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 from experience that there are 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.

But the concept of degrees of belief is fatally logically flawed. Indeed, closer analysis reveals that our experience of belief is 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

According to one 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 a 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. But, contrary to what might be thought, it isn't a feeling about whether the claim is true.

A feeling about whether claim X is true is a level of confidence in, or doubt about, the claim X is true. And, as I explained in Why Do We Believe What We Believe?, the claims X and X is true are different claims. So a feeling about whether X is true 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.

The difference between our level of confidence in, and belief of, a claim is apparent from the analysis in the previous article, How Beliefs Form – in three ways.

First, as I said in that article, 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, different kinds of mental states.

Second, whereas, as I pointed-out in the previous article, the mental state of believing claim X doesn't in itself actually involve thinking about claim X, our level of confidence in claim X is a feeling about X.

For example, the mental state of believing the claim There are six eggs in the fridge involves thinking about simply the existence of six eggs in the fridge, rather than about this claim about the existence of six eggs in the fridge. But our level of confidence in the claim There are six eggs in the fridge is a feeling about whether there are six eggs in the fridge, which is a feeling about the possibility that there are six eggs in the fridge, which is a feeling about the claim There are six eggs in the fridge.

Third, I 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 the existence of X's content in our mind as an aspect of reality isn’t in itself a feeling of confidence.

Of course, 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 be a degree of belief of that claim. That is, this conception of degrees of belief is fatally logically flawed.

It might be thought that this conception of degrees of belief 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.

Another conception of degrees of belief

According to another conception of degrees of belief, our belief about the probability of claim X's content is, when that apparent probability is high enough, a degree of belief of X. For example, our belief that it’s highly likely that there are six eggs in the fridge is a high degree of belief of the claim There are six eggs in the fridge.

But our belief that it’s highly likely that there are six eggs in the fridge is belief of the claim It’s highly likely that there are six eggs in the fridge. And the claims There are six eggs in the fridge and It’s highly likely that there are six eggs in the fridge are different claims. So our belief that it’s highly likely that there are six eggs in the fridge isn't in itself belief of the claim There are six eggs in the fridge.

So this conception of degrees of belief is also fatally logically flawed. Our belief about the probability of claim X's content can't be a degree of belief of X, however high that apparent probability, and is instead belief of the different claim There's probability P that X.

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 a level of confidence in X and belief about the probability of X's content. But this conception is also 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.

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, believing X consists solely of X's content existing in our mind as an aspect of reality.

But this means that the concept of different degrees of believing X is the concept of X'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 X's content or it doesn’t.

For example, the concept of different degrees of believing the claim There are six eggs in the fridge is the concept of 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 concept of a claim's content existing in our mind as an aspect of reality to different degrees, the concept of the picture's content existing in our vision as a real outdoor scene to different degrees is nonsensical. 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.

Consider also the fact that thinking claim X is the mental state of believing X. The concept of different degrees of believing X is the concept of different degrees of thinking X, which is nonsensical. That is, logically, we're either thinking X or we're not.

And consider that, as I said in the previous section, a 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 concept of different degrees of understanding in the sense of degrees of belief is nonsensical. That is, 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 are no degrees of belief, whereas there are levels of confidence in claims, is another way in which our level of confidence in a claim is a different kind of mental state from our belief of a claim.

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

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

If our level of confidence in X is sub-certain then we’ll believe that X's content is sub-certain. That is, 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 confidence, which is a mental state, the second use refers to sub-certainty in the sense of below the maximum level of probability, which is a mathematical concept.

And our particular level of sub-certain confidence in X means that we’ll believe that X's content has a corresponding level of sub-certain probability. For example, our high but sub-certain confidence in the claim There are six eggs in the fridge means that we’ll believe the claim It's highly likely that there are six eggs in the fridge.

And our belief of the claim There's sub-certain probability P that X is incompatible with belief of X, however high P is. For example, our belief of the claim It's highly likely that there are six eggs in the fridge is incompatible with belief of the claim 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 high but sub-certain confidence in the claim There are six eggs in the fridge, and our corresponding maximum degree of belief of the claim It's highly likely that 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.

In sum, if our level of confidence in claim X is sub-certain then, however high that confidence, we won’t believe X – we’ll merely believe the claim There's sub-certain probability P that X.

Now consider the maximum level of confidence in X – certainty. 

If our level of confidence in claim X is certainty then we’ll believe that X's content is certain. That is, we’ll believe the claim It’s certain that X. For example, if our level of confidence in the claim There are six eggs in the fridge is certainty then we’ll believe the claim It's certain that 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 just as the claims X and X is true are different claims that merely imply each other, so the claims X and It's certain that X are different claims that merely imply each other.

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.

So these two sentences are making 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 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 our belief of the claim It’s certain that X, when our level of confidence in X is certainty, isn’t in itself belief of X, although it implies our belief of X. For example, our belief of the claim It's certain that there are six eggs in the fridge, when our level of confidence in the claim There are six eggs in the fridge is certainty, isn’t in itself belief of the claim There are six eggs in the fridge, although it implies our belief of this claim.

In sum, if our level of confidence in claim X is certainty then we’ll believe X, as well as believing the claim It's certain that X.

Although our feeling of certainty in claim X is a separate mental state from our belief of X, there's 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 confidence, so, by the same logic, it can't in itself involve 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 our belief of 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 belief involves certainty about X not in the sense of a mental state concerning claim X, but in the sense of the absence of a mental state concerning claim X.

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

Our apparent experience of degrees of belief

As I said in the introduction, it can seem obvious from experience that there are 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.

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

In the example, our apparent maximum degree of belief that we’re eating an apple will actually either be the maximum level of confidence – certainty – that we're eating an apple or the undegreed belief that it's certain that we're eating an apple. And, separately from this mental state, we'll have the undegreed belief that we're eating an apple.

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

Note that although 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 end of our belief of X, we may then at least believe that X's content is highly likely.

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.

An alternative argument for credulism?

The fact that belief involves, in itself, certainty in the sense of an absence of doubt 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 a particular 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 an absence of doubt, an absence of doubt about a claim, X, that exists in our mind isn’t in itself the mental state of believing X. That is, an absence of doubt about X is in itself simply the absence of a particular mental state involving X, whereas the mental state of believing X is the existence of X's content in our mind as an aspect of reality. Indeed, the mental state of feeling certainty in X involves in itself an absence of doubt about X, but without in itself being the mental state of believing X.

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 instead lead us to think of a reason to be uncertain about X, and so we’d then not conclude X.

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 actually has a doubly counterintuitive conclusion. In revealing not only that there aren't degrees of belief, but 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 – nothing about the human brain prevents degrees of belief. 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 using mathematics to merely describe the current understanding of nature, based on observation – was controversial. But today it’s accepted as a legitimate way to further our understanding of nature. Indeed, in physics the use of mathematics to develop and refine theories is a whole branch of this field – theoretical physics.

The logical impossibility of degrees of belief shows that using logic alone is likewise 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 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 that I presented, which I developed, doesn't just support the certainty of belief, it proves it, by showing that degrees of belief are simply logically impossible, and that the certainty of belief is therefore simply logically necessary. Although, in addition to this theoretical proof, this aspect of the nature of belief must of course also be confirmed experimentally. So far, 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:

Also, our experience of belief is very similar to what we think the experience of degrees of belief would be like.

For example, consider our experience of believing that we're eating an apple. This experience of being certain, in the sense of an absence of doubt, that we’re eating an apple is very similar to what we think 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. Also, either way we'll feel certain that X, whether or not this confidence is a separate mental state.

Likewise, consider our experience of believing that it's likely that it’ll rain tomorrow. This experience of being certain, in the sense of an absence of doubt, that it's likely that it’ll rain tomorrow is very similar to what we think 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 equally positive about X's content. Also, either way we'll feel the same high but sub-certain level of confidence in X, whether or not this confidence is a separate mental state.

Why the certainty of belief matters

It might be thought that although the conclusion that there aren’t degrees of belief is obviously significantly different from the current understanding that there are, it doesn't actually have any significant theoretical or practical implications.

As I’ve just said, 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 equally positive about X's content. 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.

But, as I said at the end of the previous article, in upcoming articles in this series 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

Receive email notifications of new articles

Submit your address via Substack – you don't need a Substack account.

Help fund my work – give from $2/£2/€2

Please consider helping to fund my work – including helping me to buy books – by making a one-time or recurring donation of your choice, from $2/£2/€2.

Donate via Donorbox

If you have any questions or problems regarding donating, email me at derrick.farnell@gmail.com.

Feedback

I welcome feedback, however minor, about both my writing and its content – email me at derrick.farnell@gmail.com.

Referencing this article

The content of this article can change, and so referencing of it should include the date of reading. Also, you can save the current version in the Internet Archive and then link to the archived copy.

Article history

This article was first published 10 January 2023. Past versions are available in the Internet Archive here.