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 our understanding of human psychology – and therefore also has profound implications for our lives.

One conception of degrees of belief

There are actually three conceptions of degrees of belief. According to one, our belief about the probability of the content of claim X is, in the case of high apparent probabilities, a degree of belief of X. So our belief that the content of X has the maximum probability, certainty, is a 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-maximum 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 a 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, whereas a sub-maximum degree of belief that there are six eggs in the fridge would be a sub-maximum degree of belief of the claim There are six eggs in the fridge. These are of course different claims, given that the former states that the existence of six eggs in the fridge is uncertain, whereas the latter doesn't. And, whether or not there are degrees of belief, belief of one claim is by definition a different belief from belief of a different claim.

So if a sub-maximum degree of belief that there are six eggs in the fridge were possible then it would merely imply, rather than be, our belief that there's a high but sub-certain probability P 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 be a maximum degree of belief that there are six eggs in the fridge. The former is belief of the claim It's certain that there are six eggs in the fridge, whereas a maximum degree of belief that there are six eggs in the fridge would be a maximum degree of belief of the claim There are six eggs in the fridge. And although the sentences It's certain that there are six eggs in the fridge and There are six eggs in the fridge can seem to be just different wordings of the same claim, they're aren't.

As I explained in How Beliefs Form – Part 1, 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. In the example, 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 that article, if two sentences are just 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.

Hence, if a maximum degree of belief that there are six eggs in the fridge were possible then it would merely imply, rather than be, our belief that it's certain that there are six eggs in the fridge.

In sum, this conception of degrees of belief is fatally logically flawed. Our belief about the probability of the content of claim X can't in itself 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.

A second conception of degrees of belief

It might be thought that our belief about the probability of the content of claim X can be referred to as our level of confidence in this claim. However, these are actually two distinct mental states that merely imply each other.

Our level of confidence in claim X is a feeling about this claim. Indeed, we often refer to how confident we feel about a claim.

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 the claim X is true. So this feeling isn't in itself a level of confidence in claim X. Our level of confidence in claim X is a feeling not about whether X is true, but 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.

And 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. Likewise, our belief that there's probability P that there are six eggs in the fridge is our understanding that there's probability P 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 – Part 2, 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 the content of X existing in our mind not as the content of X – which would involve thinking about X – but as an aspect of reality. And the content of X existing in our mind as an aspect of reality isn't in itself a feeling.

Hence, our belief about the probability of the content of claim X, and our level of confidence in this claim, are distinct mental states that merely imply each other.

This distinction enables a second conception of degrees of belief, although the above analysis implies that it too is fatally logically flawed.

That second conception is that our level of confidence in claim X is, in the case of high levels, a degree of belief of X, with the maximum level of confidence, certainty, being a maximum degree of belief of X. 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 a maximum degree of belief that there are six eggs in the fridge.

Note that whereas I'm using the term certainty here to refer to the maximum level of confidence, which is a mental state, I was using it in the previous section to refer to the maximum level of probability, which is a mathematical concept. Also note that an unawareness of the distinction between the mental states of confidence in a claim and belief about the probability of its content will lead to an unawareness of the distinction between this conception of degrees of belief and the previous one.

Of course, if a level of confidence in a claim is a different kind of mental state from belief, then no level of confidence in a claim, even certainty, can in itself be belief of that claim. Hence, 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.

A third conception of degrees of belief

The remaining conception of degrees of belief simply states that a degree of belief of claim X is a strength of belief of X. That is, unlike the previous two conceptions, it doesn't actually define strength of belief. But even this much vaguer conception is fatally logically flawed.

By definition, any strength of belief of X, however low, would constitute 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 constitute belief. However weakly we believed X, weak belief of X would still constitute belief of X. So any degree of belief of X, however low, would constitute belief of X.

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

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

For example, different degrees of believing the claim There are six eggs in the fridge would necessitate the content of this claim existing in our mind as an aspect of reality to different degrees, which is illogical. 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 content of the picture can exist in our vision are analogous to the two ways that the content of a claim can exist in our mind.

As with the idea of the content of a claim existing in our mind as an aspect of reality to different degrees, the idea of the content of the picture 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 so believing – X or we're not.

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

Also consider that, as I said in the previous section, 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 knowledgeable we are about something.

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 the content of claim X, our level of confidence in X, and our belief of X

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

If we believe that the content of X is sub-certain then our level of confidence in X will be sub-certain. 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 apparent 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 a high but sub-certain level of 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, given that the first claim implies that the content of X is uncertain, whereas the second claim implies that it's certain. For example, our belief It's 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 a maximum degree of belief of the claim It's likely that there are six eggs in the fridge, and a corresponding high but sub-certain level of confidence in the claim There are six eggs in the fridge, would imply, and so be compatible with, a sub-maximum degree of belief of the claim There are six eggs in the fridge.

Now consider believing that the content of X is certain.

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

So if we believe X, doubting X, however slightly, will in itself be the end of our belief of X. However, we may then at least believe that the content of X is almost certain.

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 our experience of either our belief about the probability of the content of X or our level of confidence in X.

In the example, our apparent experience of a 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 apple – or 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 certainty of belief

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, the mental state of believing claim X involves, in itself, an absence of doubt for two reasons. First, a feeling of doubt about a claim is a low level of confidence in that claim, and our accompanying feeling of certainty in X obviously excludes the possibility of having a low level of confidence in X. Second, any level of confidence in a claim is a different kind of mental state from belief.

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.

Therefore 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.

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

The logical necessity of the certainty of belief

As with the argument for credulism I presented in the previous article, the argument I've presented for the certainty of belief has a highly counterintuitive conclusion in two ways. That is, it reveals not only that there aren't degrees of belief, but also that this is due not to the nature of the human brain but simply 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 ultimately 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 the human mind, but in the case of the mind of any intelligent entity.

As I also said in the previous article, at the dawn of modern science in the 16th and 17th centuries using mathematics to develop theories, as opposed to using mathematics simply to describe and predict the world in a way that's compatible with current theories, was controversial. Such reliance on abstract reasoning rather than observation was considered unscientific by most scholars. But in science today it's accepted as a legitimate form of theoretical research.

Similarly, the logical impossibility of degrees of belief shows that using logic alone to develop theories about human psychology is scientifically legitimate, even though it's currently almost universally considered unscientific by scientists. 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.

By showing that degrees of belief are logically impossible, and that the certainty of belief is therefore logically necessary, the argument that I've presented in this article doesn't just support the certainty of belief, it proves it. Although, in addition to this theoretical proof, the certainty of belief must of course also be confirmed by observation – that is, 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 explained by the following factors:

• Belief of the claim It’s likely that X, and a high but sub-certain level of confidence in claim X, and belief of X, are all mental states that are implicitly positive about the truth of X, but the first two are less so than the third. So we can mistake each of the first two for a lower degree of belief of X than the third.

• Both belief that the content of claim X is likely to some extent, and a high level of confidence in X, are mental states that are implicitly positive about the truth of X, and they are so to the same degree, and so we can conflate them. And given that the former mental state is a level of confidence in X, and the latter is a belief regarding the content of X, we can then in turn conclude that this imagined single mental state is a level of belief of X – that is, a degree of belief of X.

• If we believe that the content of claim X is likely to some degree, and we think about the fact that we therefore don’t disbelieve X, we may conclude that we must therefore believe X, albeit to some sub-maximum degree, forgetting that it’s actually possible to neither disbelieve or believe a claim.

• We can refer to the belief that the content of claim X is likely to some extent as the belief that the content of X is likely to some degree. And we can refer to a high level of confidence in X as a high degree of confidence in X. And such uses of the word degree can contribute to us concluding that such mental states are degrees of belief of X.

• We can refer to a high level of confidence in claim X as a high level of confidence that X. For example, we can refer to a high level of confidence in the claim There are six eggs in the fridge as a high level of confidence that there are six eggs in the fridge. And, likewise, we can refer to belief of X as belief that X. For example, we can refer to belief of the claim There are six eggs in the fridge as belief that there are six eggs in the fridge. And this identical grammar can contribute to us concluding that that high level of confidence is a degree of belief.

• Our experience of belief is very similar to what we think the experience of degrees of belief should be like, unaware of the logical impossibility of degrees of belief.

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 we think it should be like to experience having a 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 a maximum degree of belief that X, we believe X with an absence of doubt, and are therefore implicitly positive about the truth of X to the maximum degree.

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 we think it should 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 about the truth of X to the same sub-maximum degree.

An alternative argument for credulism?

The fact that belief involves 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 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 so believe X upon that entrance. But this much simpler argument unfortunately doesn't work. 

Although belief involves certainty in the sense of an absence of doubt, it doesn't merely involve an absence of doubt. 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 the content of X 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 explain our belief of X upon that entrance.

Of course, if immediately after X entering our mind we consider our immediate 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

Knowing that there aren’t degrees of belief, contrary to the prevailing understanding, obviously matters in the sense that knowing the truth about the world, including our psychology, matters intellectually. But it might be thought that this truth doesn’t itself actually have significant implications for human psychology – and therefore also doesn't have significant implications for our lives

As I 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 implicitly positive about the truth of X to the same sub-maximum degree. And whether we have a 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 about the truth of X to the maximum degree.

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 ultimately due to the nature of the human brain.

But cognitive biases are actually 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 so for most of human irrationality.

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

How Belief Works