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I found this Euler diagram representing a definition of knowledge problematic.

First, propositions should only express beliefs. When there is no belief, there should be no proposition. But the diagram shows that some propositions are not beliefs.

Secondly, there should be pre-verbal beliefs that have not been expressed in propositions, but the diagram shows that all beliefs are propositions.

Finally, truth should be a property of beliefs. There should be no truth independent of beliefs, but the diagram shows that some truths are not beliefs.

The source of this diagram is wikipedia. I wonder how widely accepted this definition is.

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    What this presents is a classical JTB ("justified true belief") account of knowledge. Contemporary accounts have to deal with Gettier-style edge cases too... – Joseph Weissman Sep 4 '14 at 18:33
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    I would say that all Truths must be beliefs, and that the area labelled Propositions should simply be unlabeled, as "unknown", not considered yet, as if outside the light-cone of the flow of awareness. I would also divide belief into conscious (thought of, reasoned about), and unconscious (not consciously examined), like when I push a door but should have pulled it. – user16869 Feb 9 '16 at 0:23
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First of all, as Joseph Weissman pointed out, since Gettier pointed out a significant problem with this definition several new definitions were proposed. The case is not closed yet as it seems that every new definition has some quirky consequences or can still be gettierized.

Addressing you questions:

  1. There are some propositions no-one believes. Some of them because they are plainly wrong, some of them because no-one ever came up with them like 16783-465²*12.85/log(2)=100.12345. It's not only wrong, I'd think it's something no-one ever thought to be true - it was never a belief.

  2. The word 'belief' here is somewhat of a technical term: Only propositional structured things can be beliefs, because the entity needed needs the ability to have a truth value assigned to it. If you are very convinced of something, but fail to express it this 'it' is the thing we need: The proposition, which can be true or false.

  3. Truth is a property of beliefs. But also of propositions. And some of them happen to be true without any-one believing it. For example: Do you think the number of hairs on your head: are the odd or even? One of the both is true, and has been true even before you thought about it.

  • Thanks, @Einer. I read in some other other context that propositions are incomplete symbols. Until they are asserted or appear in other asserted propositions, they have no meanings by themselves. I confess I don't quite understand it. I only remembered it as rule. – George Chen Sep 5 '14 at 3:07
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    @GeorgeChen It kinda depends on who you are asking, but beginning with Frege it's somewhat common to assume, that proposition indeed don't have a meaning; they are the meaning (or way more precisely: the sense ;-)) – Einer Sep 5 '14 at 8:47
  • According to this theory(1st ED of W&R's Principia Mathematica), judgement is a part of proposition, and the phrase expressing a proposition is an incomplete symbol. The explanation smacks of sense-data which Russell later abandoned. – George Chen Sep 5 '14 at 17:05
  • Re: your point 3: There is no truth until someone frames a question (proposition). Truth is something that humans consider, it is in their minds. Outside the mind, things simply ARE, there is no "thing that is not", so there is no "untruth" outside of our thoughts. Perhaps this is the OP's line of inquiry? – user16869 Feb 9 '16 at 0:13
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Your question includes some mistakes.

You write:

First, propositions should only express beliefs. When there is no belief, there should be no proposition. But the diagram shows that some propositions are not beliefs.

This is wrong. It is common to put a whole load of facts into a computer program and to let it work out many complicated implications of those facts that people want to use but not to work out. Indeed, the person might not even want to know those implications. He might design another program to take the output of the first program and follow a routine for fixing the problem without him ever knowing the specifics of the problem. Also a lot of stuff in books isn't believed by anyone because the point of writing it down in a book is so that people don't have to remember it.

Finally, truth should be a property of beliefs. There should be no truth independent of beliefs, but the diagram shows that some truths are not beliefs.

So the world didn't exist before there were people around to know about it. Dinosaurs didn't exist.

The wikipedia diagram is wrong, but so are you. As others have commented it seems to represent the idea that knowledge is justified true belief. I have already explained that knowledge need not be belief.

As Popper pointed out in Chapter I of "Realism and the Aim of Science" justification is impossible, unnecessary and undesirable. If you assess ideas using argument then the arguments have premises and rules of inference and the result of the argument may not be true (or probably true) if the premises and rules of inference are false. You might try to solve this by coming up with a new argument that proves the premises and rules of inference but then you have the same problem with those premises and rules of inference.

It is perfectly possible for an idea to solve some problem and so to constitute knowledge without it being true, so knowledge need not be true either. Knowledge is not justified true belief.

  • Regarding the second mistake, there are facts before there were people. Truths are not the same as facts. – George Chen Sep 5 '14 at 9:14
  • I need some time to think what a proposition is. Thanks, anyway. – George Chen Sep 5 '14 at 9:14
  • Originally, belief is defined as a state of organism. Whether rocks and chairs have beliefs is unknown. Now we can say computers definitely have beliefs, because its internal state changes according to some external facts. – George Chen Sep 5 '14 at 9:40
  • A book doesn't change its state. – alanf Sep 5 '14 at 9:43
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    Shakespeare's propositions are never asserted; they are not nonsense either. I need some more time to think about this. – George Chen Sep 5 '14 at 15:35

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