Theories
Persephone and the seasons
This theory explains why winter is cold: Mother Earth is sad and shows her sadness by making it cold. Yet, is this the right explication? If not, then why is this explication wrong? Can we prove that it is wrong?
It turns out that it is very difficult to prove this explication false. To date, no proof has been found that Hades and Persephone do not exist. Also, the theory of Hades and Persephone makes astonishingly correct predictions to date: Every summer, the weather is hotter than in the preceding winter. Does this not prove that Persephone is still under the spell of Hades?
This chapter will develop a theory of truth to address this question (as well as the question of truth in general).
It is certain that nothing is certain. And even that is not certain.
Statements
For the purpose of this book, a statement is any declarative sentence. For example, the following sentence is a statement:- Questions (“Is the Earth flat?”)
- Commands (“Go home!”)
- Interjections (“Ouch!”)
- Incomplete sentences (“Well, you know...”)
How strangely do we diminish a thing as soon as we try to express it in words.
Predictive Rules
For the purpose of this book, a predictive rule is a statement of the formIf it rains and it is sunny, then there is a rainbow.
If a stone is left in the air without support, then it will fall down.
The left part of the rule is called the premise of the rule, and the right part is called its conclusion. The nature of an accurate rule is that if we find a case where the premise is true, then the conclusion will also be true. In our example, if we find a day where it rains and it is sunny, then on that day there will be a rainbow.
A predictive rule can fail to predict something true as well. The following rule is poor at delivering accurate predictions:
A pragmatic perspective on rules
For this book, a predictive rule is a statement of the form “If A and B and C..., then Z”. In everyday discourse, rules are framed in informal ways without a strict form. Examples are as follows:- “Lions roar” is an abbreviation for “If something is a lion, then that something will roar from time to time.”
- “When it rains, the street gets wet” is an abbreviation for “If at some point of time it rains, then the street will be wet in the moments after that point of time.”
- “When it rains, you get wet” can be more precisely stated as “If at some point of time it rains, and you are outside without anything above you, then you get wet.”
It is better to be roughly right than precisely wrong.
Examples for predictive rules
Let us now look at some examples of predictive rules. Indeed, many phenomena can be formulated in this way:- Scientific theories
-
Scientific laws can be stated as rules. For example, the law of gravity can be written as follows:
If there are two physical bodies then the force between them is proportional to the product of their masses, divided by the square of their distance.
- Regulations
-
Regulations can likewise be formulated as predictive rules. For example, the rule that pupils are admitted to the A-levels if and only if they pass the exams in Math and English can be formalized as the following set of rules:
If someone does pass the exam in Math and the exam in English then they are admitted to the A-levels.
If someone does not pass the exam in Math then they are not admitted to the A-levels.
If someone does not pass the exam in English then they are not admitted to the A-levels.
- World knowledge
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Common sense knowledge such as “every human will die” can likewise be formulated as a predictive rule. For this purpose, the implicit assumptions of the sentence become the conditions of the rule:
If some entity is a human then that entity will eventually die.
The most formidable weapon against errors [...] is reason. I have never used any other, and I trust I never shall.
Perceptions
For some statements people can immediately and incontestably say whether they are true. These are statements about their own perceptions and impressions. Here are some examples:- I feel hungry
- I perceive something blue
- I hear a noise
These statements are not statements about the world. Rather, they are statements about someone’s perceptions. For example, someone can have the impression of seeing something blue even if there is nothing blue within their field of vision and instead they are experiencing a hallucination. In that case, the statement about the impression will still be incontestably true — even if the impression in the visual field does not correspond to an object in the world, the person is still experiencing that perception.
The theory of truth that we will discuss in this chapter is a tool to structure and predict these impressions.
My universe is my eyes and my ears. Anything else is hearsay.
Definitions and auxiliary statements
So far, we discussed only very basic statements, namely those about perceptions. We will now build more complex statements by joining perception statements together to form definitions. A definition is a predictive rule whose premise contains only perception statements and previously defined statements.Here is an example of a definition:
in Tunisia, with permission
A rose by any other name would smell just as sweet.
The meaning of auxiliary statements
We have seen that auxiliary statements appear in the conclusions of rules. They can also appear in the premise of rules:If it is raining, and the Sun shines, there will be a rainbow.
If it is raining, and I am outside, and I do not have an umbrella, then I will get wet.
...
Some of these conclusions simply reflect the definition of the auxiliary statement: When it rains, water droplets fall from the sky. Others enumerate more consequences of the premise: We might get wet, we might see a rainbow, etc.
This book posits that these rules are tantamount to the meaning of the auxiliary statement. In other words, when people ask “What does it mean, it’s raining?”, we reply to them that water falls from the sky, that there might be a rainbow, that they might get wet, etc. The conclusions of predictive rules are again statements with a meaning. We can always ask for the meaning of a statement until we reach a perception statement. In this fashion, each statement can be ultimately grounded in a perception.
From now on, we will assume that there is a set of rules that defines uncontroversial everyday statements such as “It’s raining” and gives them a meaning. You can think of this set as the rules that we learn during our childhood. Examples are:
If you sleep, your eyes are closed.
etc.
The paralysis of thought that comes to philosophers: One saying to the other: “You don’t know what you are talking about!” The second one says: “What do you mean by talking? What do you mean by you? What do you mean by know?”
Theories
For the purpose of this book, a theory is a predictive rule, together with the necessary definitions and meanings.Examples for theories are:
- The rule “If it rains and the Sun shines, then there is a rainbow”, together with the definitions and meanings of the words.
- The theory of gravity, with all its rules and formulas.
- The theory that anything that is alive has been created by God, together with the necessary definitions.
Such theories are not necessarily true. Rather, they are our object of study in what follows.
Definitions are native to a theory, such that each theory can define its auxiliary notions in the way it wishes. In the following, we will refer to a theory by its main predictive rule, and assume that the theory contains the definitions and meanings of all common English statements. For example, in the case of the theory of gravity, the main rule says “If there are two physical bodies and their distance is R, then the force between them is so and so”. The accompanying definitions will define the statements in the premise of this rule. For example, “the distance between them is R” is defined to hold if we can take a tape measure, connect it to the first object, lead it in a straight line to the second object, and find that second object at the mark R. This definition would need again rules to define “taking a tape measure”, etc.
Applying a theory
A theory consists of a main rule together with the necessary definitions and meanings to make sense of it. As an example, consider the following theory:Definition 1: If there are droplets of water coming from the sky, then it rains.
Definition 2: If there is extremely bright round shape on the sky, then the Sun shines.
Meaning of “rainbow”: If there is a rainbow, then I can see colors in the sky.
The question is how to arrive at your opinions and not what your opinions are.
Grounded theories
We have seen that a theory can be used to make predictions. This works only if- all statements in the premise are well-defined, i.e., for every statement, there is a rule (or a sequence of rules) that allows deducing this statement from perception statements; and
- the conclusion is meaningful, i.e., there is a rule (or a sequence of rules) that allows deducing perception statements from the conclusion.
I never can catch myself at any time without a perception, and never can observe any thing but the perception.
Truth in a case
A theory can be applied to a particular scenario and yield a prediction. If the theory is grounded, this prediction can always be reduced to a perception statement. There are two cases:- The predicted perception corresponds to our actual perception. We call the theory true in this case.
- The predicted perception does not correspond to our actual perception. We call the theory false in this case.
Both notions of truth and falsehood apply only to one particular case at hand. It can also happen that the theory makes no prediction whatsoever. This occurs, for example, when the premises of the theory are not fulfilled. For example, an inhabitant of the central Sahara may never observe a drop of rain. Thus, she or he will not be able to apply the theory that sunshine with rain yields a rainbow. In this case, the theory can be called neither true nor false.
All sentiment is right; because sentiment has a reference to nothing beyond itself, and is always real, wherever a man is conscious of it.
True Theories
Truth
A theory is a rule with accompanying definitions and meanings. We have seen how to determine the truth of a theory in one particular case. Now a theory is called true if there are cases where it is applicable and if it is true in all these cases. This means all three of the following hold:- There are cases where the theory is applicable.
- The theory is grounded.
- All of its predictions in all cases where the theory is applicable are true.
The problem is, of course, that we do not know which rules are true in eternity. Even if we have observed quite some number of cases in our lives, and others have also observed quite some number of cases, a theory that seems to be true can still turn out to be false one day. Take, for example, the theory “If water is heated to 100 degrees Celsius, it boils”. This theory is generally considered true. However, it may happen that the theory makes a false prediction. For example, in environments with higher air pressure or more gravity, the theory may turn out to be incorrect. There is no guarantee that reality behaves according to the theories we build.
This is the main insight of this chapter: We do not actually know the truth. We just approximate it by our theories. Once a theory behaves nearly perfectly, we assume that it is the truth — but we can never be sure.
Objective truth is an aspiration that no mortal can ever claim to have attained. But the conviction that it is out there licenses us to approach the truth collectively in ways that are impossible for any of us individually.
Proving the truth of a theory
A theory is true if it will always make correct predictions. As an example, take a naïve version of the theory of gravity: If an object is not held in place, it falls down to the floor. This theory is most likely true — in the sense that it will most likely make correct predictions in eternity.And yet, we cannot prove that the theory is true. This is because there could be a case, one day, where the rule makes a false prediction. For the naïve theory of gravity, this has actually happened: Objects do not fall “down to the floor” in outer space. The theory that objects fall “down to the floor” is actually not true in general.
For this reason, scientists never prove a theory. They just validate it (i.e. they confirm that it applies in all extant instances).
No matter how many instances of white swans we may have observed, this does not justify the conclusion that all swans are white.
Validation
We cannot prove that a theory is true. All we can do is observe whether it makes correct predictions. If the theory has made a number of correct predictions in the past, and has made no incorrect predictions, we say that the theory is validated. We assume that it is part of the truth.Consider an example: We hire a violinist to play at an art exhibition. We know that all of her performances so far have been very well received. The theory is thus “If this violinist gives a performance, then that performance will be great”. This theory predicts that the violinist will do a great job at the exhibition. However, even if all of the performances so far have been great, our trust in this prediction will depend on how many performances the artist already has already given. If, for example, the artist has given only 2 performances so far, then it does not mean much that all of them have been well received. If, on the contrary, the artist has a long career of hundreds of performances (all of them great), then we are more likely to trust the prediction. The rule “This violinist gives great performances” is validated by a large number of correct predictions.
If a theory has delivered a large number of correct predictions in the past, we say that the theory has been validated. Even a validated theory can make false predictions in the future. However, the frequency of accurate performances in the past turns out to be the best predictor for the performances in the future. (The attentive reader will have noticed that this claim itself is already a theory that has been validated by positive examples in the past, but is not necessarily true.)
The effort to understand the Universe is one of the very few things which lifts human life a little above the level of farce and gives it some of the grace of tragedy.
Rejection
in New Zealand
In principle, a single false prediction shows that a theory is false. In practice, however, we know that all of our theories are mere approximations of the truth. Thus, we should be willing to accept a few false predictions. In the example of the violinist, we are willing to tolerate some number of not-so-great performances of the violinist, and still say that “This violinist gives great performances” — if the vast majority of the performances were well received.
I would never die for my beliefs because I might be wrong.
Unknown truth
If a theory always makes correct predictions, we assume that the theory is a good approximation of a true theory. Some theories have not yet made any confirmed any predictions at all. Consider again the theory that a certain violinist would give only great performances. Now assume that this violinist has so far not given any performances at all. In this case, the theory has not yet been applicable. There has not yet been a case where we could see whether the theory makes true predictions or false predictions. Thus, we cannot say that the theory is true, because its truth depends on true predictions. The theory is not false either, because the falsehood of the theory follows only from false predictions. Thus, it is just unknown whether the theory is true or not. The truth of a large number of theories is actually unknown.In the absence of facts, the wise man suspends his judgment.
Useless theories
One condition of a true theory is that it must be applicable, i.e., there must be cases where the theory makes true predictions. The theory may not yet have made a prediction in the past, but it has to make a prediction at some point in time. If the theory cannot make any prediction on principle, then the theory is useless.As an example, consider again the violinist whom we wanted to hire for our art exhibition. Assume that the violinist has trouble with rapid passages in the music, but promises that “If only I had 6 fingers, my performance would be great”. This theory is not applicable. The violinist does not have 6 fingers, and there is (currently) no way to grow such supernumerary fingers. Hence, the theory of our violinist is not true (because it does not make true predictions). It is not false either (because it does not make false predictions). It is not of “unknown truth” either, because truth presupposes applicability, and we know that the theory is not applicable. The theory is just useless.
The most important feature of an insight is that it shall allow us to predict future events and adapt our actions accordingly.
Unfalsifiable theories
A theory can only be true if it is grounded, i.e., if all premises of its rules are well-defined, and if their conclusions are meaningful. It is not always easy to see whether a theory is grounded. Consider again the example of the violinist whom we want to hire for playing at our art exhibition. Assume that she says:This reasoning is known as the No true Scotsman fallacy. Technically speaking, the premise is not well-defined: We cannot determine whether someone is a “true music connoisseur” purely from data that is known beforehand. Thus, the premise of the rule is ill-defined, and the rule is not grounded.
A simple way of checking whether a theory is grounded is to see whether we can imagine a situation in the future where the theory makes a false prediction. For a grounded theory, we can imagine a situation where the rule does not hold in a particular case: The theory of gravity, for example, says that masses attract each other. We can easily imagine a situation where two masses do not attract each other (even if this is unlikely to happen). If there could be, at least hypothetically, a case where the theory makes a false prediction, we say that the theory can be falsified. Now consider the theory with the violinist and true music connoisseurs. It cannot be falsified. If this theory ever predicts that someone loves the performance and it turns out not to be the case, then that person was simply not a true music connoisseur in the first place. Thus, the theory cannot be falsified.
Falsifiability explained
The concept of falsifiability was advocated by Austrian-British philosopher Karl Popper3. A theory is falsifiable if we can imagine a case in the future where the theory makes a false prediction. If a theory is falsifiable, then it is grounded. To see how it’s grounded, consider the rule “If A then B”. To falsify this rule, we have to find a perception statement X such thatIf B then X is false.
The second rule, “If B then X is false” asks for what we can deduce when B becomes true. In other words, it asks for the meaning of B. Falsifiability is thus equivalent to having (1) a well-defined premise and (2) a meaningful conclusion. Thus, falsifiability is a way to ensure that the theory is grounded in perceptions.
If a theory is not falsifiable, it is not grounded: Either its conclusion has no meaning, or its premise has no definition. We say that the theory is meaningless. Such meaningless theories typically take one of the following forms:
- If someone is a real hero/Christian/American/..., then...
- If you do it right, then...
- If ..., then something that cannot be seen/understood happens (such as “it will please God”).
I wish to propose for the reader’s favourable consideration a doctrine which may, I fear, appear wildly paradoxical and subversive. The doctrine in question is this that it is undesirable to believe a proposition when there is no ground whatever for supposing it true.
Generality
So far, we have talked only about the truth of a theory. However, there is something more that we expect from a good theory besides being true. To see this, consider the high tides of the ocean as an example. There was a high tide in New York on Saturday the 3rd of June 2017 at 17:02, and at 5:16 the next day. Hence, we can build the following theory:Now consider the following theory instead:
A general theory compresses information, in the sense that it expresses the same information as a list of events, but in a much shorter form. In our example, the times of thousands of high tides are captured by a single rule. In this way, we do not have to memorize the events, but just the theory, because the events can be reconstructed from the theory. Compression is so quintessential to a good theory that Argentine-American mathematician Gregory Chaitin has suggested that it is tantamount to comprehension4.
Brevity is the soul of a pattern: The pattern must be shorter than the dataset itself.
Useless premises
A theory is general if it covers a large number of events. One way to make a theory more general is to remove premises that are not necessary. Consider for example the theory “If I clap my hands, and press the light switch, the lamp will light up”. This theory is obviously true. However, the theory is just as true without the clapping. The clapping does not make it any more likely that the lamp will light up. Therefore, the clapping is a useless premise. Removing the premise will make the theory more general, and thereby more compressive.In the Central Park in New York, a man jumps around on one foot.
People ask him why he’s doing that.
“To scare away the rhinoceroses!”, he answers.
“But there are no rhinoceroses here!”, people tell him.
“You see”, he says, “it works!”
Testability
A theory is testable if we can trigger its premises on our own initiative. Consider for example the theory that Peter has a crush on Sarah. This theory says:Testable theories have a huge advantage over non-testable ones: We can repeatedly trigger the premises, and thus see whether we can validate the theory.
If the facts won’t fit the theory, let the theory go.
Causality
It is difficult to distinguish causality from mere correlation. For example, assume that we submit a random sample of high school students to a math test. We may well find that those students who possess a motorbike do better in the test than those who don’t. This, however, is not because riding a motorbike would make them intelligent. Rather, those who have a motorbike tend to be older than those who don’t have one, and they will thus perform better in the test. If we consider only students of the same age, then those who have a motorbike might well fare worse than the others.The trick to distinguish this correlation from a causation is perform a test. That is, we have to test the theory “If someone has a motorbike, then she or he performs better in the math test”. For this, we give people a motorbike, and see if this improves their performance in the math test. If it does, we have found a causal relationship. There is a caveat here: If we give the motorbike only to the older students in our test, then the cause of the better performance in the math test could still be the age. Hence, the test has to be performed in what is called a randomized controlled trial: The choice of giving someone a motorbike must be completely random (determined, for example, by a coin flip).
In our framework, this means: A causes B, if and only if
- The theory “If A then B” is testable, i.e., we can trigger its premises at will;
- the theory is true, i.e., it makes only correct predictions, on cases that are chosen randomly; and
- the theory contains no useless premises, i.e., B is not just true by itself.
The reason why the rooster does not cause the Sun to rise, even though one always follows the other, is that if the rooster had not crowed, the Sun would still have risen.
Explanation
We can now formally define what constitutes a good explanation for a statement. We say that a theory “If A then B” is an explanatory theory if it fulfills the following three conditions:- the theory is a causal theory;
- the theory is not trivial, i.e., “If B then A” is not true; and
- the theory is general.
We say that a theory explains a statement if it fulfills all three conditions and predicts the statement. An explanation thus consists of an explanatory theory and observations that make the premises of that theory true. As an example, consider again the scenario of high tides. An ideal explanatory theory for the high tides is:
The Moon is a large mass.
The ocean is a mass.
Any fool can know.
The point is to understand.
In summary: theories
This chapter argues that we perceive ourselves in a continuous stream of perceptions. Our goal is to predict these perceptions. For this purpose, we build theories of the form- Grounded
- A theory is grounded if all statements in the premise are well-defined, and the conclusion is meaningful. Only grounded theories can reasonably be evaluated for their truth. One way to check groundedness is to check falsifiability.
- Applicable
- A theory is applicable if there is (or will be) a case where the theory makes a prediction.
- Validated
- A theory is true in a case at hand, if it predicts something that actually happens. A theory is validated if it has been found to be true in the vast majority of cases. Validated theories tend to make correct predictions, and are thus useful.
- True
- A theory is true if (1) it is applicable, (2) it is grounded, and (3) all of its predictions in all cases are correct. The set of true theories makes up what we call the truth. Obviously, we cannot always verify whether a theory is true, because we cannot apply it exhaustively to all cases in the past, present, and future. Therefore, we resort to validating the theory. If a theory has been validated, we believe it to be true.
- General
- A theory is more general if it encompasses more cases. In particular, the theory should not contain useless premises. Generality is akin to compression of information, and makes a rule intellectually satisfying.
- Testable
- A theory is testable if we can trigger its premises at our own initiative. This makes it possible to test the theory, i.e., to validate it by triggering the premises.
- Causal
- A theory “If A then B” establishes that A causes B, if the theory is testable, it is true, and it does not contain a useless premise.
- Explanatory
- A theory is explanatory if it is causal and general, and not trivial.
In the absence of other evidence,
the best predictor of the future is the past.
True Statements
Evidence
For some statements we can immediately see or feel whether they are true. These are perception statements. For other statements we cannot. For example, the truth of the statement “There will be a solar eclipse in 2030” cannot be observed in the year 2021. For these statements, we have to use evidence to confirm the prediction.Evidence for a statement is a combination of (1) a true theory and (2) true premises that, together, predict the statement in question. In the example: If we have a true theory that tells us how the Earth orbits around the Sun and how the Moon orbits around the Earth, and if we know that the Moon, Sun, and Earth are in a certain position today, and if the theory predicts from this information that the Sun, Moon, and Earth will be in line in 2030, then this theory counts as evidence for the statement that 2030 will see a solar eclipse. If a true theory predicts a statement from true premises, we say that the statement is true.
Whatever is not deduced from the phenomena is to be called a hypothesis.
Counter-evidence
A statement is true if it is predicted by a true theory. A statement is false if there is a true theory that predicts the negation of the statement.Take for example the following theory:
There is an alternative way to prove a hypothesis false: A hypothesis is false if it appears in the premise of a true rule, and if that rule makes a false prediction. As an example, take the following rule:
Beliefs do not change facts.
But facts should change beliefs.
Modus Tollens
Suppose that the rule “If it rains, the street gets wet” is true. Then the following rule is also true:Joint inference
A statement is true if it is predicted by a true theory, and it is false if its negation is predicted by a true theory. Now assume that there is a theory that predicts the statement and another one that predicts its negation. By definition, this cannot happen if all theories are true. However, in reality, we do not know which theories are true. Therefore, we often find ourselves in situations with contradictory evidence. In that case, we can only hypothesize about the truth of the statement.If, however, two theories predict the same thing, then the conclusion is more likely to be true. For example, if a suspected murderer has left DNA traces at the site of crime as well as announced his deed on his social media account, then he is more likely to have committed that murder than if only one of these conditions applied.
Sometimes, a more special theory overrides a more general theory. For example, birds can generally fly. So the theory is “If something is a bird, it can fly”. Penguins, however, cannot fly. Here, a more special theory overrides a more general one. We see that if we want to believe in a conclusion, it is not sufficient that one theory predicts it. We also have to check that no other theory contradicts or overrides it.
This means that, in order to predict a future event, we have to know lots of theories, weigh them against each other, and combine their conclusions. Predicting the future is a complicated business.
Prediction is very difficult, especially about the future.
Joint inference for definitions
A definition is a rule that has an auxiliary statement as its conclusion, as in “If a man is not married, he is a bachelor”. With such rules, we can define new notions that serve to simplify our discourse. In reality, definitions are rarely as clear-cut as we would like them to be. Canadian-American cognitive psychologist Steven Pinker gives examples of things that are surprisingly difficult to define9. For example: Is the pope a bachelor? What about the male half of a couple that never bothered to marry? The definition of “woman”, likewise, can get one into trouble these days, he notes. Or take the concept of vegetables: There is no Linnaean taxon that includes carrots, fiddleheads, and mushrooms; there is no one trait that applies to all vegetables and not to other objects (such as fruits). Rather, there are several veggie-like traits (being green, being not too sweet, etc.), and the more such traits an edible part of a plant accumulates, the more we agree that it is a vegetable. In this reasoning, lettuce is a vegetable by excellence, parseley a lot less, and garlic still less. In the terminology of this book, these traits are captured in rules that define the auxiliary notion of a vegetable, as in “if some edible part of a plant is green, then it is more of a vegetable”. (Technically, we enter the realm of Fuzzy Logic10, and this chapter’s theory of truth can indeed be extended to a Fuzzy Logic interpretation.) Vice versa, other rules say that something is rather not a vegetable (for example, if something is sweet, we tend to call it a fruit). Determining whether something is a vegetable or not then boils down to joint inference over these rules.Steven Pinker goes on to illustrate that this type of joint reasoning can be performed very well by neural networks (simplified models of the human brain): The more traits of a vegetable the model identifies, and the less counter-evidence the model sees, the more weight it gives to the assertion that the object at hand is a vegetable. In this spirit, the human brain can be seen as a giant joint reasoning machine over such fuzzy rules.
Reason is our innate ability to discover laws and apply them.
The unknown
A statement is true if it is predicted by a true theory, and it is false if its negation is predicted by a true theory. Now suppose that there is a statement about which no theory at hand makes a prediction. For example, assume that we have theories about physics, chemistry, and biology. These theories contain the law of gravity, the chemical reactions of substances, and the functioning of human organs. Now suppose that our question is “How does Sarah feel about Peter?”. None of our theories can deliver a conclusion that would be remotely relevant.In such cases, something very simple happens: The answer to the question is unknown. This means that there could be an answer but that we do not know it. Thus the right thing to say is “I do not know”. That is not surprising; in fact, the vast majority of things are unknown to us.
Honest ignorance is better than false knowledge.
Falsifiability
We have already seen how to apply the concept of falsifiability to theories. It can also be applied to statements: A statement is falsifiable if we can imagine a situation in the future that contradicts the statement or its meaning. For example, the following statement is falsifiable:This view can even be extended to the realm of human emotions. Consider the following claim:
Falsifiability is about the possibility that an instance could be found that runs counter to the claim being made. Thereby, falsifiability is a purely theoretical property of a statement — it does not actually require actually proving or disproving something to be true or false.
Unfalsifiable statements
From our examples above, it might appear that every statement is falsifiable. Yet this is not the case. Take for example the following statement:Unfalsifiable statements cannot be proven false. This has an interesting consequence: It means that we can come up with several non-falsifiable statements that contradict one another. For example, I can say “Reality is a dream“, and you can say “No, reality is a trick that is being played to our mind by extra-terrestrials”. These statements are contradictory. However, since they are both non-falsifiable, there is no way to show that one of them is false (based on the truth of the other). Hence, we have two statements that cannot be true together, and yet neither of them is false. Therefore, each of us can defend our respective statement without ever giving in. We can literally argue forever. This is indeed what people do.
Falsifiability and meaning
A statement is falsifiable if we can imagine a situation in the future that would prove it false. Formally, a statement is falsifiable if there is a perception statement X about the future, such that the following rule holds:Non-falsifiable statements typically take one of the following forms:
- It is claimed that something is the case, but this something cannot be seen, understood, or verified.
- It is claimed that something is the case in a very abstract form.
- It is claimed that something is the case, but not in the usual meaning of the word.
Falsifiability in a nutshell:
If it cannot be false, then it cannot be true.
In summary: statements
Among the meaningful statements, we distinguish:
- True statements
- A statement is known to be true if (1) it is an accurate perception statement or (2) if there is a true theory that predicts it (what we call evidence).
- False statements
- A statement is known to be false if (1) it is an inaccurate perception statement or (2) there is a true theory that predicts its negation (what we call counter-evidence).
- Unknown statements
- If a statement is neither known to be true nor false, its truth is unknown. The vast majority of statements belong to this class.
A wise man proportions his belief to the evidence.
In summary: the concept of truth
This chapter has argued that we perceive ourselves in a continuous stream of perceptions: We smell something, we see something, we feel something, we hear something. Our goal is to use past perceptions to predict future perceptions. For this purpose, we build theories of the formThe problem is, of course, that we never know whether a theory will always correspond to our observations. This is why we humans take a rather humble stance in this universe: We try to describe our reality by theories; we validate them, refine them, and hope that they are the truth — but in the end we have no guarantee that nature behaves as we predict. We thus find ourselves in a continuous search for the truth. In this search, we prefer those theories that are more general: The more phenomena the theory covers, and the less premises it has, the better it is.
When a true theory makes a prediction, we call that theory and its premises evidence for the prediction. For example, the evidence for the hypothesis that the stone probably landed in that position (even if you can’t see it) is the theory of ballistics plus the fact that you threw it at this angle. In everyday life, evidence for a hypothesis is often elusive: The definite truth of theories is hard to establish, concepts are often poorly defined, different theories can make different predictions, and new evidence can appear at any moment. This is why we find ourselves in a continuous process of what is called joint inference: We evaluate reasons for and against a statement and try to estimate its likelihood. Even with joint inference, though, we cannot predict everything. Some things remain unknown — and that is okay.
There is one particular class of theories that merits our attention. Consider the following theory: “If you pull the lever of the slot machine in the right way, then you win”. This theory sounds enticing, because it seemingly tells you how to win with the slot machine. Yet, as soon as you try out the theory, you find that it does not work: You don’t know how to pull the lever in the “right way”. You know whether you found the right way only afterwards, when you see the outcome. Then, however, you do not actually need the theory. A theory is useful and meaningful only when it tells you about the outcome beforehand. At the same time, the theory cannot be proven wrong: If you don’t win, then you simply did not pull the lever in the right way. And if you do win, and you try the same way of pulling the lever again, and it does not work, then this means that you still don’t know how to pull the lever in the right way, and you merely won by chance on your first attempt. Thus, no matter what happens, you cannot conclude that the theory is wrong. This means that the theory is what we call unfalsifiable. Any theory that is unfalsifiable is automatically meaningless: If there is any event that the theory predicts, then the non-occurrence of that event would prove the theory wrong. Now, if the theory cannot be proven wrong, then this can mean only that the theory predicts no event. And if the theory predicts no event, then it is thus literally meaningless. Unfalsifiability and meaninglessness are two sides of the same coin.
Don’t be afraid of perfection. You will never achieve it.
Theories in Real Life
The theory of truth
Technically speaking, the theory of truth that we have presented in this chapter is a definition of the word “true”. It holds that if the conclusions of some theory consistently correspond to our experiences, we call that theory “true”. And indeed, this is by and large what the average person regards as “truth”.For the purposes of this book, we have defined theories as sets of logical rules. Such theories of logical rules are useful when we want to explain our reasoning, justify a choice, or convince another. However, in general, these rules are just a model for the fact that people use past experience to predict future perceptions. They do so not by computing conclusions of rules but by running parallel neural processes in their heads. Indeed, American philosopher Rick Grush has argued that the brain is constantly generating and updating a mental model of its environment with the goal of making predictions from sensory input in a theory known as Predictive Coding12. This theory has been further developed into the Predictive Processing Model, a theory in cognitive and computational neuroscience that sees the brain as a complex, multi-layered prediction engine13.
All models are wrong, but some are useful.
Learning theories
We build theories about everything around us, possibly starting as a baby. On this view, the baby’s job is to discover the patterns that govern how the physical world works, and encapsulate them in theories that can be used to predict what will happen in new situations15. By the time we are adult, we have built up a corpus of theories. These theories can pretty accurately give a structure to our past and predict much of the immediate future.
All people are smart — some before, and others afterwards.
Correction
If a theory has consistently yielded correct explanations in the past, we trust its predictions also for the future. Now what happens if a theory has made thousands of correct conclusions in the past, but then one day delivers a false prediction? For example, consider the theory that speed adds up. If I walk at 6 km/h in a train that runs at 100 km/h, then my speed relative to the ground is 106 km/h. This theory makes lots of true predictions. Now assume that there is a screen at the end of the rails, some kilometers down in the direction of travel. Assume that I point at that screen with a laser pointer from the train. You stand at the station and also point at the screen with a laser pointer. Then my laser beam should travel faster than yours. If we both switch on the laser pointer at the same time, in the very moment the train passes the station, then my dot should show up on the screen slightly earlier than yours. Yet it does not. Both dots appear at the same time. The speed of both laser beams is the same, no matter whether it is measured relative to the ground or relative to the train. This is confusing and contradicts the theory that speeds add up.In such a case, the prediction of the theory is false. This is surprising, because the theory has made zillions of correct predictions in the past. Still, its prediction in the case of the laser beam is false. In fact, most of its predictions are false when the speed approaches the speed of light. Therefore, no matter how plausible this theory sounds, and no matter how many correct predictions it has made in the past, it is false in the general case. Importantly, it is not reality that needs fixing. It is the theory that needs fixing.
This is indeed what happened to the theory. German-born physicist Albert Einstein had the courage to say that, if the theory does not correspond to the facts, then it had to be changed. He thought up a new theory that permits light to always have the same speed. This theory entails all kinds of weird things, such as objects gaining infinite weight when they approach the speed of light. Despite its counter-intuitive conclusions, this theory turned out to make consistently correct predictions in realms where the earlier model did not. It is known as the special theory of relativity16.
The ability to see that a theory is wrong, and the readiness to abandon it if it does not correspond to the facts, is one of the cornerstones of rational thinking. In many cases, a theory is not actually wrong in itself. It just needs additional premises. The theory that speed values add up, for example, is not outright wrong. It provides very good approximations of reality under the condition that the speeds we are talking about are small — to a degree that the average human is unlikely to bother with special relativity in real life. The problem is just that it is very hard to identify the conditions (i.e., the additional premises) under which a theory is true. This issue is known as the Duhem-Quine problem17.
Life can only be understood backwards
but it must be lived forwards.
Science
This does not mean, however, that the theory would be the truth. A theory can always make a false prediction one day. This is why scientific theories are called theories. Even universally accepted laws, such as the law that all objects with mass or energy gravitate towards each other, are still called theories. If, one day, any theory is found to make a false prediction, then the theory will no longer be considered an approximation of truth. This entails that science never proves that a theory is true. Science just builds models of reality.
Science is the profession of induction.
The theory of truth in practice
Our theory of truth is hard to apply literally in practice: People do not carry a set of logical rules with them, and even if they did, many rules would be ill-defined, contradictory, not validated, or outright false. This does not mean, however, that our theory of truth would be useless. Quite the contrary, the theory allows for some important steps in our quest for finding the truth. First, it instructs us to exclude unfalsifiable statements from our discourse. These can never predict anything, and they are thus meaningless in the sense of this book. If we accepted such statements, we would open the door to accepting all other kinds of bogus theories, like conspiracy theories and pseudo-science. Both and their ilk all have in common that we would never be able to find out whether they are false. Falsifiability is the bulwark against such nonsense.The second thing that our theory of truth allows us to do is to exclude theories that have consistently clashed with our perceptions — these are false theories. In our search for truth, false theories are as important as correct theories. Knowing them allows us to reject invalid claims and to avoid drawing false conclusions.
Finally, and most importantly, the theory tells us at least what we would have to do if we wanted to check the truth of a hypothesis: We have to find evidence, i.e., we have to find a true theory that predicts the statement. This is a fundamental insight: It allows us to weigh our confidence in a statement by the evidence that we have for it. We should not believe strongly in a statement for which we have only weak evidence. In particular, we should not make life-changing decisions based on weak or no evidence.
Ignorance is of a peculiar nature. Once dispelled, it is impossible to re-establish it. Though man may be kept ignorant, he cannot be made ignorant.
Common Mistakes
Common Mistakes
We will now look into common mistakes that people make in their search for truth. We will illustrate each mistake by an example. We will then apply our theory of truth to find out what the problem is in the example scenario, and, as we shall see, the theory of truth allows us to identify and rebuke the mistakes reliably.The examples that we will use are quite abstract — even absurd you may say. And yet, as we will argue, they appear in very similar form in the world’s major religions.
Doubt is the origin of wisdom.
It makes sense
A first common mistake that people make is that they assume that something is true because it “makes sense”. Let us look at an example. In historical times, people thought that certain illnesses were caused by “bad blood”. Hence, they reasoned, the illness could be cured by removing the bad blood. Therefore, they punctured the arm of the ill person and let the blood come out. This practice (bloodletting, as it was known) was widespread from the ancient Greeks to the 18th century19. It made a lot of sense to people. However, it did not have any positive effect. On the contrary, in the overwhelming majority of cases, the historical use of bloodletting was harmful to patients.This shows us that a theory can be false even if it “makes sense to us”. A theory does not become true if “it makes sense”. It becomes true if it makes correct predictions. To check if a theory makes correct predictions, we have to systematically validate it. In our example, we would have to recruit a number of patients with the same symptoms, and submit a randomly chosen half of them (with their consent) to bloodletting. If the group that underwent the procedure fares better than the other group, the practice can be assumed to be helpful — but if it does not, the method has to be abandoned. We will argue later that the failure to see this, and the reliance on the intuitive “sense” instead of on concrete and correct predictions is one of the pillars of religious belief.
One accurate measurement is worth a thousand expert opinions.
Ghostification
Another type of mistake people make is what we will call ghostification. For example, many ancient peoples believed that inanimate objects were inhabited by spirits. Let us assume that one particular people believed that fire was kept alive by fire demons. Based on this, they could have built the following theory:If the fire demon is chased away, the fire dies.
If you are not kind to them, they may refuse to be chased away.
We will later argue that modern religions, too, piggy-back on otherwise reasonable theories in a similar fashion. For example, prayer works because it helps us reflect our lives — not because there would be a god who listens. A religious community is fulfilling because of common chants and activities — and not because of some magical connection with the supernatural. In this way, religion infuses its own supernatural entities into otherwise reasonable theories. Thus, religion lives from the ghostified versions of these theories.
Let nature tell you how nature works. It knows the answers, not you.
Unclear theories
To illustrate our next point, imagine a people with an elaborate system of myths and collective wisdom. One of their omens is as follows: “And that day will be a great day. After that day, no one will be as they were before. And that day will show the grandness and glory of the God of the Sun.” This theory seems to be making predictions. Yet, it is completely unclear when the theory refers to. Was today a great day? Should we expect the God of the Sun? And if so, what does it mean that the day will “show the grandness” of that god? Will he actually appear? Or is the grandness rather shown implicitly? The theory is just not clear.The criterion that helps us here is falsifiability: The theory is not falsifiable. There is nothing that we would accept as a proof that the theory is false: Even if no great day has arrived so far, adherents of the omen can still claim it will arrive one day. With this, the theory cannot make any concrete predictions about the real world — it is nonsensical. Such theories can be poetic and beautiful, but they cannot be true. And yet, many people like adhering to such theories. For example, many people believe that “The end times are near”. As we shall see later, this belief is as ill-defined as the omen.
As simple as possible, but not simpler.
Postdictions
For our next common mistake, consider the story of Asterix and the Seer21. It goes as follows: During a thunderstorm, a man finds shelter in the village of Asterix. The man says that he knew that the thunderstorm would be coming, and he knew that he would find shelter in that village. When the perplexed villagers ask him how he knew all of this, he replies: “I’m a seer”.However, the seer just claims he foresaw whatever happened. He waits until something happens, and then says he knew that it would happen. Therefore, the seer is not actually making predictions. He is making postdictions.
The real challenge is, of course, to build theories that predict not just the known, but also the unknown. The essential quality of a true theory is that it is true not just for the case at hand but also for the cases in the future.
There is an analogy to religion here: As we shall see later, some people believe that their religious book saw something coming. However, never has a religious person made a new invention by carefully studying scripture, sold their home because a holy book correctly predicted the date of a war, or invested in the right company because of a prediction found in the Bible. The predictions are made only ever after the fact. As we will argue, this puts such predictions on the same level as the predictions of the seer.
Science is the journey to the truth.
Religion is pretending you’re already there.
The narrative fallacy
To illustrate our next common mistake in reasoning, let us look at the war between the United States and Iraq in the early 2000’s: When the United States captured the Iraqi dictator Saddam Hussein in 2003, the oil price initially fell. Hence, newspapers ran stories explaining that the capture gave confidence that the war would be over, which would lower the price of oil in the long run. Shortly thereafter, however, oil prices rose. Hence, newspapers ran stories explaining that the capture has created new conditions for the mission in Iraq, and that this insecurity has driven up oil prices. In reality, there was possibly no causal relation whatsoever, as oil prices tend to fluctuate in any case22. However, humans tend to add causality to unrelated events. This tendency is what Lebanese-American statistician Nassim Taleb calls the narrative fallacy23.This tendency to “storify” may have a very simple reason, Taleb hypothesizes: It is easier to remember events if they are connected in some way. As an example, consider the sentence “The king died and then the queen died”. Now compare it to “The king died, and then the queen died of grief”. It is somehow easier to remember the second story, because it seems to contain just a single piece of information with its consequence, rather than two unrelated events. Rephrasing the initial story as a causal sequence may thus be a technique of “dimensionality reduction”, i.e., of data compression. This may explain why people have a tendency to consider the second sequence more probable than the first — even though the second actually implies the first.
The same phenomenon appears in religion: People have a tendency to add supernatural causality to the events of life. For example, they might say: "This and that happened because God wanted it this way because he has a grander plan with us" — not because there would be any evidence for such a causality, but because it makes the events of life more palatable. We will discuss this way of thinking in the Chapter on the God of the Gaps and in the Chapter on the Abrahamic god.
No one ever made a decision because of a number. They need a story.
Abductive reasoning
For our next point, let us consider one of the biggest unsolved conundrums in the history of aviation: the fate of Air Malaysia flight MH370. This airplane disappeared shortly after take-off in 2014, and has never been found. Despite the most expensive search in aviation history, nobody knows what happened to MH37024.However, imagine that your friend (let’s call her Patricia) has long been suspecting aliens behind the disappearance. She comes up with the following rule:
Patricia will defend her conclusion as follows: “You don’t believe my claim? Well look at it: The airplane disappeared. Isn’t that enough evidence? Nobody can explain how MH370 disappeared. Only I can explain it! And aliens are the simplest explanation for the disappearance. Therefore, my reasoning must be right!” Is that correct?
It is true that nobody can explain how MH370 disappeared. However, that does not make Patricia’s hypothesis true. If we argue that an explanation must be true because we can imagine no other, we commit a fallacy known as the argument from ignorance: Something does not become true just because we cannot imagine it any other way. In order to accept a hypothesis as true, we need a rule that has the hypothesis not in the premise, but in conclusion. That is, we need a rule of the form “If ... then aliens shot down MH370”. In other words, we need evidence for the hypothesis. Until such evidence appears, the correct answer to “What happened to MH370?” is “We do not know”.
It is easy to see that our example bears similarities to the question “Who created the world?”. From the fact that we do not know the answer, believers of different religions deduce that it was their particular gods who created the world. This, however, does not make their hypotheses true.
Our example analogy can be continued: Your other friend Paul has long suspected the Russians behind the disappearance of MH370. Hence, he comes up with the rule
Of course, neither argument is correct, because in order to assert a hypothesis, we need evidence for it, i.e., a rule that implies it. However, if people do not subscribe to the necessity of evidence for hypotheses, they can go on arguing forever. This is indeed what we observe in discussions about religious truth.
Irvin: I am sure I have liver disease!
Doctor: That’s impossible. If you had any kind of liver disease, you’d never know it — there is no discomfort of any kind.
Irvin: Those are my symptoms exactly!
Counting the hits
Imagine a woman called Chelsea, who drives to work this morning. Her dad warns her that he has the bad feeling that Chelsea might have an accident. Indeed, that day Chelsea has an accident. Predictably, his dad says that he told her so.Chelsea reminds him that he has been warning her about accidents every morning for the past 2 years. Thus the theory that her dad can predict accidents was correct in only 1 out of around 660 cases. That ratio is too bad to call his theory true. The theory is in fact false.
Technically speaking, the theory has failed the test of validation. It has delivered numerous false predictions in the past, and has hence to be rejected. Even a false theory can make a true prediction from time to time, but that still does not make the theory true. Looking only at the cases where the theory makes a true prediction is a fallacy called counting the hits. Again, this fallacy can also be encountered in religion: When it comes to the efficiency of prayer, for example, many people are willing to count the hits only, and to disregard the cases where prayer did not work.
Neither people nor organisations can learn if they deny that an error has happened.
Useless premises
Imagine that your superstitious friend Sara has the theory that on Friday the 13th there will be car accidents. This theory has made correct predictions in the past, and it continues to do so. Is Sara finally right with her superstitions?Yes and no. The theory that there are car accidents on Friday 13th is true, but it is unsatisfactory for a different reason: The prediction “There are car accidents” is true no matter whether it is Friday 13th or not. The premise of the theory does not add any insight. For such theories, we can develop arbitrary variations. For example, we can construct the theory that “on Tuesday the 17th, there will be car accidents”. This theory is a true as the Friday the 13th theory. It is clear that with this argument we can take any number as an unlucky number (or as a lucky number for that matter). And indeed, in some countries residents view 17 as an unlucky number, and in others 13 is viewed as unlucky.
Technically speaking, Sara’s theory contains a useless premise: The condition of Friday the 13th can just be removed. The resulting theory is:
Again, we can see a link with religious beliefs: Some people believe that prayer cures illnesses. They uphold this claim even for illnesses that disappear by themselves. In such cases, prayer is a useless premise in the sense of this book.
The greatest ideas are the simplest.
Wishful thinking
Imagine that you have a friend (let’s call him Bill) who is a very happy person. One day you talk to him, and you ask him why he is always so happy. He pulls you aside and whispers: “It’s because I am rich!” You are surprised, because Bill doesn’t look particularly rich, and from what you can tell, he seems to possess only a single pair of trousers. You ask: “What do you mean, you are rich?” Bill whispers: “I believe that I have a treasure hidden in my garden!” You say that this is great, and you ask him whether he checked. He says no, he didn’t check. You ask then why he believes he has a treasure in the garden if he didn’t even check. Bill replies: “Would you want to be poor?”You are perplexed. From the fact that Bill does not want to be poor, it does not follow that there is a treasure in his garden. Such a rule is just plain wrong, meaning that it produces conclusions that do not correspond to reality. Such thinking is called wishful thinking. Absurd as such thinking may seem, it plays a role in the popularity of religions as well: People believe in something (like the afterlife, the power of prayer, or a supposed ultimate justice) merely because they want it to be true.
Reality is not negotiable.
Happy thinking
Your friend Bill believes that he has a treasure hidden in his garden. As this is obviously very unlikely to be true, you are worried, and you talk with a common friend of yours. She tells you: “So what, if this makes Bill happy, just let him believe he has a treasure in the garden!” You ask: “So if that makes him happy, is it true that he has a treasure in the garden?” And your friend is wise enough to say: “If a belief makes you happy, it doesn’t mean it’s true.”While you are still pondering this wisdom, your friend suggests: “You look as if you could use some happiness in your life, too! Why do you not also start believing that there is a treasure in your garden?” You are perplexed: It does not make sense to believe something that you know to be false just to be happy. Believing in a false theory will lead to false conclusions. Such belief is thus ultimately treacherous. And yet, as we shall discuss, this argument is frequently brought forward in favor of faith as well: Belief makes people happy, so why don’t you start believing as well?
If someone has truly lost touch with reality and still claims to be happy, we worry that they might be incorrect even about this, and might need to be rescued from a state they misperceive as being bearable.
Meaningless statements
Let us suppose you are inviting your friend Sandra over for dinner. Sandra is very happy to join you. However, as soon as she steps into your place, she starts looking around and seems to be worried. You ask whether everything is alright. Sandra replies yes, everything is alright, but says: “There is something here.” You ask: “What do you mean, there is something here?”. Sandra says: “I don’t know, I just feel there’s something here”. She continues to walk around, looks behind the door, checks the kitchen sink, and verifies that there’s nothing under the table. You are worried and ask her: “Sandra, what is it? Are you okay? Do you need anything? Did you see anything? Is there a smell?”. But she just says that yes, she’s okay, she doesn’t need anything, she didn’t smell anything, and she didn’t see anything. It’s just that, as she keeps saying, “there is something here”. Later, to your relief, she seems to have forgotten about it and the evening turns out to be really nice.After she left, you still wonder what she meant by “There is something here”. It did not mean that she was uncomfortable, that she would need something, or that she would smell or see something particular. In fact, the statement didn’t seem to mean anything.
You try to fit Sandra’s statement into the theory of truth that you read in this book, and you find that, from the fact that “There is something here”, you cannot deduce anything about your place: nothing to see, nothing to smell, nothing to worry about. There is no rule that has this statement in its premise. Thus, the statement is literally meaningless in the sense of this book. It does not have any meaning outside Sandra’s head.
We will later argue that many statements in the spiritual domain fall in the same category. Examples are “There exists a universal principle of the Universe”, “There is a conscious being behind everything”, and even “God is the first cause of the Universe”. We will later argue that these claims are on the same level as Sandra’s “There is something here” — they allow for no concrete predictions about the real world either, and are thus ultimately meaningless.
Empty statements
Let us assume that Andy is your friend from high school and you still meet him from time to time. One day, Andy pours out his heart to you, and tells you how he suffers in his relationship. You listen attentively and comfort your friend. Andy is very grateful and tells you that you are really the best friend he has. You are happy that you could help and that he appreciated it.A few days later, you go with Andy to a bar. Andy is still sad and orders a scotch. The waitress sees that Andy is having a difficult time and pities him. She brings peanuts on the house and tells him to cheer up. Andy smiles at her and tells her that she is really the best friend he has. She smiles back. Later that evening, a client who has drunk too much gets rude and acts aggressively towards the waitress. Andy seems visibly uncomfortable and proposes that the two of you should leave the bar quickly.
At this point of time, you understand two things: First, Andy actually calls everybody “best friend”. Second, this phrase does not mean anything to him, as he does not even consider helping the waitress. Technically speaking, you cannot deduce anything from the fact that Andy calls you his best friend: neither that you are the only “best friend”, nor that Andy will do anything more for you than he does for a random person in the street. There is no rule of the form “If you are Andy’s best friend, then ...”. Thus, the statement “You are Andy’s best friend” is technically meaningless.
This is different from the usual semantics of the phrase “best friend”. If a normal person calls you “best friend”, then you can deduce a number of things from this. First of all, you are the only best friend. Second, that person will give you special treatment and help you when you are in trouble. Andy, in contrast, uses the phrase in another way. The way Andy uses the word gives it no meaning at all. It becomes an empty word.
We will later argue that some religious statements such as “Every person is a sinner”, “God loves you”, or “We are the chosen people” are of the very same nature as being Andy’s best friend: They carry a strong suggestive meaning — but boil down to senselessness when questioned.
Unfalsifiable statements
Let us suppose that your friend Chris is convinced that one of his colleagues fell in love with him. However, the woman appears to treat him just like any other colleague. When you challenge Chris on his conviction, Chris argues:Does that mean that Chris’ hypothesis is true? Does the woman really love him, but is just too shy to act on it (after all, this claim cannot be proven false)? As the attentive reader will have noticed, this immediately makes the hypothesis unfalsifiable. That is, we cannot find a condition X such that
It is interesting to note that our example bears noticeable similarities to phrases such as “God loves us but does not show it in an obvious way” — or even to statements such as “God exists but you cannot see him”, “Heaven will punish you in the afterlife, but you cannot prove it”, or “You will be reborn, but you cannot know it” are unfalsifiable. All of these statements are unfalsifiable, and hence meaningless. They are on the same level as "The woman loves me but does not show it". And yet, they are the bedrock of religions, as we shall see in the Chapter on Memes.
The essence of the independent mind lies not in what it thinks, but in how it thinks.
Truth and Atheism
Revisiting Persephone
We started our discussion of truth from the ancient Greek story of Persephone. Persephone is the Goddess of Spring, and she goes to the God of the Underworld, Hades, each year. This leaves her mother, Demeter, the Goddess of Earth, sad and so she makes it cold. This explains why winter is cold. Is that a good explanation for the temperature drop in winter?To answer this question, let us look at the story of Persephone in detail. It goes roughly as follows:
If someone is forced to go somewhere, then that person goes there.
If Persephone is in Hades, her mother Demeter is sad.
If Demeter is sad, then it is cold.
The supernatural
Based on our theory of truth, we can now formally define the supernatural. It is tempting to define the supernatural as anything for which there is no evidence. For example, we could argue that God is supernatural because there is no evidence for his existence. The issue with this definition is that many believers insist that there is actually evidence for their supernatural entities (i.e., that there is a true theory that predicts the existence of these entities, and that this theory has not yet been found, has not yet been sufficiently validated, or has not yet been applied correctly). In fact, many “proofs” for the supernatural argue with such theories. Thus, simply agreeing that “God is supernatural” would require solving one of the biggest conundrums of humanity: proving that there is no evidence for God’s existence.Quite fortunately, the converse notion (that God’s existence cannot be disproven) is both sufficient to define the supernatural and easier to establish. Believers will typically agree easily that God’s existence cannot be disproven: Whatever happens, it does not question the existence of God. And for an atheist, assuming that God’s existence cannot be disproven amounts to unfalsifiablity, and thus to meaninglessness — which is indeed one of the ways to define atheism.
In this spirit, we define a supernatural statement as a statement that, on principle, cannot be falsified (or, by extension, such a theory). Note that, with this definition, being supernatural is a property of statements and theories, and not of entities. In other words, it is not God himself who is unfalsifiable, but the statement “God exists”. Other supernatural statements and theories are “There is life after death” or “If you have done only good things in life, then you will go to Heaven” — and these are, likewise, unfalsifiable.
Not every unfalsifiable theory is supernatural, because not every theory that is unfalsifiable is so on principle. For example, the theory “If you pull the lever of the slot machine in the right way, then you will win” is unfalsifiable, but not so on principle. Once we endeavor to properly define “pulling the lever in the right way” (i.e., to ground the theory in reality, for example by describing the parameters of speed and direction), the theory becomes falsifiable. A theory is supernatural only if such grounding cannot be achieved on principle, and the theory is thus shielded from human perception. We may regret that this definition of the supernatural is not something objective, but depends on whether the interlocutors agree on the principled unfalsifiability of some statement, i.e., on the impossibility of grounding it in perceptions. While this objection is justified, our definition is in line with the entire theory of truth that we develop in this chapter: It holds that the models of reality that humans build are inherently dependent on their own perceptions — and hence usually imperfect. We will later argue that this does not hamper mutual agreement on truth.
[Religion is a complex issue.] This is why it takes several chapters to approach a question that many people, in my experience, can solve to their entire satisfaction in a few seconds of dinner-table conversation.
Truth and atheism
Atheism is the rejection of belief in the supernatural. Before, we only had a very vague description of “the supernatural”. With our new definition of the supernatural, we can now explain what atheism means: When we say that atheists reject belief in the supernatural, we mean that supernatural statements are not part of the statements that atheists believe.Positive atheism goes further, and holds that the supernatural does not exist. Technically, positive atheism is a theory of the form “If T is a time point, then the supernatural does not exist at time point T”. The meaning of “the supernatural does not exist at time point T” is that prayer has no effect at time point T, that no god will reveal himself in a verifiable way, that no proof for reincarnation or life spirits is found, etc. We will later argue that this theory is not just falsifiable, but also validated: It has made exclusively true predictions so far. Therefore, positive atheists assume it to be true.
What evidence would prove that some occurrence was supernatural? None — because if it was observed, it was natural; and if it was not observed, we have no reason to believe that it occurred.
Questions
Is truth subjective?
The theory of truth put forward in this chapter says that we judge the truth of a theory by comparing its conclusions to our experiences. If the conclusions correspond to our experiences, we call the theory true. Since nothing guarantees that other people have the same perceptions as everyone else, this means that someone else can have a completely different truth.Indeed, it is a long-standing philosophical conundrum whether truth is absolute or subjective. With the concepts that we have seen in this chapter, however, we can find a way out: We can build the theory that if a healthy person has some impression (be it a noise, a smell, or a visual perception), then other healthy people in the same spot at the same time will have that same impression. This theory makes generally true predictions — to a degree that people who do not have the same impressions as everyone else are considered to have a health problem, almost by definition.
Now comes the interesting part: If people perceive the same way as everyone else, then they will evaluate the truth of a given theory in a given case in the same way as everyone else as well. For example, if some theory predicts a rainbow, and if someone observes that the theory makes a correct prediction on a given day, then other people in their vicinity will also make this observation. Thus, if we wanted to determine whether a given theory is true, we would just have to validate or reject the theory together by making the same observations together. This in turn entails that truth itself, as defined in this book, is objective.
The claim “Everything is subjective” must be nonsense, for it would itself have to be either subjective or objective. But it can’t be objective, since in that case it would be false if true. And it can’t be subjective, because then it would not rule out any objective claim, including the claim that it is objectively false.
Science is not everything!
The approach to truth that this chapter puts forward may look rather scientific. It is based on theories, evidence, and testability. Indeed, this chapter’s notion of truth is just an extension of the scientific principle itself. And yet, as the reader might object, science is not everything.This book does not claim that everybody should have to follow the proposed way of thinking, let alone that it is the only one. It does claim, however, that most people consider trustworthy those theories which have made true predictions in the past. And that most people mistrust theories that have made no predictions or false predictions in the past. To see this, consider an example: In a casino, someone tells you that he can predict the outcome of the roulette. You trust him and bet the money as he says. It does not work. You try again. Again, it does not work. You have already lost a substantial amount of money. Would you trust his predictions again? Probably not. This guy was just wrong. But by not trusting him, you act “scientific”: You decided to reject a theory based on false predictions.
The bottom line is that for the most part we all operate according to this principle whenever it comes to trusting people or theories. It is true that we are vulnerable to bias and error. However, clearly not all of us are vulnerable all the time — otherwise we would not even be able to say that we are vulnerable to bias and error.
The human brain is capable of reasoning, given the right circumstances. The goal is to identify those circumstances, and to put them more firmly in place.
You can’t prove everything!
This chapter proposes that evidence for a hypothesis is a true theory that predicts this hypothesis. Now, unfortunately, we do not have evidence for all hypotheses that we make. In fact, for the vast majority of hypotheses, we do not have any evidence at all. Even if we have evidence, there may be counter-evidence. This means that, strictly speaking, we would have to go through a process of joint reasoning before deciding the truth of that statement. And yet, in everyday life, we rarely do that. Therefore, is the theory of truth put forward in this chapter not just absurd?On a Humanist view, everyone is free to believe what they want. There is no moral necessity to prove anything.
However, we rarely believe just “anything”. Rather, we usually judge the likelihood of a hypothesis by our experience — as in the previous example of the roulette player. Technically speaking, this “experience” is nothing else than the set of theories that we have accumulated in our life time. It’s just that we rarely make these theories explicit. But when we are asked why we believe in something, then we will certainly have our reasons. Again, technically speaking, these reasons are explanations in the sense of this book. Thus, by and large, we do use the methods of this book — just in a watered down variant that is much less formal than presented here.
This book will argue in the Chapter on Criticism of Religion that anything that restricts people’s lives drastically, anything that impacts the lives of others, and anything that claims absolute truth (read: religion) falls in the second class of cases: It requires examination via the formal type of methods. At the very least it should be falsifiable. That is the minimal condition for even just being considered in earnest.
Mayor: “Your new bridge sure is beautiful. But are you sure that it can support the weight of cars and trucks?”
Builder:“Yes, I am.”
Mayor: “Great! Let’s see you drive all your company’s trucks across it.”
What is the meaning of love?
in Lima, Peru
if you love someone, you feel happy when that person is around; and
if you love someone, you want to help that person wherever you can.
Love is putting someone else’s needs before yours.
What is the meaning of mathematical concepts?
Mathematical concepts are labels for real-world phenomena. For example, when I say “There are 5 apples”, I mean “There is one apple for each finger of my right hand”. Thus, “5” is simply an abbreviation for the fact that a group has as many elements as we have fingers. When we say “5+5=10”, then this is an abbreviation for “When you have as many apples as you have fingers on your right hand, and you have as many bananas as you have fingers on your right hand, then you have as many fruits as you have fingers on both hands”. It is way more convenient to say “5+5=10”.Based on this, we can come up with plenty of rules in which these labels appear:
If ｘ＋ｘ＝ｙ, then ２ｘ＝ｙ
etc.
These theories make predictions. In the simplest case, they make predictions about the real world, as in “If you have 2 apples and you add 1 more, you will have 3 apples”. In the cases of more abstract mathematics, the theories still make predictions. However, these may concern the process of calculus only. As an example, consider the rule “If ｘ＝ｌｏｇ（ｙ）, then １０ｘ＝ｙ”. This rule predicts that if ｘ is the logarithm of ｙ, and if I compute １０ｘ, then I will obtain as result ｙ. This prediction rarely finds its uses in everyday life, but it is undoubtedly a prediction — and a true one at that.
You can’t believe in math. You have to understand it.
You can’t understand religion. You have to believe it.
What about historical facts?
This thing, however, may turn out to be false. For example, it was widely assumed that Cleopatra, the queen of ancient Egypt, committed suicide by an asp bite. Nowadays, some historians are questioning this version of history, based on the fact that an asp bite would have caused severe bleeding that is not mentioned in sources25. As the reader may have noticed, we have just presented evidence that could suggest that the common assumption about Cleopatra’s death is incorrect. Thus, we have shown that a historical conviction can be falsified. Therefore, historical facts are not meaningless. They have their role in the physical course of time.
There are two possible outcomes: If the result confirms the hypothesis, then you've made a measurement. If the result is contrary to the hypothesis, then you've made a discovery.
This theory of truth limits one’s view of the world
This chapter defines truth by building on perception statements. It cuts away anything that does not ultimately talk about perceptions. This raises the question whether this view of truth is not too limited. All metaphysical concepts, for example, are cut off.It turns out that there are theories that are grounded in perceptions and theories that are not. The first class of theories can make tangible predictions, whereas the second one cannot. This holds no matter how you define truth. The first class of theories is the subject of science. This holds by definition, because science is what is concerned with the natural world. Science encompasses not just physics. It encompasses also psychology, biology, history, and sociology.
The second class of theories is not about perceptions. This does not mean that this class of theories would be useless. Such theories can make up stories, metaphors, or poems. These can provide entertainment, consolation, or inspiration. However, they cannot provide true predictions.
Truth is a false concept
It has been argued that we cannot be sure whether the physical system in which we perceive ourselves really exists (most famously maybe in the movie The Matrix). There could be no physical system at all. There could be also multiple such systems. Then our sensations would be just impressions that have nothing to do with any physical world. Reality would not exist, and truth itself would be a false concept.This is certainly a fascinating theory (as the popularity of the movie shows). However, what would be concrete predictions that this theory allows for? It turns out that no single perception can be predicted. When we assume that reality does not exist, or that truth is a false concept, we are no wiser than before. Thus, the theory that reality does not exist is literally meaningless. It is completely irrelevant for our life. We could as well claim that you are dead and what you think is your life are in fact just hallucinations of some surviving neurons in your brain. Such theories lead nowhere. They do not talk about our perceptions, and thus do not even qualify to be considered for being true.
This theory is not only not true.
It is not even false.
How does this theory relate to formal logic?
(Less theoretically inclined readers, who would not have asked this question, do not miss anything by not reading the answer.)From a logical point of view, a rule in the sense of this chapter is a clause of first order logic with positive and negative literals, only universally quantified variables, and no function symbols. This means that any first order logic formula that has only universal quantifiers in its prenex form and that does not have function symbols can be translated into a set of rules. For this purpose, the formula has to be brought to disjunctive normal form. Then, one has to pick one literal in each clause. That literal becomes the conclusion of the rule, and all others are negated and go to the premise. The system can deal with disjunctions in the premise and in the conclusion as follows:
- ａ ∨ ｂ ⇒ ｃ can be transformed to two rules, ａ ⇒ ｃ and ｂ ⇒ ｃ.
- ａ ⇒ ｂ ∨ ｃ can be transformed to ａ ∧ ¬ｂ ⇒ ｃ, for any of the conclusions.
- ａ ⇒ ｂ ∧ ｃ can be transformed to two rules, ａ ⇒ ｂ and ａ ⇒ ｃ.
The system as presented above treats statements as atomic units and does not delve into their grammatical structure. This approach has its limitations, but is still sufficiently powerful to make meaningful deductions2. If we need to reason about rules themselves, we have to resort to formalisms that can express properties of formulas26.
It’s in the very nature of an argument that people stake a claim to being right. As soon as they do, they have committed themselves to reason — and the listeners they are trying to convince can hold their feet to the fire of coherence and accuracy.
References
- David Deutsch: “A new way to explain explanation”, in TED talks, 2009
- Zacchary Sadeddine and Fabian M. Suchanek: “Verifying the Steps of Deductive Reasoning Chains (VANESSA)”, in Findings of the Association of Computational Linguists, 2025
- Karl Popper: The Logic of Scientific Discovery, 1959
- Gregory Chaitin: On the intelligibility of the Universe and the notions of simplicity, complexity, and irreducibility, 2002
- Diderik Batens: “Relevant implication and the weak deduction theorem”, in Studia Logica, 1987
- Judea Pearl: The Book of Why, 2018
- Chadi Helwe, Simon Coumes, Chloé Clavel, and Fabian M. Suchanek: “TINA - Textual Inference with Negation Augmentation”, in Findings of the Conference on Empirical Methods in Natural Language Processing, 2022
- Encyclopedia Britannica: “Positivism”, 2025
- Steven Pinker: Rationality - What It Is, Why It Seems Scarce, Why It Matters, 2021
- Yiwen Peng, Thomas Bonald, and Fabian M. Suchanek: “FLORA: Unsupervised Knowledge Graph Alignment by Fuzzy Logic”, in International Semantic Web Conference, 2025
- Michael Potegal and Gerhard Stemmler: “Constructing a Neurology of Anger”, in International Handbook of Anger, 2010
- Rick Grush: “The emulation theory of representation - Motor control, imagery, and perception”, in Behavioral and brain sciences, 2004
- Andy Clark: “Consciousness as Generative Entanglement”, in The Journal of Philosophy, 2019-12
- Nedeljko Radulović, Albert Bifet, and Fabian M. Suchanek: “Confident Interpretations of Black Box Classifiers”, in International Joint Conference on Neural Networks, 2021
- Ian H. Witten, Eibe Frank, Mark A. Hall, and Christopher J. Pal: Data Mining: Practical Machine Learning Tools and Techniques, 2016
- Encyclopedia Britannica: “Special relativity”, 2024
- Encyclopedia.com: “Underdetermination Thesis, Duhem-Quine Thesis”, in Encyclopedia of Philosophy, 2025
- Encyclopedia Britannica: “Problem of induction”, 2025
- Encyclopedia Britannica: “Bloodletting”, 2025
- Live Science: “How does water put out fire?”, 2023-06-26
- René Goscinni and Albert Uderzo: Astérix et le Devin, 1972
- Daniel Kahneman: Thinking, Fast and Slow, 2011
- Nassim Taleb: The Black Swan - The Impact of the Highly Improbable, 2007
- Encyclopedia Britannica: “Malaysia Airlines flight 370 disappearance”, 2024
- François Peter Retief and Louise Cilliers: “The death of Cleopatra”, in Acta Theologica, 2006
- Simon Coumes, Pierre-Henri Paris, François Schwarzentruber, and Fabian M. Suchanek: “Qiana - A First-Order Formalism to Quantify over Contexts and Formulas”, in International Conference on Principles of Knowledge Representation and Reasoning, 2024