Persephone and the seasonsAs a motivating example for our study of truth, let us consider a story from the Greek mythology (inspired by a TED talk by David Deutsch). To explain the seasons of summer and winter on Earth, the Greeks had the following story: Hades, God of the Underworld, kidnaps Persephone, the Goddess of Spring, and negotiates a forced marriage contract, requiring her to return regularly, and lets her go. And each year, she is magically compelled to return. And her mother, Demeter, Goddess of the Earth, is sad, and makes it cold and barren.
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?
Some people think that truth is a matter of strength of belief. Since “The earth’s rotation around the sun produces the seasons” is strongly held now but “The stories of the Greek Gods (including Persephone)” were strongly held in their day, both appear to be equally defensible statements on this view of truth.
And yet there is a fundamental difference between these statements about the seasons: While the first rests solely on the strength of the belief of its adherents, the second rests on evidence. This makes the truth of the first statement debatable but not resolvable, limiting the usefulness of the statement, while the truth of the second statement is verifiable, making it both useful and predictive.
This chapter will develop a theory of truth to address this question (as well as the question of truth in general).
StatementsFor 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...”)
Predictive RulesFor the purpose of this book, a predictive rule is a statement of the form
If 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 the conclusion of the rule. 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. For 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 rulesFor this book, a predictive rule is a statement of the form “If A and B and C and..., then D”. 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.”
Examples for predictive rulesA predictive rule consists of a premise and a conclusion. Many phenomena can be formulated in terms of predictive rules. Here are examples:
- Scientific theories
- These include (for example) the law of gravity. This law can be stated as a rule:
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
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.
PerceptionsFor 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 statementsAbove 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:
Auxiliary statements allow us to simplify our predictive rules. For example, using the auxiliary statement “It is raining” as it is defined above, the rule
This predictive rule about rain does not necessarily correspond to the meteorological definition of rain (it could be the case that I am being sprayed with water). Thus, we should rather have defined the conclusion statement “I experience the impression that it is raining”. Yet, it does not matter what we call this feeling of water droplets from above. We could equally well have called it “XYZ” instead. After such a definition, the impression of water droplets falling from the sky would be called “XYZ”.
A rose by any other name would smell just as sweet.
The meaning of auxiliary statementsWe 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.
The theory of truth of this chapter 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 has to 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.
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?”
TheoriesFor 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 A and B, and A has mass m(A), and B has mass m(B), and the distance between A and B is R, then the force between A and B is so and so”. The accompanying definitions will define the statements in the premise of this rule. For example, “the distance between A and B is R” is defined to hold if we can take a tape measure, connect it to A, lead it in a straight line to B, and find B at the mark R. This definition would need again rules to define “taking a tape measure”, etc.
Applying a theoryA 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:
Def. 1: If there are droplets of water coming from the sky, then it rains.
Def. 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 theoriesWe 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.
- 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 caseA 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, e.g., 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. Even when the premises of a theory are fulfilled, the theory can fail to make a prediction when its conclusions are not grounded. In all of these cases, the theory can be called neither true nor false.
Truth of a theoryA 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, e.g., 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 that 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 theoryA 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, where other masses are too far away. 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.
ValidationWe 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, e.g., 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 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.
RejectionA theory can make correct and incorrect predictions. If the theory makes lots of correct predictions, we call the theory “validated”, and we assume that it is part of the truth. If, on the contrary, the theory makes lots of incorrect predictions, then it cannot be part of the truth. We call the theory “false”, and reject it. We have already seen that a theory cannot be “proven” true. However, it can be proven false: False predictions mean that the theory is false.
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.
Life can only be understood backwards
but it must be lived forwards.
Unknown truthIf 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 theoriesOne 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 promises that “If she gives a performance with Vivaldi’s original violin, her performance will be great”. This theory is not applicable. Nobody knows where Vivaldi’s violin is, and even if it turns up somewhere, it would be too expensive to lend it to our violinist. 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.
Unfalsifiable theoriesA 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:
Technically speaking, the premise is not well-defined: We cannot determine whether someone is a “true music connoisseur” purely from known data. We cannot even arrive at this conclusion after the performance, because someone may like the performance even if he’s not a music connoisseur. 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 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. The theory cannot be falsified.
Falsifiability explainedThe concept of falsifiability was advocated by the philosopher Karl Popper. A theory is falsifiable if we can imagine a case 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”. We are looking for a statement X such that
If 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, and thus 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.
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.
GeneralitySo far, we have only talked 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 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. Thus, 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 Gregory Chaitin has suggested that it is tantamount to comprehension.
Brevity is the soul of a pattern: the pattern must be shorter than the dataset itself.
Useless premisesA 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!”
TestabilityA 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.
CausalityIt 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 have 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.
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.
- The rule 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.
ExplanationWe 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
- the theory is general
We say that a theory explains a statement if it fulfills all three conditions and predicts the statement. As an example, consider again the scenario of high tides. An ideal explanation 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: TheoriesThis 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
- 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.
- A theory is applicable if there is (or will be) a case where the theory makes a prediction.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
EvidenceFor 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 true theory that predicts this statement. 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 this theory predicts that moon, sun, 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 the statement, we say that the statement is true.
Whatever is not deduced from the phenomena is to be called a hypothesis.
Counter-evidenceA 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 tollensSuppose that the rule “If it rains, the street gets wet” is true. Then the following rule is also true:
If the original rule has several conditions in the premise, then we obtain as many rules as there are conditions.
Joint inferenceA 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 Facebook 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 definitionsA definition is a rule that has an auxiliary statement as its conclusion. 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. Stephen Pinker gives examples of things that are surprisingly difficult to define2. Take the concept of a bachelor: if a bachelor is an unmarried man, 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”. (This brings us to Fuzzy Logic, 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 (e.g., 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.
Stephen 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.
The unknownA 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.
I was gratified to be able to answer promptly. I said I don’t know.
FalsifiabilityThe concept of falsifiability was advocated by the philosopher Karl Popper , and we have already seen how to apply it to theories. It can also be applied to statements: a statement is falsifiable if we can imagine a situation in the present or future that contradicts the statement or its meaning. For example, the following statement is falsifiable:
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 statementsFrom 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 meaningA statement is falsifiable if we can imagine a situation that would prove it false. Falsifiability asks for a perception statement X, 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: StatementsA statement is a declarative sentence. The meaning of a statement is the set of rules in which it appears in the premise. If there are no such rules, then the statement is literally meaningless. We have seen that being meaningful is tantamount to being falsifiable.
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.
Theories in real life
Learning theoriesWhen we wander through life, we see things, we hear things, and we feel things. We perceive ourselves in a continuous stream of perceptions. We would like to structure these perceptions, to understand them, and (most importantly) to predict them.
And so we start to build theories about these perceptions. These theories are rarely verbal and explicit. Rather, they are a body of knowledge about the patterns that we observe in this world. For example, we pretty quickly build the theory that if we touch a hot oven we will then feel pain. This theory helps explain past instances of pain and is very effective in preventing future instances of pain. And this is the goal: We want to build theories that explain past experiences and that predict future experiences.
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 situations 3. 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.
Wisdom is not the product of schooling, but of the lifelong attempt to acquire it.
CorrectionTheories yield explanations and predictions. 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 6km/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. No matter how plausible this theory sounds, and no matter how many correct predictions it has made in the past, it is false. Importantly, it is not reality that needs fixing. It is the theory that needs fixing.
This is indeed what happened to the theory. 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 relativity.
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.
I would never die for my beliefs because I might be wrong.
ScienceThis chapter argues that throughout our life, we build up theories that help us structure our experiences. We seek those theories that are “true”, i.e., that will in principle eternally make correct predictions (though of course we cannot know this). The more correct predictions a theory makes, the more likely it is that the theory is true. In that sense, science is more systematic variant of that process. Science continuously builds theories and tests them to see if their predictions are consistently correct. When scientists propose a new theory, they conduct experiments to show that the theory makes predictions that correspond to reality. Other scientists try to build experiments that show that the theory makes false predictions. If a theory turns out to make consistently correct predictions, it is accepted into the corpus of scientific theories.
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 may be described as the art of systematic over-simplification — the art of discerning what we may with advantage omit.
The theory of truthThis book argues that we constantly build theories to explain our past perceptions and to predict our future perceptions. In this view, we humans have a rather humble position in this universe: we perceive, and we try to model what we perceive. At any moment, our theories may turn out to be incorrect. Then we have to abandon what we thought was true.
Interestingly, this view of truth is itself nothing else than a theory. This theory says that we judge the truth of a theory by comparing its conclusions to our experiences. If the conclusions consistently correspond to our experiences, we trust the theory — and we assume it to be part of “the truth”. In such cases, we will just say “The theory is true”. This theory of truth predicts what a person in the street calls “true”. This theory is true, because by and large this is indeed what the person in the street calls “true”. Then again, the theory is only true by the theory’s own views regarding the nature of truth.
We have seen that this theory of truth can be interpreted as a generalization of science. Other parallels can be found: Karl J. Friston has argued that the brain is constantly generating and updating a mental model of the environment, with the goal of generating predictions of sensory input. This theory is known as Predictive coding, and it bears obvious resemblance to the theory of truth presented in this chapter. 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-layer prediction engine 4. Another parallel can be found in the domain of artificial intelligence, and more precisely in machine learning: Statistical learning approaches try to find a function that, given some input data, predicts some output data. Such a function is called a model. Good models predict the output data that was expected, they generalize beyond the input data, and they are simple 5 — in the very same way that good theories are validated, general, and short .
All models are wrong, but some are useful.
Using the theory of truthOur theory of truth says that a statement is true if we have evidence for it — i.e., if it is predicted by a true theory. A statement is false if we have counter-evidence for it — i.e., if there is a true theory that predicts its negation. In real life, our evidence for or against a statement is often very weak: we do not have a set of theories at hand that predicts our hypothesis. And even if we had, we would probably not have the statistics to show that these theories are validated. And even if we did, a validated theory is not necessarily true. So we would have to perform careful joint reasoning to arrive at the most likely conclusion. Since we usually cannot do all of this, we often just believe a statement.
This does not mean, however, that the theory of truth elaborated in the present chapter would be useless. Quite the contrary. The first thing that our theory of Truth allows us to do is to exclude unfalsifiable statements. 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 meaningless statements, 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 the 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 bogus claims and to avoid drawing false conclusions.
Finally, and most importantly, the Theory of Truth tells us at least what we would have to do if we wanted to check the truth of a hypothesis: we know that we have to find evidence, i.e., we know that 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 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 MistakesUp to this point in the chapter we have elaborated a theory of truth. Humans build hypotheses to describe and to predict what happens around them. The perfection of this endeavor (the hypotheses that always make correct predictions) is what we are calling “the truth”.
We will now look into common mistakes that people make when trying to approach the truth. We will illustrate each mistake by an example. We will then apply the theory of truth to find out what the problem is in the example scenario. As we will 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 they appear in very similar form in each of the world’s major religions, as we shall see in the Chapter on Proofs for Gods and in the Chapter on Gods.
It makes senseIn 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 century. 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. 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 .
Know how to rank beliefs not according to their plausibility, but by the harm they may cause.
GhostificationMany ancient peoples believed that inanimate objects were inhabited by spirits. They might well have 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.
Throughout history people have tried to explain nature by personifying it. Then these personifications developed a life of their own.
We will later argue that religions piggy-back on otherwise reasonable theories in a similar fashion: prayer works because it helps us reflect our lives — not because there would be a god who listens. Faith healing works because of the placebo effect — and not because of some magical connection with the supernatural. Religion lives from the ghostified versions of these theories.
Unclear theoriesImagine 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. Therefore, the theory 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 “For every bad event, something good will eventually happen”. As we shall see later, this belief is as ill-defined as the omen.
PostdictionsConsider the story of the seer in the book of “Asterix and the Seer” 6. The seer finds shelter in the village of Asterix during a thunderstorm. He 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 “pre-dictions”. He is making “post-dictions”.
The real challenge is 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 for all 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, when such a prediction is actually made only 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.
Counting the hitsImagine 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, as we shall discuss: When it comes to the efficiency of prayer, for example, many people are willing to count the hits only.
Useless premisesImagine 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 — or indeed as true as the theory “There are car accidents”. In some countries residents view 17 as an unlucky number, and in others 13 is viewed as unlucky. It is clear that with this argument we can take any number as an unlucky number (or as a lucky number for that matter).
Technically speaking, the theory contains a useless premise. This entails that the theory is not as general as it could be. The following theory compresses many more events:
Again, we can see a link with religious beliefs: Some people believe that prayer cures illnesses, and uphold this claim even for illnesses that disappear by themselves.
Narrative fallacyWhen the US captured Saddam Hussein in Iraq 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 case 7. However, humans tend to add causality to unrelated events. This tendency is what Nassim Taleb calls the narrative fallacy in his book The Black Swan 8.
This tendency to “storify” may have a very simple reason, as 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.
It is interesting to note that, in the same spirit, people sometimes add supernatural causality to whatever events happen, as we will discuss in the Chapter on the Founding of Religion and in the Chapter on the Meaning of Life.
Abductive reasoningThe Air Malaysia flight MH370 was an international passenger flight scheduled from Kuala Lumpur to Beijing on March 8th, 2014. The airplane disappeared shortly after take-off, and has never been found. Despite the most expensive search in aviation history, nobody knows what happened to MH370.
However, imagine that your friend (let’s call him Robert) has long been suspecting aliens behind the disappearance. He comes up with the following rule:
Robert will defend his 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! Therefore, my reasoning must be right!” Is that correct?
It is true that nobody can explain how MH370 disappeared. However, that does not make Robert’s hypothesis true. In order to accept a hypothesis as true, we need a rule that has the hypothesis in conclusion and not in the premise. In other words, we need evidence for the hypothesis. Until then, 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 god(s) who created the world. This, however, does not make their hypotheses true. We discuss this in the Chapter on the God of Gaps.
The analogy can be continued: Richard 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 a rule that implies it. However, if people do not subscribe to the necessity of evidence for hypotheses, they can go on arguing forever.
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!
Wishful thinkingBill 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”. As we will discuss, it has played a role in the popularity of religions as well: Sometimes people believe something merely because they want it to be true.
Happy thinkingYour 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. He 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?
Meaningless statementsYou 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 OK? Do you need anything? Did you see anything? Is there a smell?”. But she just says that yes, she’s OK, 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. There is no rule that has this statement in its premise. Thus, the statement is literally meaningless, as far as your place is concerned. 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 something greater than us”, “There is a conscious being behind everything”, and even “God is the first cause of the universe”. We will argue that these allow for no concrete predictions about the real world either.
Empty statementsAndy 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 week later, you meet another acquaintance, and you find out that you both know Andy. The other person tells you that he gets along really well with Andy, and that Andy calls him his best friend. You are surprised, but do not say anything. 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 that has “X is Andy’s best friend” in its premise. 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” are of the very same nature: They carry a strong suggestive meaning, but boil down to senselessness when questioned.
Unfalsifiable statementsLet’s 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
Our example bears noticeable similarities to phrases such as “God loves us”. Even 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. Such statements are the bedrock of religions, as we shall see in the Chapter on Memes.
Truth and Atheism
Revisiting persephoneWe started our discussion of truth from the ancient Greek story of Persephone. Persephone is the Goddess of Spring, and she returns to Hades each year. This leaves her mother sad and so she makes it cold. This explains why winter is cold.
Now let’s 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.
Summary on the concept of truthThis book argues 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 predict future perceptions. For this purpose, we build theories of the form
The problem is, of course, that we never know whether a theory will always correspond to our observations. This is why we humans should take a rather humble position in this universe: we try to describe our reality by theories, we validate the theories, and we hope that they are the truth — but in the end we have no guarantee that nature behaves as we predict. Therefore, we find ourselves in a continuous process of improving our theories.
When a true theory makes a prediction, we call that theory “evidence” for the prediction. Evidence distinguishes myths from facts. But our theories cannot predict everything. Some things are just unknown. In that case, it does not help to just invent an explanation. Even if the explanation is beautiful, or psychologically comforting, that does not make it true. As we will argue, it is better to believe only the validated theories, and to continue searching for such validated theories.
In everyday life, our evidence for or against a statement is often weak. In such cases, we tend to just believe the statement. This is fine, as long as our confidence in the statement is proportional to the evidence.
Believing in truthThis book defines truth as the set of theories that make accurate predictions. The question remains why we should believe in these theories and not in others.
The reasons why it is advantageous to believe in this theory of truth are as follows:
- If a theory is “true”, then you can use it to predict parts of the future.
- This is an immense advantage over alternative definitions of truth. It is a very constructive and useful property.
- Given two theories, it is easy to see which theory is more likely to be true.
- Any theory can be measured by what it predicts. Thus, there is a single yardstick to compare theories. This implies that two people who search for truth will converge onto the same theories — which is what happens in science.
The criterion of validated evidence helps us distinguish between myths such as Persephone (whose existence is not predicted by a validated theory), and objective facts such as the spherical Earth (whose truth is established by validated theories).
You are entitled to your own opinions.
But you are not entitled to your own facts.
Believing in falsehoodsBelieving theories that are true in the sense of this book has a number of advantages. At the same time, believing in theories that are not true in the sense of this book has a number of disadvantages. If we abandon the requirement for validated evidence for theories, we run into the following problems:
- We make false predictions.
- If we no longer judge a theory by the proportion of correct predictions, then we may make predictions that do not correspond to reality. If our mental model of the world is not accurate to predict what happens in the world, then we cannot achieve our goals effectively.
- We open the floor to arbitrary theories.
- If our theory is non-falsifiable, or if we abandon the need for evidence, then anybody can claim anything. For example, instead of claiming that the Abrahamic God created the world, we can also claim that the goddess Gayatri created the world. If we abandon the requirement for falsifiability and evidence, then this theory is as good as any other theory. The result is endless disputes with no way to judge which theory is better.
- We blur the distinction between reality and nonsense.
- If we accept unfalsifiable theories as readily as true theories, then this is a sign that we are unable to distinguish between the two. This means that we will accept other nonsensical theories as well. We are then prone to develop justifications, explications, reinterpretations, or view points to justify these theories.
- We inhibit the search for truth.
- If we accept a false or unfalsifiable theory as an explanation, we remove the need for scientific enquiry, and thus prevent ourselves from finding out the true explanation. If we do not know something and assume we know it, then we will never know it.
- We give undeserved credit to the inventor of such a theory.
- If we believe in theories not because they are validated, but because they are presented convincingly, then we give a blank cheque to any charlatan to tell us whatever he wishes. He may use this trust for his own advantage and/or for our disadvantage.
- We risk suffering needless harm
- If the false theory requires us to do or abandon something, and if we follow, then we are needlessly wasting our time, restricting our liberty, renouncing part of our property, or worse.
- We risk harming others.
- If the false theory requires us to reprimand other people, to harm them, or to attack them, then we are unjustly causing damage to other people. When actions are informed by beliefs, then false beliefs can give rise to dangerous or harmful actions 9.
- We perpetuate the problem for the next generation.
- If we do not teach our children to distinguish between validated evidence and myths, then the very same problems will be passed on to the next generation (← this is a recursive reference).
It is always better to have no ideas than false ideas;
to believe nothing than to believe what is wrong.
The supernaturalBased on our theory of truth, we can now formally define the supernatural. A supernatural statement is any statement that, on principle, cannot be validated by a direct observation and that cannot be predicted by a validated theory. Note that with this definition, being supernatural is a property of statements rather than entities. Supernatural statements can be of the form “God exists” or “There is life after death”.
A statement about the supernatural is unfalsifiable. However, not every unfalsifiable statement is about the supernatural. For example, the statement “A true music connoisseur will love this violin performance” is unfalsifiable, but not a statement about the supernatural. A statement is supernatural only if it is, on principle, shielded from our observations and from grounded theories. Religions subscribe to the existence of supernatural entities, and thus to supernatural statements about them.
Truth and atheismThis book has laid out a theory of truth that is based on perceptions. Now what does this have to do with atheism?
Atheism is the rejection of belief in the supernatural. Before, we only had a very vague description of “the supernatural”. With our definition of the supernatural, we can now explain what atheism means: When we say that atheists do not believe in the supernatural, we mean that atheists will not believe supernatural statements.
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.
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. There is no reason to assume a priori that other people perceive like everyone else. The idea that truth is objective is just a theory. This theory, however, has made an impressive number of correct predictions in the past. In fact, for every single case that everyone comes across, other people have perceived physical input just the way they did. There may be people with different perceptual capabilities or limitations (such as color blindness or deafness), but while they may observe different sensitivities to stimuli, they rarely observe contradictions: when they see a mountain, other people also see a mountain; when they hear noise, other people hear noise as well. Therefore, these observations constitute a validation of the theory “Everyone perceives physical observations the same way”.
Now comes the interesting part: if this assumption is true, then other people will evaluate the truth of a given theory in a given case in the same way 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?
Everyone is free to believe what they want. People can believe what is true, what is false, or what is unknown. There is, on a Humanist world view, no 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. 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.
That is all fine of course. But things are different when it comes to more serious questions. For example, when it comes to constructing a skyscraper it would be foolish to rely just on the intuition of the architects. You would want engineers to thoroughly verify that the building is stable and safe. In that case, you would insist on formal methods. The same is true when it comes to anything that influences your life. Say, for example, that your doctor prescribes you a new medicine. If he tells you that this medicine has never been tested, you would certainly hesitate to take it. It would be of poor comfort to you if the doctor assures you that “If you really deserve the healing, this medicine will not be lethal”. Even if you are not familiar with the principle of falsifiability, you would realize that there is something wrong with this doctor. You would insist on taking only medicine that has been tested thoroughly beforehand. And so this is the compromise that we use: In our everyday lives, we use some informal methods to decide what we believe in. These informal methods are vaguely related to the theory of truth of this chapter. For anything that engages our lives crucially, we insist on formal methods that align with the approach spelled out in this chapter. The distinction between these two modes of reasoning is what the psychologist and economist Daniel Kahneman has called “System 1” vs “System 2” in his 2011 bestseller “Thinking, Fast and Slow” 7.
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 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.
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.
I don’t want theories, I want facts!This chapter puts forward the idea that everything we do is building theories about our perceptions. The reader might wonder where this view accommodates facts.
If the reader wishes, they can view their elementary perceptions as facts. Statements such as “I feel hungry”, “I see red color”, or “I feel happy” are undoubtable facts. This chapter says that everything that goes beyond these elementary perceptions are theories that we build to explain these perceptions: the concepts of hunger, of red objects, and of happiness are theories that we build on top of these perceptions in order to structure them and to predict them.
What about moral truths?This chapter says that truth is whatever makes correct predictions. This leaves us to wonder whether a moral statement such as “Theft is wrong” is true.
In fact, according to this chapter, such a statement is not true. It is not even false. This is because it is not falsifiable. For this book, moral statements are not absolute truths. Rather, they are subjective opinions about behaviors. Thus, we can never say “Theft is wrong”. We can only say “I find theft wrong”. That makes sense: there are cultures where theft is not considered wrong. In an ideal communist world, for example, the notion of theft is entirely meaningless. Thus, “I find theft wrong” is on the same level as “I find Alice beautiful”. Such theories do make predictions: For example, if you steal something from me, I will get angry, because I think that theft is wrong. Thus, these theories are falsifiable.
On this insight, we can build an entire moral theory, which we discuss in the Chapter on Morality.
What is the meaning of love?The meaning of a statement is the set of rules in which it appears. Consider the following then as a starting point:
If you love someone, you feel happy when that person is around.
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
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.
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?How do we know that Cleopatra was queen of Egypt? Is that falsifiable?
Historians reconstruct past events from human artefacts, written accounts of events, archeological sites, and other sources. Technically speaking, they have a number of theories, such as “If a person appears on a coin, then that person was a ruler of a state”, or “If a contemporary historian wrote a book, then things happened as written in that book”. These theories are not always correct. They may make false predictions. For example, a person on a coin could be a deity instead of a ruler. However, in a large part of the cases, the theories are known to be correct. Thus, historians basically have a set of imperfect rules. From these, they try to deduce what most likely happened in the past. If several rules predict the same thing, and no rule predicts the contrary, then the historian accepts this thing as the most likely course of history.
This thing, however, may turn out to be false. For example, it was widely assumed that Cleopatra committed suicide by an asp bite. Nowadays, some historians are questioning this version of history, based on the fact that an asp bite could not have caused the quick death claimed by most sources. 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.
This theory of truth limits one’s view of the worldThis 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, or 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. It is useful to keep this distinction in mind.
There is no society in human history
that ever suffered because its people became too reasonable.
There is divine truth!Some people define truth through God. Truth is what God says.
The problem with this definition is that it never allows making any tangible predictions. If you assume that what God says is the truth, then you do not know anything more about this world than if you don’t. To see this, assume that what God says is the truth. Now, where did the Air Malaysia flight MH370 disappear? You don’t know. How many species exist on Earth? You don’t know. Who will win the lottery? You don’t know. It turns out that you know exactly as much as everyone else, as far as concrete predictions are concerned. And what everyone else knows is what we commonly call “the truth”. There is no need to use God for this.
We may say that you know at least that the Earth was created by God, that all species were created by God, and maybe even that the lottery winner was chosen by God. However, you can know this only after the fact. Whatever happens, you say that it was God’s will. But you can never know in advance what in fact will turn out to be God’s will. Thus, such a theory of truth is just a story that is put on top of whatever happens. Technically, it is a ghostfication.
If the test tube turns green, God wanted it. Oh, sorry, it actually turned blue. No worries, God decided that in the first place.
Truth is a false conceptIt has been argued that we cannot be sure whether the physical system in which we perceive ourselves really exists (most famously 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. We could be living in a dream world.
We note that this assumption cannot be falsified. There is no way to prove that we are not living in a dream. This entails that the theory does not tell us anything about our perceptions. When we assume that we are living in a dream, we are no wiser than before. Thus, the theory 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 ａ ∧ ¬ｂ ⇒ ｃ, ａ ∧ ¬ｃ ⇒ ｂ
The disadvantage is that the system cannot express existential quantifiers in the conclusion. The system cannot deduce “There exists x such that...” — unless the x is an entity that does not depend on any other variables in the rule. In that case, we can simply give a name to that entity and have it appear in the conclusion. This holds in particular if x is an entity for which we already have a name (such as “God”). If we need existential variables that depend on other variables, we have to use Skolem functions.
If we need to reason about rules themselves, we have to resort to higher order logic.
- Judea Pearl: The Book of Why, 2018
- Stephen Pinker: Rationality — What It Is, Why It Seems Scarce, Why It Matters, 2021
- Ian H. Witten, Eibe Frank, Mark A. Hall, Christopher J. Pal: Data Mining: Practical Machine Learning Tools and Techniques, 2016
- Andy Clark: “Consciousness as Generative Entanglement”, in The Journal of Philosophy, 2019-12
- Nedeljko Radulović, Albert Bifet, Fabian M. Suchanek: “Confident Interpretations of Black Box Classifiers”, in International Joint Conference on Neural Networks, 2021
- René Goscinni, Albert Uderzo: Asterix et le Devin, 1972
- Daniel Kahneman: Thinking, Fast and Slow, 2011
- Nassim Taleb: The Black Swan: The Impact of the Highly Improbable, 2007
- Armin Navabi: Why there is no god, 2014