How Belief Works

THE LOGICAL IMPOSSIBILITY OF DEGREES OF BELIEF

How Belief Works is an ongoing series of articles on the psychology of belief that's best read in sequence.

It can seem obvious that we experience different degrees, or strengths, of belief. For example, it can seem obvious that our belief that we’re eating an apple is stronger than our belief that it’ll rain tomorrow.

However, the concept of degrees of belief is actually fatally logically flawed. Indeed, closer analysis of our experience of belief reveals that it's actually equally compatible with there not being degrees of belief.

And the non-existence of degrees of belief has profound implications for human psychology, and therefore also our lives.

One conception of degrees of belief

There are actually three competing conceptions of degrees of belief. According to one, belief about the probability of claim X's content is, in the case of high probabilities, a degree of belief of X, with belief that X's content has the maximum probability, certainty, being the maximum degree of belief of X. For example, our belief that there's a high but sub-certain probability P that there are six eggs in the fridge is a sub-certain degree of belief that there are six eggs in the fridge. And our belief that it's certain that there are six eggs in the fridge is the maximum degree of belief that there are six eggs in the fridge.

But consider our belief that there's a high but sub-certain probability P that there are six eggs in the fridge. This is belief of the claim There's a high but sub-certain probability P that there are six eggs in the fridge. And the claims There are six eggs in the fridge and There's a high but sub-certain probability P that there are six eggs in the fridge are, although close, ultimately different claims, given that the latter states that the existence of six eggs in the fridge is uncertain, whereas the former doesn't.

And belief of one claim is by definition a different belief from belief of a different claim, however close the two claims. So our belief that there's a high but sub-certain probability P that there are six eggs in the fridge actually can't in itself constitute a sub-certain degree of belief that there are six eggs in the fridge.

Even our belief that it's certain that there are six eggs in the fridge can't in itself constitute a maximum degree of belief that there are six eggs in the fridge. That is, this is belief of the claim It's certain that there are six eggs in the fridge, and even this claim is different from the claim There are six eggs in the fridge.

The sentences There are six eggs in the fridge and It's certain that there are six eggs in the fridge can seem to be just different wordings of the same claim, but they're aren't. As I explained in Why Do We Believe What We Believe?, the claims X and X is true are different claims that merely imply each other. Similarly, the claims X and It's certain that X are different claims that merely imply each other. That is, if there are six eggs in the fridge then the probability that there are six eggs in the fridge is certainty, and vice versa.

As I said in Why Do We Believe What We Believe?, if two sentences are different wordings of the same claim, then, by definition, neither sentence will refer to something that the other doesn’t refer to. But whereas the sentence It's certain that there are six eggs in the fridge refers to the probability of there being six eggs in the fridge, the sentence There are six eggs in the fridge doesn’t – it simply refers to there being six eggs in the fridge.

As with the claims X and X is true, the sentences making the claims X and It's certain that X can seem to be just different wordings of the same claim because each claim follows so obviously from the other that we can fail to notice the very basic logical step separating them.

And also as with the claims X and X is true, the claims X and It’s certain that X being different claims that merely imply each other means that our belief of them are by definition different beliefs that merely imply each other.

So believing It’s certain that X isn’t in itself believing X, although it implies our belief of X. For example, believing It's certain that there are six eggs in the fridge isn’t in itself believing There are six eggs in the fridge, although it implies the latter belief.

In sum, our belief about the probability of claim X's content can't in itself constitute a degree of belief of X, however high that probability, and is instead belief of the different claim There's probability P that X.

So this conception of degrees of belief is fatally logically flawed.

Another conception of degrees of belief

According to another conception of degrees of belief, different high levels of confidence in a claim are different degrees of belief of it, with the maximum level of confidence, certainty, being the maximum degree of belief. For example, our high but sub-certain confidence that there are six eggs in the fridge is our sub-maximum degree of belief that there are six eggs in the fridge. And our certainty that there are six eggs in the fridge is the maximum degree of belief that there are six eggs in the fridge.

Our level of confidence in a claim is a feeling about that claim. Indeed, we often refer to how confident we feel about a claim. But, contrary to what might be thought, our confidence in a claim isn't a feeling about whether the claim is true.

Again, the claims X and X is true are different claims. And a feeling about whether X is true is a level of confidence in, or doubt about, the claim X is true. So this feeling isn't in itself a level of confidence in, or doubt about, claim X.

Our level of confidence in X is therefore instead a feeling about whether X. For example, our level of confidence in the claim There are six eggs in the fridge is a feeling not about whether this claim is a true claim, but about whether there are six eggs in the fridge.

However, whereas a feeling is an emotional mental state, belief isn't. Belief is an understanding. For example, our belief that there are six eggs in the fridge is our understanding that there are six eggs in the fridge. And an understanding is in itself a purely intellectual, and so unemotional, mental state. Of course, a belief can arouse an emotion, but that’s one mental state leading to another.

As I said in the previous article, How Beliefs Form, thinking claim X is the mental state of believing X. For example, thinking 'There are six eggs in the fridge' is the mental state of believing that there are six eggs in the fridge. And thinking and feeling are, famously, distinct mental states, albeit intimately interconnected.

I also explained in the previous article that, by logical necessity, thinking/believing X involves X's content existing in our mind not as X's content – which would involve thinking about X – but as an aspect of reality. And X's content can exist in our mind as an aspect of reality without this mental state in itself involving a feeling of any kind, although it may lead to a feeling.

And if our level of confidence in a claim is a different kind of mental state from our belief of a claim, then no level of confidence, even certainty, in a claim can in itself constitute belief of that claim. So this conception of degrees of belief is also fatally logically flawed.

It might be thought that this conception could be revised so that the mental state of belief involves the combination of an understanding and a feeling of confidence. But given that understanding that there are six eggs in the fridge is in itself believing that there are six eggs in the fridge, the mental state of belief doesn’t in itself also involve a feeling of confidence. Indeed, by definition, two different kinds of mental states can't together form a single mental state, and so can only ever be seperate mental states.

The remaining possible conception of degrees of belief

The remaining possible conception of degrees of belief is simply that a degree of belief of claim X is a mental state that's distinct from both belief about the probability of X's content and a level of confidence in X. But even this conception is fatally logically flawed, because degrees of belief are actually in themselves logically impossible.

Our supposed degree of belief of X is how supposedly strongly we believe X. And, by definition, any strength of belief of X, however low, would be belief of X. That is, given that strength of belief would by definition be a property of belief, any strength of belief would by definition be belief. However weakly we believed X, weak belief of X would still be belief of X. So any degree of belief of X, however low, would be belief of X.

Also, again, by logical necessity, thinking/believing X involves X's content existing in our mind not as X's content, but as an aspect of reality. And given that X's content existing in our mind as an aspect of reality is by itself believing X, the mental state of believing X solely involves X's content existing in our mind as an aspect of reality as we think X.

But, logically, X's content can't exist in our mind as an aspect of reality to different degrees. That is, logically, X's content either exists in our mind as an aspect of reality or it doesn’t – just as it either exists in our mind as X's content or it doesn’t.

For example, different degrees of believing the claim There are six eggs in the fridge would involve this claim's content existing in our mind as an aspect of reality to different degrees, which is nonsensical. That is, logically, that content either exists in our mind as an aspect of reality or it doesn’t – just as it either exists in our mind as the content of the claim There are six eggs in the fridge or it doesn’t.

Consider the fake window thought experiment that I presented in the previous article, and the fact that the content of the picture in the fake window can exist in our vision as either the content of a picture of an outdoor scene or a real outdoor scene. Again, these two ways that the picture's content can exist in our vision are analogous to the two ways that a claim's content can exist in our mind.

As with the idea of a claim's content existing in our mind as an aspect of reality to different degrees, the idea of the picture's content existing in our vision as a real outdoor scene to different degrees is illogical. That is, logically, that content either exists in our vision as a real outdoor scene or it doesn’t – just as it either exists in our vision as the content of a picture of an outdoor scene or it doesn’t.

Equally logically, we can't think X to different degrees. That is, logically, we're either thinking – and thus believing – X or we're not.

Also consider that, as I said earlier, belief is an understanding. Again, our belief that there are six eggs in the fridge is our understanding that there are six eggs in the fridge. But the terms degree of understanding and strength of understanding are only used to refer to how well we understand something. Indeed, the idea of different degrees of understanding in the sense of degrees of belief is illogical. For example, logically, it’s either our understanding that there are six eggs in the fridge or it isn’t.

In sum, degrees of belief are in themselves logically impossible.

The fact that there aren't degrees of belief, whereas there are levels of confidence in claims, is another way in which our confidence in a claim is a different kind of mental state from our belief of a claim.

The true correspondence between our belief about the probability of claim X's content, our level of confidence in X, and our belief of X

To understand the true correspondence between our belief about the probability of claim X's content, our level of confidence in X, and our belief of X, consider first believing that X's content is sub-certain.

If we believe that X's content is sub-certain then our level of confidence in X will be sub-certain. Note that whereas the first use of the term sub-certain in the previous sentence refers to sub-certainty in the sense of below the maximum level of probability, which is a mathematical concept, the second use refers to sub-certainty in the sense of below the maximum level of confidence, which is a mental state.

And the particular level of sub-certain probability will correspond with the particular level of sub-certain confidence. For example, if we believe that it's likely that there are six eggs in the fridge then we'll have high but sub-certain confidence in the claim There are six eggs in the fridge.

And our belief There's sub-certain probability P that X is incompatible with believing X, however high P is. For example, our belief It's highly likely that there are six eggs in the fridge is incompatible with believing There are six eggs in the fridge, given that the first claim implies that the existence of six eggs in the fridge is uncertain, whereas the second claim implies that it's certain.

If there were degrees of belief then our maximum degree of belief of the claim It's highly likely that there are six eggs in the fridge, and our corresponding very high but sub-certain confidence in the claim There are six eggs in the fridge, would imply, and so be compatible with, our high but sub-maximum degree of belief of the claim There are six eggs in the fridge.

Now consider believing that X's content is certain.

If we believe that X's content is certain then our level of confidence in X will be certainty. For example, if we believe It's certain that there are six eggs in the fridge then our level of confidence in the claim There are six eggs in the fridge will be certainty.

And although believing It’s certain that X isn’t, as I explained earlier, in itself believing X, it implies our belief of X. For example, although believing It's certain that there are six eggs in the fridge isn’t in itself believing There are six eggs in the fridge, it implies the latter belief.

To recap: If we believe that claim X's content has a sub-certain probability then we'll have a corresponding sub-certain level of confidence in X, and we won’t also believe X, however high that probability and confidence. And if we believe that X's content is certain then our level of confidence in X will be certainty, and we'll also believe X.

Although our feeling of certainty in X is a separate mental state from that of our belief of X, there's actually a sense in which the latter does in itself involve certainty.

Certainty involves, by definition, an absence of doubt. And we often refer to certainty by referring to that absence of doubt. For example, we may refer to our certainty that there are six eggs in the fridge by saying ‘I’ve no doubt that there are six eggs in the fridge’.

Also, just as the mental state of belief can't in itself involve a feeling of confidence, so, by the same logic, it can't in itself involve a feeling of doubt. Although, given the non-existence of degrees of belief, our belief of claim X is anyway incompatible with doubt about X, just as it's incompatible with sub-certain confidence in X.

So we can say that the mental state of believing X involves, in itself, certainty in the sense of an absence of doubt. That is, to believe X is in itself to be certain, in the sense of an absence of doubt, that X. For example, to believe that there are six eggs in the fridge is in itself to be certain, in the sense of an absence of doubt, that there are six eggs in the fridge.

Likewise, to believe It’s certain that X is in itself to be certain, in the sense of an absence of doubt, that it’s certain that X. And to believe There's sub-certain probability P that X is in itself to be certain, in the sense of an absence of doubt, that there's sub-certain probability P that X.

The fact that the mental state of believing X involves, in itself, certainty about X isn't incompatible with the fact that thinking/believing X is different from thinking about X, because our certainty about X isn't a mental state concerning X, but the absence of a mental state concerning X.

So we can refer to the non-existence of degrees of belief as the certainty of belief.

The fact that there aren't degrees of belief, and that belief involves, in itself, certainty in the sense of an absence of doubt, implies that doubting claim X, however slightly, will in itself constitute the end of our belief of X. However, we may then at least believe that X's content is almost certain, or at least highly likely to some degree.

I’ll address at the end of the article why the certainty of belief has profound implications for human psychology, and therefore also our lives.

Our apparent experience of degrees of belief

As I said in the introduction, it can seem obvious that we experience different degrees of belief. For example, it can seem obvious that our belief that we’re eating an apple is stronger than our belief that it’ll rain tomorrow.

But our apparent experience of a degree of belief of claim X is actually either our belief about the probability of X's content or our level of confidence in X.

In the example, our apparent experience of the maximum degree of belief that we’re eating an apple will actually be either our belief that it's certain that we're eating an apple – that is, our belief not of the claim I'm eating an apple, but of the claim It's certain that I'm eating an appleor the maximum level of confidence – certainty – that we're eating an apple. And, separately from this mental state, we'll experience believing that we're eating an apple.

And our apparent experience of a sub-maximum degree of belief that it’ll rain tomorrow will actually be either our belief that there's a high but sub-certain probability P that it’ll rain tomorrow – that is, our belief not of the claim It'll rain tomorrow, but of the claim There's a high but sub-certain probability P that it’ll rain tomorrow – or a high but sub-certain level of confidence that it’ll rain tomorrow. And we won't experience, separately from this mental state, believing that it’ll rain tomorrow.

The logical necessity of the certainty of belief

As with the argument for credulism that I presented in the previous article, the argument that I presented for the certainty of belief has a doubly counterintuitive conclusion. In revealing not only that there aren't degrees of belief, but also that this is because they're logically impossible, it reveals that the non-existence of degrees of belief is due not to the nature of the human brain, but to logical necessity.

As I said in the previous article, the central tenet of the field of psychology today is that how the human mind works is due solely to the wiring and chemistry of the human brain, which are in turn determined by a person’s genes and experiences.

But although our capacity for belief is obviously due to the nature of the human brain – as is our capacity for reason, emotion, language, perception, and so on – the non-existence of degrees of belief isn't. Instead, degrees of belief are simply logically impossible, and the certainty of belief is therefore simply logically necessary – and not just in the case of human intelligence, but in the case of any form of intelligence.

As I also said in the previous article, another tenet of the field of psychology today is that using logic alone to reach conclusions about human psychology is unscientific, given the absence of observational evidence in such reasoning.

But, as I pointed-out, at the beginning of modern science the use of mathematics, a form of logic, alone to reach conclusions about nature, rather than in conjunction with observational data, was controversial. And yet in science today this is accepted as a legitimate way to further our understanding of nature. For example, much of theoretical physics is the use of mathematics, and non-mathematical logic, alone to develop and refine theories.

The logical impossibility of degrees of belief shows that using logic alone is a legitimate way to further our understanding of human psychology. Indeed, that logical impossibility means that using logic alone is the only way of fully understanding this aspect of the nature of belief. That is, although we could find observational evidence against degrees of belief by studying brain activity, no amount of such evidence could show the logical impossibility of degrees of belief.

The argument I presented, which I developed, doesn't just support the certainty of belief, it proves it, by showing that degrees of belief are logically impossible, and that the certainty of belief is therefore logically necessary. Although, in addition to this theoretical proof, the certainty of belief must of course also be confirmed experimentally. The idea of degrees of belief has seemed so obviously true that it has never actually been tested experimentally.

The origin and prevalence of the idea of degrees of belief

The origin and prevalence of the idea of degrees of belief is likely due to the combination of the following factors:

For example, our experience of believing that we're eating an apple involves being certain, in the sense of an absence of doubt, that we're eating an apple, which is very similar to what it would be like to experience having the maximum degree of belief that we're eating an apple. Whether we have certainty, in the sense of an absence of doubt, that X, or we have the maximum degree of belief that X, we believe X with an absence of doubt, and are therefore implicitly positive to the same degree about X's truth.

Likewise, our experience of believing that it's likely that it’ll rain tomorrow involves being certain, in the sense of an absence of doubt, that it's likely that it’ll rain tomorrow, which is very similar to what it would be like to experience having a sub-maximum degree of belief that it’ll rain tomorrow. Whether we have certainty, in the sense of an absence of doubt, that there's a high but sub-certain probability that X, or we have a sub-maximum degree of belief that X, we're implicitly positive to the same degree about X's truth.

An alternative argument for credulism?

The fact that belief involves, in itself, certainty merely in the sense of an absence of doubt, as opposed to a high level of confidence, including even a feeling of certainty, might seem to lead to an alternative argument for credulism – the theory, that I presented in the previous article, that the mere entrance of a claim into our mind causes us to believe it. That is, given that any level of doubt about claim X must arise from our consideration of X, which obviously can only occur after X has entered our mind, we must have an absence of doubt about X upon X entering our mind, and thus believe X upon that entrance. But unfortunately this argument doesn't work. 

Although belief involves, in itself, certainty merely in the sense of an absence of doubt, it doesn't merely involve such certainty. That is, an absence of doubt about claim X is in itself simply the absence of a particular mental state concerning X, whereas the mental state of believing X is the existence of X's content in our mind as an aspect of reality. Indeed, our feeling of certainty in X involves, by definition, an absence of doubt about X, but isn't in itself the mental state of believing X.

So our absence of doubt about X upon X entering our mind doesn't in itself explain our belief of X upon that entrance.

Of course, if immediately after X entering our mind we consider our absence of doubt about X we might rashly conclude X. But that belief would therefore form after the entrance of X into our mind. Also, that consideration could equally lead us to think of a reason to be uncertain about X, and so we’d then not conclude X.

Why the certainty of belief matters

It might be thought that although the conclusion that there aren’t degrees of belief obviously differs significantly from the current understanding, and matters in the sense that knowing the truth about the world, including our psychology, matters intellectually, it doesn't actually have any significant theoretical or practical implications.

As I said earlier, whether we have a sub-maximum degree of belief that X, or we have certainty, in the sense of an absence of doubt, that there's a high but sub-certain probability that X, we're implicitly positive to the same degree about X's truth. And whether we have the maximum degree of belief that X, or we have certainty, in the sense of an absence of doubt, that X, we believe X with an absence of doubt, and are therefore implicitly positive to the same degree about X's truth.

But, as I said at the end of the previous article, in subsequent articles I'll show that credulism and the certainty of belief together explain cognitive biases – universal biases in human cognition which account for most of human irrationality.

Again, there are many theories about why cognitive biases exist. Some only concern a single bias, whereas others try to unify several biases with a single explanation. All involve the cognitive bias or biases being due to the nature of the human brain.

But I'll show that all cognitive biases are simply manifestations of credulism and the certainty of belief, and so are simply logically necessary aspects of cognition, for any intelligent entity. That is, these two aspects of the psychology of belief together provide a unifying explanation for all cognitive biases, and thus for most of human irrationality.

Next article: The Origin of Cognitive Biases (in progress)

How Belief Works

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This article was first published 26 September 2023. Past versions are available in the Internet Archive here.