative to space. This fact (i.e., that kinds do not exist) does, of course, also apply on the class "Higgs boson" (given that it does not consist of a single object - "The True Higgs boson". Physicists' attempt to "prove the existence" of this class (or of any class) is nothing but ridiculous. even if performed in the high-tech context at Cern. It is, none-the-less, nothing but a case of what I would like to call "blinding by belief" - a turning of deduction into induction under influence of a subjective belief that reality must be rational, although both theory and facts contradict this belief. Reality is, on the contrary, obviously, irrational, although following some basic principles that allows us to discuss it rationally. Reality isn't rational, only we are (at least some of us). (The) Higgs boson is thus, of course, just a tool that we (objectivists) use to be able to talk about reality both consistently and rationally. Without it, not only we (objectivists), but all of us are totally lost. The only alternative is the subjective (i.e., believing) contradiction (see Cladistics) that presently lead some physicists into a vain chase for (the) Higgs boson.

When will subjectivists stop chasing their own classification (i.e., "what really is")? I can give them the answer: reality is, and it is irrational. That's why it can't stop. We can only discuss it, not find out what it "really is". (If we could prove it to be something else than what it is, then we could also prove this something to be something else, and so on in an endless transformation between representations of representations.)

These physicists are like donkeys chasing the carrot in front of their eyes (just like cladists). The subject for their chase is their own assumption, and their own assumption is contradictory as a conclusion. They are simply contradictory from the beginning to the end.

Cladistics is the fundamentalism of science, ie the belief that concepts are real

Conceptualization began mankind's intellectual development. Today, cladistics is trying to end this development with reverse conceptualization. Unfortunately for cladistics, reverse conceptualization is contradictory, i.e., has no definite solution. Cladistics will thus never be able to finish mankind's intellectual development (luckily), but will instead continue its vain search for the solution that could forever. Meanwhile, cladistics is the fundamentalism of science, ie the belief that concepts are real.

Biological systematics - the vain quest for The Consistent Classification

Biological systematics has been chasing the idea "The Consistent Classification" since at least the ancient Greeks, that is, for at least 2500 years. The goal is a classification that breaks even, ie, is free from contradictions.

Today, we have, however, obtained the knowledge to figure out that this goal is actually an interface between subjectivity and objectivity (traditionally called realism and nominalism, respectively) and thus not a "thing", but a passage between two things, ie, between subjectivity and objectivity. The goal is the difference between subjectivity (realism) and objectivity (nominalism). If we analogize subjectivity and objectivity with railway stations, the goal is the rails between them. It is possible to pass, but it is not possible to transform to a single station (or several stations) in both approaches, because it in one view (subjectivity) has to be transformed into types of stations (ie, into several stations of the same kind) whereas it in the other has to be transformed into single stations. The two approaches thus can't see (or can't recognize) the same thing (s). The passage between them, that is, the goal for biological systematics, will thus remain an unattainable idea as long as the two approaches exist. This passage was in the early 1900's named Russell's paradox in objectivity, and about 50 years later named "clade" in subjectivity.

Biological systematics is thus closing up on understanding of the context, but right now too much of it (perhaps most) is too busy trying to find the ultimate clade, ie, Russell's paradox, which it calls the Tree of Life, to be able to think clearly. These biological systematists think they are so close to the goal that thinking is finished and only the practical chase remains. They can almost see the goal.

Biological systematics is thus approaching understanding of the context, but right now it is too occupied with the vain chase for "The consistent classification" to be able to think clearly. Ironically, the approach called evolutionary taxonomy has already formulated the solution (although not the explanation).  

On the relation between "clade" and "Russell's paradox"

The term (i.e., class) clade is contradictory: if clades can be unambiguously distinguished, then entities (like organisms or biological species) can't, because clades are incompatible with entities. If, thus, clades are entities, then they are contradictory entities, which, thus, can't be unambiguously distinguished. Clades thus can't be unambiguously distinguished if they are entities (and, of course, neither if they are not entities).

This contradiction is defined in Wikipedia in terms of Russell's paradox as (slightly modified by replacing set with clade):
(The axiomatic definitions are that any definable collection is a set (i.e., naive set theory) and that clade is a set). So, let R be the clade of all clades that are not members of themselves (i.e., the ultimate clade of clades, or the True Tree of Life). Now, if R qualifies as a member of itself, it contradicts its own definition as a clade containing all clades that are not members of themselves. On the other hand, if R does not qualify as a member of itself, it qualifies as a member of itself by the same definition. R does thus neither qualify as a member of itself nor as not a member of itself. This contradiction is a true paradox (i.e., with two impossible and contradictory states) called Russell's paradox.

Symbolically:

The term clade thus denotes the same "thing" (i.e., contradiction) as Russell's paradox, that is, the interface between objectivity and subjectivity, An entity of this kind is thus just a passage between objectivity and subjectivity. The difference between it and an entity is analogous to the difference between a passage through a door and the door. Never can they fuse, and the door is, obviously, a paradox. .    

So, what does this paradox tell us? Existentially that history is inherently contradictory, theoretically that our distinctions can't be found, and practically that theorists can screw matters up all the way to contradiction. What it definitely does not tell us is The Truth. On the contrary.   
 

Cladistics is the practical search for Russell's paradox

The class clade is the subjective aspect of Russell's paradox (link to Wikipedia).

The (subjective) approach called "cladistics" is thus the practical search for Russell's paradox (per definition).

Strangely enough, Wikipedia doesn't accept that this is the proper definition of "cladistics" (link to Wikipedia). Doesn't Wikipedia understand what it explains?

On the third alternative approach to the question of the origin of Life (i.e., evolutionary taxonomy)

Cladistics claim to BE phylogenetics. However, if cladistics (i.e., classification of biodiversity into only clades) would be possible, then evolution would be impossible, and thus also a Tree of Life, and much more the single True Tree of Life cladistics acknowledges. Cladistics is thus contradicted by denying, in turn, evolution, Trees of Life and a single Tree of Life, that is, itself.

Cladistics thus makes phylogenetics appear ridiculous. If its self-contradiction would be the only alternative to creationism, then humanity would be doomed to self-contradiction - either as a self-contradictory belief in a single True Tree of Life (i.e., cladistics) or as a self-contradictory belief in that a God created Life.

Luckily, there is a third alternative - belief in a single origin of Life combined with understanding that this origin can't be described unambiguously, that is, understanding that there is no single True Tree of Life to be found (i.e.,the scientific approach called evolutionary taxonomy). The scientific alternative thus bypasses the contradiction between the self-contradictions of cladistics and creationism by acknowledging Darwin's idea of a single origin of Life combined with understanding that a description of the origin of Life has to be either ambiguous (which it acknowledges) or contradictory (which it does not acknowledge, but which cladistics only acknowledges). The scientific alternative thus distinguishes itself from cladistics by denying cladistics' idea "a single True Tree of Life". It understands that this idea is practically impossible.

Evolutionary taxonomy is thus the preferable approach for people that acknowledge Darwin's idea of a single origin of Life, but which at the same time understand that such origin can't be unambiguously described. They understand that the fundamental obstacle for an unambiguous description of reality is that reality is distinct from descriptions of it. They do not fall into the cladistic trap of confusing reality with descriptions of it.

Evolutionary taxonomy is thus the only approach that combines belief in Darwin's idea with practical possibility. The system it uses for classification is the orthogonal Linnean systematics. It is thus not revolutionary, but inclusive. It includes Darwin's idea in Linnean systematics.

Cladistics is the denial that reality has two aspects, pattern and process

Cladistics is the denial that reality has two aspects, pattern and process, and the corollary claim that not every description of history thus is self-contradictory per definition, but that there instead is a single True description of history to be found, manifested in its idea "a single True tree of Life".

Fact is, however, that reality has two aspects, pattern and process, and thus that cladistics' both denial and claim are plain wrong. Instead, fact is that every description of history is self-contradictory per definition, and thus that cladistics' idea of a non-contradictory "True" description of history is an illusion, a "pie in the sky". It would not help if all historians agreed on this idea, it would still remain an illusion. Facts is not a matter of popularity contest, but of undeniable conditions. Cladistics does thus deny undeniable conditions to instead claim self-contradictory conditions. Much can be said about this proactive "move", but most important is that it is self-contradictory.  

How long can this go on?

Isn't allowing cladists to preach their belief in scientific institutions letting the fox into the hen-house?

It is impossible to find a non-contradictory dichotomous classification, because every dichotomous classification is contradictory.

The approach called cladistics is a belief in the contrary, but it is both internally and externally contradictory, and also falsified by facts.

So, how can science allow cladists to preach their contradictory belief in their institutions? Isn't it letting the fox into the hen-house?  

Cladistics is the ultimate paranoic trap for theorists that begin with the axiom that classes (instead of objects) are real

What cladists don't understand is that their idea "the class clade" is contradictory, due to that the class object and the class class contradicts each other. The reason is that objects participate in one orthogonal system with its two aspects pattern and process, whereas classes participate in another orthogonal system with its classes in classes, and that these two orthogonal systems are orthogonal to each other.Such orthogonal relation between two orthogonal systems is actually a double ambiguity between the two kinds of entities of the two systems, which, in turn, actually is a contradiction (i.e., a paradox). The class object does thus contradict the class class, and vice versa, making up a paradoxical relation.

Such double ambiguity (i.e., paradoxical relation) between classes (i.e., objects and classes) may be comprehended as an unambiguity, following the principle that "two wrongs may be comprehended as one right", but it is a (practical) illusion. The problem is that every possible specific solution of one of them is contradictory with all specific solutions of the other. There is thus no specific solution that is not contradictory between them (i.e., the two orthogonal systems); but the illusionary unambiguity is restricted to a generic level. The relation thus appears unambiguous in a theoretical sense, but is contradictory (i.e., paradoxical) in a practical sense.

Cladistics is thus the ultimate paranoic trap for theorists that begin with the axiom that classes (instead of objects) are real, like many biological systematists do. Sooner or later they are bound to arrive to the class clade, which then has to be real (i.e., unambiguous), although it factually is contradictory (i.e., paradoxical). At this point, these theorists have only two options: to reconsider their approach, or to go all the way (that is, to argue contradictory for a contradiction). I can see examples of one of them (e.g., Malte Ebach), but guess that there are examples of the other too although I don't see them. My aim is not, however, to convert cladists, but to protect young students from cladistics.