Induction Rules

Philosophical Reflections XXVII

Part A: Truth & Lies

In Philosophical Reflections 25 and 26 we looked at the validity of inductive reasoning and the nature of truth. Because induction is so important in generating our knowledge, it is clearly important to understand the rules of inductive reasoning.

We have already covered some of the fundamental rules of inductive reasoning when discussing the validity of concepts (Philosophical Reflections 24) and the basis and validity of inductive reasoning (Philosophical Reflections 25–26. Here we’ll look at the rules of evaluating evidence. What evidence can you trust, and what can’t you trust? How much weight should you put on different kinds of evidence?

We will start with general principles and move on to some specific rules.

Nature Doesn’t Lie

That nature doesn’t lie is inherent in reality. A is A is one expression of it: things act according to their nature and cannot act contrary to it.

Consequently facts must never be ignored. Contradictions cannot exist, so any fact of reality which conflicts with what you think you know shows a hole in your understanding. In any conflict between what you believe and reality, it is reality which you must go with. All induction is based on observed reality, therefore, observed reality cannot be ignored or evaded if you wish to enjoy a consciousness free of contradictions: and thus be free to pursue your values without being sabotaged by wilful blindness to reality. To ignore reality is to undercut induction’s foundation – which is to undercut the functioning of your mind – which is to undercut your primary tool for achieving knowledge, life and happiness.

How much is adherence to reality worth? That is the same as: How much is your life and happiness worth?

Senses Can Lie

In an important sense your senses cannot lie. They are physical organs which merely act according to their nature. Thus your retina responds to light and your nerves process and transmit its output according to their biochemistry: they have no volition in what they do.

However, as is well known, sensory illusions occur both within our nervous systems (e.g. geometric optical illusions, water feeling hotter or colder depending on previous temperature exposure, and the effects of certain drugs) and externally (e.g. mirrors and mirages). This is one of the reasons why it is not mindless acceptance of sensory input which is the source of knowledge, but the application of reason to that sensory input. It is reason which tells us what is real and what is illusion: indeed, that is how we know there are illusions.

The primary rule of induction is to integrate all the evidence of all your senses, past and present, into a non-contradictory whole. Your senses always tell you something about external reality: but exactly what that something is, only reason can tell you.

Feelings Just Feel

Feelings neither lie nor tell the truth: emotions are not tools of cognition. They are value-based responses built on past experience and cognition. If your emotions are not pathological or misguided, but are rooted in integrated, rational values, then they are likely to prompt you to act appropriately to the situation. But valid emotions follow valid understanding: they are not evidence of validity in themselves.

As well as emotions, under “feelings” we can include other non-logical things such as hunches, dreams, visions and revelations. Reason does not preclude these. What it precludes is taking them at face value – even more so than taking perceptions at face value. The evidence of your senses comes from the world around you, from what is. A hunch or revelation comes from within, and how do you know what it’s based on? Is it from subconscious pattern recognition – good or flawed – or from random subconscious activity throwing up wild ideas – good or flawed? In any case, they are not derived directly from reality, but from within you. To be confirmed – to become part of your knowledge – requires that they be tested against that reality: integrated without contradiction into your other knowledge. A feeling which flies in the face of the evidence of your senses has to be rejected as inconsistent with reality: as wrong.

Feelings are notoriously unreliable: you just need to observe the host of conflicting religions and the self-deceptions that come from taking feelings as facts. Look at the number of people who feelsure that their religion is right, but later fall away from it in disillusion: not to mention those who are equally sure of sundry other opposing religions. Look at the number of romantic relationships based on nothing but wishful thinking, which start with declarations of undying love (truly felt) but end within months. Feelings are to be enjoyed, but not accepted without thought. Hunches and other activities of your subconscious are a valuable part of creative problem-solving: but not a magic ticket to truth. The only way to understand reality is to look at it and identify it for what it is.

People Often Lie

People, of course, do lie and make mistakes. This is partly why unsupported human testimony is one of the least reliable sources of knowledge.

It is even worse when the testimony isn’t first hand: then you have to deal not only with the possibility of observational errors, mistaken interpretations or plain lying by the primary sources, but with the distortions that occur during serial transmission. The party game “Chinese Whispers”, in which a simple story is passed from person to person, is an instructive example of this distortion, which increases when the story is interesting. It is easily shown that people notice most what interests them or fits their biases, and like to improve it on the retelling!

However, that you shouldn’t believe everything you hear doesn’t mean you shouldn’t believe anything you hear. Indeed, to believe nothing would be to cut yourself off from one of the main benefits of living with other people: the advantages of specialisation, which includes and requires specialisation of knowledge. So how do you judge the reliability of the people on whose knowledge you rely?

The basic rule of thinking is that all knowledge must be based on reason applied to perception of reality. A corollary is the virtue of independence: let no man’s claims come between you and your own perception of reality. The two primary tests of the reliability of another person reflect these: What is the basis of their claims?, and Are their claims consistent with what you already know?

If their knowledge can in fact be classified as knowledge, because they base it ultimately on observation and reason, then you can have far more confidence in their pronouncements than if they go misty-eyed and speak of their or some guru’s feelings and mystical revelations – in which you should have no confidence at all. There is only one reason for a person refusing to prove his claims: knowing he can’t. The validity of a person’s method of knowledge is the sine qua non of giving them any credence at all. Most people realise this when they deal with mechanics and electricians. It is a pity so many are less careful when dealing with values, and the meaning and goals of their life.

If a claim passes that test, does it contradict something you already know? If so, one of you is wrong. If the truth is important in this case, you need to examine the evidence and logic – yours and theirs. Reality and reason are the final arbiters of all disagreements. That someone’s claims conflict with your knowledge indicates someone is wrong: but it isn’t necessarily them!

A secondary test of another person’s reliability is their character and intelligence. Are their claims of fact true, and is their reasoning from those facts sound? So you need to judge their honesty, accuracy as observers, and (if you need to rely on their thinking as well) their reasoning ability.

What about the opinions of “mankind in general”? What is the cognitive value of the beliefs of the majority? Opinion polls are useful if you need to know what people think, but are of little use in determining the truth. By definition, when any new truth is discovered the vast majority haven’t even heard of it. In a society where people are taught how to think, the results of opinion polls would have a better chance of coinciding with the truth than in a society where irrationality is encouraged. But truth is determined by reference to reality, not by reference to other people’s opinions. As Einstein once commented on a book entitled “101 Scientists Against Einstein”: “If I was wrong, one would have been enough!”

How Extraordinary!

A specific part of the rules of human testimony concerns extraordinary claims. The principle here is well known: extraordinary claims require extraordinary evidence.

An extraordinary claim is one which flies in the face of existing knowledge, for example, a claim to be in communication with aliens or to have invented an antigravity device. The reason that extraordinary evidence is required derives from the basic principle that your knowledge must be integrated into a non-contradictory whole. If you are rational and your existing knowledge is not held lightly or loosely, but has been derived by proper principles of reasoning and evidence, then a claim which contradicts it – or is highly unlikely in its context – is up against all the prior evidence upon which your beliefs were based.

Thus, if your neighbour tells you “a car drove past 5 minutes ago”, then usually you can take his word for it. If cars drive by often and your neighbour is honest, there’s no reason not to believe it. But if he says “a flying saucer flew past 5 minutes ago”, he’d need more proof!

On the other hand, as the first rule of induction is that facts cannot be ignored, if extraordinary proof is supplied, then you must modify your beliefs accordingly! Like everything else, the principle is not dogmatism but reason. Perhaps the best statement of how to deal with extraordinary claims is Hume’s maxim:

That no testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous than the fact which it endeavours to establish.

Did You Hear the One About…?

This brings us to anecdotal evidence.

Anecdotal evidence is simply the things people report having happened. It has little cognitive value as a means for proving general truths (though it can have value as evidence for specific occurrences).

Its lack of value is not primarily a matter of doubting what people say, though that is part of it. Much anecdotal evidence is true. The issue is mainly one of coincidence. Trillions of events happen in the world every day, and while rare coincidences might be rare in any individual’s life, they are not rare at all when summed over the experience of all mankind. “A followed B, and that is so unlikely to happen by chance that there must be a link”, is invalid reasoning. The probability that it happened by chance has to be calculated, not assumed. And that is precisely what anecdotal evidence cannot do.

For example, an old medical scam is to sell a “cure” for an “incurable” disease such as cancer or arthritis, quoting “success stories” as proof of its efficacy. Even if those stories are all true, they prove nothing: inevitably, there are cases where remission or improvement happen by the natural course of events. Only a proper scientific study can demonstrate an actual causal link.

To rely on anecdotal evidence is to commit the logical fallacy of post hoc ergo propter hoc: that is, concluding from B follows A that A causes B. For example, a person might have seen with their own eyes that a person with Ebola got better after eating a snail. But 5% of people with Ebola get better anyway. The question is not “does B sometimes follow A?”, as B can often follow A purely by chance. The question is “does B follow A more often than would happen by chance?” And if the answer is yes, the further question is does A cause B, or is there some other explanation?

This does not mean that anecdotal evidence has no value. It can be the first clue to a causal link or even to the existence of a new phenomenon. But it cannot itself prove anything: the hypothesis of a link has to be tested by proper experimental, observational and/or statistical methods. An hypothesis based on anecdotal evidence can be worth testing. But an actual claim to a causal link based only on anecdotal evidence is invalid and worthless.

Part B: Numbers Games

Lies, Damned Lies, and Statistics

We ended Part A with the limitations of anecdotal evidence. That leads us to statistics.

Disraeli famously said, “there are three kinds of lies: lies, damned lies and statistics”: but that is only fair comment on the misuse of statistics. Statistics are a valid and useful tool for discovering or proving causal links, especially where phenomena are complex and there are multiple causal factors involved. As we saw when analysing anecdotal evidence, we need a method to determine whether an apparent connection between events has any meaning. Statistics is one of those methods: it comprises procedures and mathematical formulae for determining (given certain assumptions) the probability that observed relationships have not occurred merely by chance.

However, there are many things to beware of when dealing with statistics.

Validity

Is the method used valid? As noted above, statistical formulae depend on certain assumptions, such as the variations following a “normal” (bell-curve) distribution. Different statistical tools are needed for different types of variation, sample sizes and methods of data collection. Using an inappropriate tool can give false results.

(In)significance

“Statistically significant” just means that there is a low probability that a correlation is due to chance: it doesn’t mean that the linkage is significant in the sense of “important”. For example, if with a huge sample size we get a statistically significant but tiny real difference (say a variation in height of 0.1%), it is not very interesting or “significant” in the sense of something we need to care about. The questions to ask are “is the magnitude of the effect significant compared to all the other things that influence it?”, and “is the effect actually important?”

Coincidence

Isolated studies have to be treated with extreme caution. Statistical significance is generally expressed as the probability that the results are due to chance, and a conventional cut-off level is a probability of 5%. That is, results are usually called statistically significant if the probability that they happened by chance is less than or equal to one in twenty.

Unfortunately, the corollary is that one in twenty experiments with statistical significance at the 5% level are mere coincidences! On average, if you believe every such test you will be wrong one in twenty times. More generally, statistical results significant at a probability of 1/n will happen purely by chance once in every n studies, on average. And there are so many thousands of statistical studies done each year that we can expect there to be false positives in some that are significant to 0.1% or less. Thus, further studies are critical: all studies based on statistics must be treated with caution until and unless the probability level is reduced to a very low level.

A related issue concerns “shotgun” studies that look at many variables. For example, people looking for side effects of drugs or environmental conditions often include every illness they can get figures on, in order to increase their chance of catching any effects. But consider the danger here: if you look at 100 diseases, by chance alone you would expect to see five of them significant at the 5% level and one at the 1% level, with a one in ten chance of finding one significant at the 0.1% level – all meaning nothing. You might have noticed how frequently some strange and rare illness is linked to some supposed risk factor or location, and wondered why they looked at that particular illness. Often – they didn’t. They looked at everything they could get figures for, and a few “passed” the test of statistical significance. So that is a further caution: if a study looks at multiple possible links, the required probability for any one of them has to be divided by the total number looked at. On principle, finding a few things significant to 5% in a study of 100 different conditions is expected and therefore means nothing. The most it can mean is “this particular relation is worth a further look to see if we can confirm it.”

A similar consideration applies to “clustering” of illnesses in particular locations. If you look at 100 locations, chances are 5 of them will show something that is significant at the 5% level yet means nothing. It is the nature of probability that purely by chance, some places will have an unusual frequency of certain illnesses. So pointing to such a “cluster” means nothing until proper investigations have been made.

Biological Relevance

It often happens that a statistically significant effect, e.g. an increase in cancers, is linked to a particular chemical – but the level of chemical used in the study is thousands of times greater than any reasonable level of exposure in the real world.

The usual justification given for this is that a high exposure was needed to cause enough problems to measure; and that one can extrapolate back by assuming a linear response (i.e. proportional to the dose). For example, if 1000 units causes one cancer in 100 animals, then 1 unit is predicted to cause one cancer in 100,000. However this assumption cannot be accepted without independent evidence that the extrapolation is valid. The simple proof of this is that there are many chemicals and other factors (e.g. molybdenum, copper, iodine, Vitamin A, water and sunlight) which are toxic or carcinogenic at excessive levels but actually necessary at lower levels.

Correlation versus Causation

If A and B are linked, does A cause B, B cause A, or are they both caused by something else? The last possibility is one of the reasons why statistical studies must go to great lengths to make the comparison groups equivalent in every respect except the one being examined. This is especially a problem when the effects are small, because then the causal agent, whatever it is, must have a subtle action.

Trends in Results

As Richard Feynman noted in The Meaning of It All, a real phenomenon will stand out more and more from the background as further experiments are done. Conversely, if the apparent effect keeps on shrinking, it is probably unreal. Feynman gave the example of ESP studies. In the early years, quite spectacular results were achieved, but as experiments were refined to weed out fraud, subtle clues, and other problems identified by critics, the improvements in scores over chance shrunk steadily from 200% to 2%. As Feynman wrote, as the experiments got better they disproved all their own previous experiments, down to a residual effect so low as to be suspect itself.

Summary

There are a number of critical pieces of information needed to evaluate any statistical study: and these are almost always omitted from press reports. The most critical are: What was the actual probability level measured? A level of 0.001% is far more likely to be real than one of 5%. And what is the probability that this result itself was achieved purely by chance (which will depend on sample size, number of candidate associations looked at in this study, and number of related studies carried out)? These items of information tend to be left out to avoid making it “too complicated for the public”: at the expense of making the report completely meaningless to the public or anyone else.

The Smoking Gun

The link between smoking and lung cancer is a good example of the process required to turn a statistical link into real knowledge.

Statistical studies indicated a link between smoking and lung cancer. Further statistical studies strengthened the link, and eliminated other possible causal links such as socio-economic factors. This and the fact that smoking precedes cancer indicated a causal link. However, clearly not a strong causal link: many non-smokers get lung cancer, many smokers never get cancer, and those who do usually succumb only after decades of smoking. Scientific study found that cancers are caused by the chance accumulation of genetic damage in cells, leading ultimately to uncontrolled, invasive cell multiplication; and that cigarette smoke contains a cocktail of mutagens that cause such damage.

Thus the combination of statistical studies and scientific investigation produced a consistent and convincing explanation of how smoking “causes” (in the sense of increasing the risk of) cancer. Consistent, confirmed statistical studies proved the link, and scientific studies showed both why the link exists, and why smoking increases the chance of contracting cancer without being a necessary or sufficient cause.

Risk Analysis

Since a common use of statistical studies is to identify agents of harm, a related issue is an appreciation of relative risk. Nothing in life is risk free: neither doing things, nor refraining from them. Both exercise and lack of exercise have their own risks. Even smoking has subjective benefits.

The question is not is there any risk? The question is, is the risk worth worrying about? The latter is actually made of three parts: is it more or less risky than the alternatives; is it more or less risky than all the other things you do without worrying; and is the risk greater than the benefit?

For example, some decades ago the artificial sweetener saccharine was alleged to be cancer-causing after animal studies with huge doses. However it was calculated that even if the dubious extrapolation to normal doses was true, the resulting “risk” was less than that of driving to the shop to buy a drink in the first place, or of crossing the road to buy a saccharine-free drink at another shop. Not to mention the alternative risks of drinks full of sugar.

Similarly, natural foods contain a wide range of carcinogens and toxins – yet lettuce isn’t banned. True, eating is quite risky: but less risky than the alternative!

When combined with the principle noted previously on how to evaluate human evidence, this leads to a principle of evaluating studies of risk which I think should be enshrined as a maxim:

If a study labels something as a risk, but does not put that risk in the proper context of alternative, background and accepted other risks: then it is fundamentally flawed by failing the tests of honesty and/or demonstrated reasoning ability.

I encourage all editors of scientific and medical journals – and newspapers – to return such studies to the authors with a firm note to fix that deficiency.

Part C: Principles & Fallacies

If a thing exists, its existence is felt: anything that exists affects other things that exist, directly or indirectly. To postulate an entity which exists but cannot ever affect the rest of existence is the essence of an arbitrary claim: by definition, no evidence for such an entity can ever be obtained.

This basic principle has a number of specific consequences for inductive reasoning.

Onus of Proof

Asserting the positive is asserting that something exists. Therefore it is asserting that there will be an effect on other existents. Therefore it is asserting that there will be evidence for it. This is why the onus of proof lies on those asserting the positive: such an assertion requires predicting an effect. Denying a claim is merely saying there will be no effect because there’s nothing to cause one: that is, we won’t see anything we can’t explain by what we already know. And that’s alreadyconsistent with the evidence!

Asserting the positive while refusing to offer predictions or evidence is an admission that the claim is arbitrary – which is worse than being disproved (see Philosophical Reflections 26). It is like pleading guilty to murder to escape a manslaughter charge.

Evidence of Absence

It is often stated that “absence of evidence is not evidence for absence.” This is true, but not in a way that lends credence to the arbitrary – which is, unfortunately, often the motive for quoting this truism.

If a belief lacks evidence because it is an arbitrary claim, then the absence of evidence is a consequence and proof of its being arbitrary and hence meaningless. One does not need evidence of absence in such cases. “Evidence” – for or against – is the one thing that has nothing to do with arbitrary claims. Proof that a claim is arbitrary is sufficient reason to dismiss it.

However, absence of evidence is not necessarily evidence of being arbitrary. For example, a physicist might posit the existence of a particle whose existence is felt only at energies higher than currently reachable, or a palaeontologist might predict the existence of certain fossil forms not yet found. While these propositions might have no proof at present, they are not arbitrary if they are consequences of theories developed to explain other facts, and if there are specific reasons why direct confirmation hasn’t yet been found. Indeed, all scientific theories are tested by their ability to make predictions about the not-yet-observed. In that context, the lack of evidence for those specific claims is a strength, not a weakness.

As always, the basic principle is testing against reality. Any claim which avoids such a test is dishonest, arbitrary and without merit. Any claim designed to be put to such a test is honest and worthwhile.

Occam’s razor

Occam’s Razor is a well known principle of reasoning which states that in explaining something, no more assumptions should be made than are necessary. Or, the simplest explanation is the best.

Occam’s Razor is just another consequence of the principle that anything which exists affects other things. Tacking on unnecessary extras to an explanation is effectively postulating existents which have no effect, making them arbitrary and without explanatory value.

Of course, a theory that assumes more existents than another might not have merely tacked them on for no reason: it might just be a more complex theory. But all else being equal, the simpler theory is still to be preferred: but not regarded as true. It is to be preferred, because each previously unheard-of existent requires some explanation as to why, if it exists, it is unheard of. So prima facie, the simpler theory has less problems. But of course reality can be complex, and the only way to really tell is by the appropriate observational tests. Occam’s Razor is a general guide, not a law.

The overriding principle here is propose no more and no fewer existents than needed to explain the facts.

Hempel’s Paradox

Hempel’s Paradox notes that since each observation of a black crow is evidence that all crows are black, so is each observation of an orange cat! This is because “all crows are black” is logically equivalent to “all non-black things are not crows.” Thus, as each non-black thing that isn’t a crow is evidence of that, an orange cat is evidence that all crows are black! (Hempel’s Paradox was described in more detail in TableAus a few years ago by Peter Bloxsom.)

The fundamental issue here is that any thing that exists is a positive existent. It exists and is what it is: that which is cannot properly be defined as “the absence of that which it is not.” Light is not “the absence of darkness,”and matter is not “the absence of nothing.” This is our principle that knowledge is positive, which has two implications for Hempel’s Paradox.

A purely logistical problem is that there are far more non-black things and things which aren’t crows, than there are either black things or crows respectively. This principle is true of all things in reality. To attempt to gain knowledge of a thing by specifically looking at everything that is not that thing is a fool’s errand, for there is so much variety in the universe that looking for a needle in a haystack is trivial by comparison.

Most importantly, the argument only works if the non-black thing might be a crow. If on a diagram of all things you draw a circle labelled “black” and a circle labelled “crow”, the question is whether the latter is entirely contained within the former. Whether yes or no, there are vast tracts of reality which cannot be crows. A crow is a bird of a specific nature. Whatever their colour, you cannot find crows by sending a probe to Venus or looking into a microscope. You cannot use your orange cat as proof that crows are black because a cat cannot be a crow: if you already know it is a cat, it is excluded from the evidence. Thus, the only way one could use Hempel’s inverted reasoning to test the blackness of crows is to send a robot probe searching in places where crows might be found, detecting non-black things, and then identifying those things.

Thus while Hempel’s Paradox is a curious logical puzzle, it has no epistemological value. The purpose of epistemology is to aid in discovering knowledge, not to work out the most impracticalways to do it.

Fallacies

Some fallacies of inductive reasoning have been touched on earlier, but it is useful to give a brief summary of common “standard” inductive fallacies (the following are based on the very useful site Stephen’s Guide to the Logical Fallacies:

• Hasty Generalisation: the sample is too small to support an inductive generalisation about a population, as in judging all men by the actions of one.
• Unrepresentative Sample: the sample used to generalise about a group is known to belong to a subgroup with different properties, as in “all the swans in Perth are black, therefore all swans are black.”
• False Analogy: two things are similar in some ways and it is argued that therefore they share another common quality: but they differ in a way which affects whether they can both have that property. For example, “men and women are all human beings, so they should wear the same clothes.”
• Slothful Induction: the conclusion of a strong inductive argument is denied despite the evidence, as in “I’ve had 15 accidents in the last 6 months, but I’m a good driver who’s just unlucky.”
• Fallacy of Exclusion: evidence which would change the outcome of an inductive argument is ignored. For example, “my team has won the last 8 games so will probably win this time”, when the last 8 games were against poor teams and this one is against the best.

The following “statistical fallacies” are related to inductive reasoning because they concern misuse of generalisations:

• Accident: a generalisation is applied when circumstances suggest that there should be an exception, as in “it is good to be honest, so you should tell the Nazis about the Jews you’re hiding.”
• Converse Accident: an exception is applied in circumstances where a generalisation should apply: that is, the generalisation is discounted because of one exception. For example, “If lying to protect a Jew from the Nazis is moral, all lying is moral.”

The following “causal fallacies” are also related to induction because they are often part of explanations based on generalisations (especially, with statistics):

• Post hoc (fully: post hoc ergo propter hoc): because one thing follows another, it is held to cause the other, as in “he died at sunrise, therefore the sun killed him.”
• Joint effect: one thing is held to cause another when in fact they are both effects of an underlying cause, as in “your fever is giving you a rash” when both are symptoms of an underlying disease.
• Insignificant: one thing is held to cause another, and it does, but it is insignificant compared to other causes, as in “exercise causes heart attacks”.
• Wrong Direction: the direction between cause and effect is reversed, as in “cancer causes smoking”.
• Complex Cause: the cause singled out is only a part of the entire cause of the effect, as in “the accident was caused by the rain” (when you were driving at 150 kph).

Inducing Principles

If you consider the derivation of why inductive reasoning is valid, and the various rules discussed above, you can see that the rules of induction can mostly be summarised under two main principles, one positive and one negative.

On the positive side, induction is induction from reality. Sir Francis Bacon wrote that “nature to be commanded must be obeyed.” To this we can add: nature to be understood must be listened to.All knowledge of the world comes from studying the world: from seeing, hearing, touching, smelling and tasting, and using reason to integrate the resulting evidence into a consistent whole. Induction consists of observation: what is? Reason: what does it mean? And experiment: if it really means what I think it does, what will happen if I do this? These must be the tools you use in your own life: and the tools used by others, if you are to have any trust in their claims.

On the negative side, the main thing to beware of when evaluating evidence is accidental coincidence. Most of the errors discussed previously, associated with anecdotal evidence, statistics and logical fallacies, can be grouped under this umbrella. It is not enough to see that B follows A. You must then show that B follows A more frequently than it would by chance alone.And then you must find out why. Only when you have reached the second stage can you say that there is any connection. Only when you have reached the third stage can you say you understandit.