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So I understand the main differences between facts and beliefs, but is there a part where they overlap? Is it possible that there is a point where one can find similarities between both concepts?

How exactly do fact and belief relate to each other?

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    A Fact is a "piece" of reality. A Belief involves a human believing. Commented Mar 2, 2022 at 9:03
  • They are not different, they refer to unrelated objects. A fact is an instance, an existence. A belief is some knowledge. The opposite of fact would be a lack, a void. The opposite of a belief would be an uncertainty, an skeptic attitude, ignorance. You might experience a fact of belief (I have felt faith) , a fact of disbelief (I have felt a lack of faith), a lack of belief (I haven't felt faith) or a lack of disbelief (I haven't felt skeptical). Or an unreal fact (a fact is not "a piece of reality": I've felt my missing arm; in this case, the belief is the same as the fact).
    – RodolfoAP
    Commented Mar 3, 2022 at 17:52
  • Added 'epistemology', 'subjectivity', 'objectivity', and 'social-epistemology'.
    – J D
    Commented Mar 4, 2022 at 11:44

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Due to the spinning of the earth we experience sunrise and sunset. This is a fact. Standing on one side of the earth we believe the sun is on the other side after sunset and before sunrise. You can find so many instances like this where fact and belief overlaps.

You believe your internal organs like stomach, liver, kidneys etc are functioning. (If you have no equipment) In your case that also is a fact as well as a belief.

A thing that is not perceivable while believing (as mentioned above) are often like this. While one may believe, another may experience the fact. So it may not be the same from someone else's point of view. (I mean this is not universal.)

How exactly do fact and belief relate to each other?

All beliefs are not facts. One can believe something even if there is no evidence of facts. One's character, likes, dislikes, and feelings only affect beliefs, never facts. Subjectivity has a role on beliefs; but not on facts. So they are related to each other only when there is a possibility of objectivity.

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A belief is an view held by an individual in the absence of overwhelming evidence to support it. Where overwhelming evidence does exist we would say that we ‘know’ something rather than believe it, although the threshold for this is a worthy subject for philosophical debate. A belief may correlate with facts although it doesn’t have to.

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Adding on to what @MauroALLEGRANZA said in the comments, there is reality as it is, then there is our mental model of said reality based on experience and inference. If fact is a piece of reality, then belief is a piece of our mental model.

[...] but is there a part where they overlap?

Imagine a computer with a wired webcam. When the camera is running, a piece of the world -- a fact -- is translated into a piece of data in the computer's memory -- a belief.

Now imagine the camera is pointed back at the screen while the screen is showing the image in real-time, giving a video feedback loop. In this scenario, the computer holds a belief about what is out there in the real world, yet this belief contains translated copies of its own prior beliefs, in continuous recursion. In a way, the computer becomes aware of its own vision through time. Moreover, the computer now holds belief of its own prior belief.

As the rate and accuracy of video feedback increase, the closer becomes belief to its corresponding fact. That is, the computer's belief about its own belief approaches current and true. Belief approaches fact.

If awareness is belief on the current state of reality, and if consciousness is real-time awareness of one's awareness, then perhaps consciousness is an approximation of convergence between fact and belief.

In conclusion, consciousness is an overlap between fact and belief, and higher consciousness is greater overlap.

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A fact is a solidified belief: a belief that we have (for one reason or another) become convinced is true and inviolate, and projected onto the world. Sometimes we solidify that belief for good reasons, sometimes for bad reasons, sometimes out of mere custom and habit (which don't really count as 'reasons' in the proper sense).

It makes some sense to talk about 'sensory' facts: observations and perceptions that we can collectively experience. This is the lynchpin that holds scientific investigation together. But even this is better conceived of as intersubjective belief, where we have convinced ourselves of the belief that others actually experience the world exactly as we do.

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Short Answer

How do fact and belief relate to each other?

A belief is one type of mental state that collectively is recognized by philosophers as a type of intentionality, and is expressed as a proposition. In contrast, a fact is a specific type of belief. A fact is a specific type of proposition that is generally understood as a collective, reasonable, and empirical belief, a type of objective knowledge. Both are studied by epistemologists, and the question of objectivity, subjectivity, and intersubjectivity as well as belief, justification, truth, and knowledge are central to metaphysics.

Long Answer

Intentionality, Belief, and Language

Human agents are posited to have a property called intentionality. Intentionality is often characterized as "aboutness", as in people can have thoughts, feelings, and so on about objects such as trees, ideas, or other people. From WP:

Intentionality is the power of minds to be about something: to represent or to stand for things, properties and states of affairs.1 Intentionality is primarily ascribed to mental states, like perceptions, beliefs or desires, which is why it has been regarded as the characteristic mark of the mental by many philosophers. A central issue for theories of intentionality has been the problem of intentional inexistence: to determine the ontological status of the entities which are the objects of intentional states.

Beliefs are perhaps the most important example of mental states because they are an intersection between experience and language. Robert Audi in his Epistemology: A Contemporary Introduction to the Theory of Knowledge offers a number of sources for belief: perception, memory, consciousness, reason, and testimony. For instance, you may believe that you are hungry because your stomach growls. You may believe that a car is moving down the street because you are aware of the change of position. You may believe your wife when she says she's angry you didn't buy a gallon of milk at the store. When your beliefs conflict, you may experience cognitive dissonance. What is belief more specifically? Here philosophers split hairs. From WP:

There are various different ways that contemporary philosophers have tried to describe beliefs, including as representations of ways that the world could be (Jerry Fodor), as dispositions to act as if certain things are true (Roderick Chisholm), as interpretive schemes for making sense of someone's actions (Daniel Dennett and Donald Davidson), or as mental states that fill a particular function (Hilary Putnam).2 Some have also attempted to offer significant revisions to our notion of belief, including eliminativists about belief who argue that there is no phenomenon in the natural world which corresponds to our folk psychological concept of belief (Paul Churchland) and formal epistemologists who aim to replace our bivalent notion of belief ("either we have a belief or we don't have a belief") with the more permissive, probabilistic notion of credence ("there is an entire spectrum of degrees of belief, not a simple dichotomy between belief and non-belief").2

Beliefs are generally expressed as propositions, and our attitudes towards propositions are called propositional attitudes. A proposition can be thought of as a complete sentence or a statement. The fancy philosophical terminology to describe the act of conveying knowledge with a statement is called predication. A simple prediction follows:

The Willis Tower (formerly the Sears Tower) is a tall building.

When an honest person utters this, at a minimum they are expressing a belief and demonstrating intentionality. That is to say, they are thinking about the Willis Tower as they experience them. Now, is it a fact that Willis Tower is tall? Things become a little more complicated.

Facts and Opinions; Objectivity and Subjectivity

As most school children are told (and some struggle to retain), generally, propositions can be divided into subjective statements, called opinions, and objective statements, called facts. Opinions are difficult to "prove" because they often reflect internal, mental states of agents. 'The Mona Lisa is creepy' is subjective or opinionated because an agency's intentionality, their feelings are not shared experience. Opinions can be popular ('Ice cream is delicious'), and they can be unpopular ('Goat poop is delicious'). But they are not public, and therefore, it's hard to justify. Remember, knowledge is justified, true belief (at least on a simple reading). Facts are different. They are easy to justify and show are true. 'Snow is white' is easy to show because if you have a group of people, few people will dispute the proposition, explanations can usually accommodate dissenters, and modern scientific theory and equipment can measure wavelengths of light. Hence, 'pure snow is white' is generally recognized as a fact. So far, it seems simple.

But take our example about the Willis Tower. Is it true that the Willis Tower is tall? This gets difficult, because it's not clear what 'tall' means. I may think it is tall, but a geologist who studies the Alps may disagree and claim the opposite. How is this dispute settled, and why does it happen in the first place? Well, philosophers have distinguished that there are some fundamental divisions among propositions, a fact-value distinction and an is-ought distinction reflect the dichotomy between the objective and subjective. The former:

The fact–value distinction is a fundamental epistemological distinction described between:1

'Statements of fact' ('positive' or 'descriptive statements'), based upon reason and physical observation, and which are examined via the empirical method.
'Statements of value' ('normative' or 'prescriptive statements'), which encompass ethics and aesthetics, and are studied via axiology.

This barrier between 'fact' and 'value' implies it is impossible to derive ethical claims from factual arguments, or to defend the former using the latter.2

and the latter:

The is–ought problem, as articulated by the Scottish philosopher and historian David Hume, arises when one makes claims about what ought to be that are based solely on statements about what is. Hume found that there seems to be a significant difference between positive statements (about what is) and prescriptive or normative statements (about what ought to be), and that it is not obvious how one can coherently move from descriptive statements to prescriptive ones. Hume's law or Hume's guillotine1 is the thesis that, if a reasoner only has access to non-moral and non-evaluative factual premises, the reasoner cannot logically infer the truth of moral statements.2

So, if an English teacher assigns an essay with the prompt "Is the Willis Tower tall, and should it be predicated as tall?" you are firmly in philosophical territory.

Intersubjectivity, Convention, and Dispute

How do philosphers and practical people of all stripes resolves these sorts of dilemma and dichotomies? Well, sophisticated thinkers invoke the explanations of intersubjectivity and convention. Simply put, intersubjectivity is a the idea that what constitutes 'objectivity' is actually just (near-)universal consent. Is this Willis Tower tall? Yes. Why? Because most people judge size relative to their own height. In linguistics, certain expressions are understood in terms of an implicit context, and this is known as deixis. Elephants are big, and mice are small because of their size relative to average people. In fact, NBA players are tall, and leprechauns are small because of their (purported) size relative to average people. Thus, it becomes a convention in language to use tall in a certain way. You shouldn't use 'short' to describe the Willis Tower because it violates convention, and in doing so violates one of Grice's maxims, probably the Maxim of Clarity. When a person uses idiosyncratic meanings instead of conventional one's, it leads to confusion.

So, is the Willis Tower tall? Yes, and it is a fact. But, anyone can challenge the facticity of the statment. Just like the claim 'The Willis Tower is 10 meters tall' is a demonstrably false fact, so too can someone claim that 'The Willis Tower is tall' is a false fact, or even an opinion. In fact, a clever person can use philosophy and language to assert something like 'There is no Willis Tower at all, so it can't be tall'! Once the conversation is in that territory, you're clearly in what philosophy calls metaphysical speculation. Deeper discussions, of course, can be had about subjectivity, beliefs, facts, and truth such as in this post on post-modernism (PhilSE).

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