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By proposition, I mean something like a statement or an argument. Such as "Sun always rises from the east." This, however, does not mean that I am moving away from the domain of mathematics, on this forum. If you can bring in arguments from epistemology, I will welcome you to do so, whole heartedly.

Vibhu
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    Please make the body of the post self-contained. Also, what sort of "something" is this that can never be negated? – Tobias Kildetoft Aug 01 '16 at 12:18
  • Yes, please wait for a moment. – Vibhu Aug 01 '16 at 12:19
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    way way way way to broad a question. – 5xum Aug 01 '16 at 12:22
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    The negation of your statement is "there is a time the sun rises from somewhere other than the east". What do you mean by "can never be negated"? In the context of your example, are you asking: "if we cannot find a time when the sun did not rise from the east, then can we conclude that the sun always rises from the east?"? – smcc Aug 01 '16 at 12:23
  • Pardon me for bringing in ambiguity. What are the materials available to understand the topic more? Thanks. @5xum – Vibhu Aug 01 '16 at 12:23
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    No, just because you can't prove something false, it doesn't make it true. That's the short answer. – kamoroso94 Aug 01 '16 at 12:23
  • @kamoroso94 yes, that had come to my mind., which is why I've posted it here: to know more. – Vibhu Aug 01 '16 at 12:25
  • Thank you for pointing it out @smcc. I've made changes to the title. – Vibhu Aug 01 '16 at 12:25
  • @smcc yes. That's what I was trying to say. – Vibhu Aug 01 '16 at 12:26
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    Several claims in number theory could not be disproved by finding a counterexample, and were believed to be true, but eventually it turned out that they were false. The most famous example : The claim $\pi(x)<Li(x)$, where $\pi(x)$ denotes the number of primes not exceeding $x$ and $Li(x)=\int_{t=2}^x \frac{1}{ln(t)}$ , turned out to be false. The smallest counterexample is probably so large that a systematic calculation would have never disproven it. – Peter Aug 01 '16 at 12:38
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    The smallest counterexample is larger than $10^{19}$ and is believed to be roughly $10^{316}$. This is a very good example of the "law of small numbers". Even if we do not find a counterexample upto a very high level, the claim can be false. – Peter Aug 01 '16 at 12:40
  • An even better example : The Goodstein-sequences are growing rapidly and seem to increase forever. But it is known that every Goodstein-sequence terminates with $0$ after a finite number of iterations. – Peter Aug 01 '16 at 12:44
  • A remark to my example above : The claim is actually $\pi(x)<li(x)$. This should not make much difference because the difference between $Li(x)$ and $li(x)$ is $1,045$ for ever $x\ge 2$. – Peter Aug 01 '16 at 12:51
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    Your question is equivalent to "Are there true statements which can't be proven true?". @user3154270 has answered this exactly. – MPW Aug 01 '16 at 13:31
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    Lots of people have been sucked in, but the OP's example about sunrise makes it quite clear that this question is not in scope for MSE. – Rob Arthan Aug 01 '16 at 21:35
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    Cosing that thread is a little bit pettifogging, don't you think? – user3154270 Aug 02 '16 at 09:05
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    The answer you've gotten is misleading, because it mentions "truth", when in actuality Godel's theorem only applies to arithmetical truth as defined in a formal system, specifically in the meta-system. See http://math.stackexchange.com/a/1873544/21820 for the full definition of arithmetical truth in mathematical logic, and note for your question that PA cannot disprove $\neg Con(PA)$ but it is in fact arithmetically false, namely that $\mathbb{N}$ does not satisfy it. – user21820 Aug 03 '16 at 05:56

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From the Gödel incompleteness theorem, we know that there is a sentence which is true but there exists no deduction for it, so there is no prove for this theorem.

So in your case, if there exists no prove that you proposition is wrong, it could still be wrong. Even if you prove that there is no deduction to make you proposition wrong, it could still be wrong.

user3154270
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    Gödel's theorems give no information whatsoever about statements about where the Sun rises. The OP needs some help with the philosophy of natural science and not with mathematical logic. – Rob Arthan Aug 01 '16 at 21:39
  • The question is, whether a statement is always true if there is no deduction (no prove) which makes it false. And the answer is no, because there exists at least one structure, the natural numbers, in which the statement might be wrong even there is no prove which makes it wrong. If you speak about that a sentence (or proposition whatsoever) is true/false not naming an explicit structure in which the sentence is true/false, you mean that the statement is true/false in all structures. So I give you one structure in which it is not true so the sentence might not be true. – user3154270 Aug 01 '16 at 22:02
  • The question makes it quite clear that the OP is concerned with statements that are outside the scope of mathematics. You can't offer $(\Bbb{N}, 0, 1, +, \times)$ as a witness to a statement about the natural world. – Rob Arthan Aug 01 '16 at 22:07
  • But if you speak about prove and proposition, you are inside the mathematical logic – user3154270 Aug 01 '16 at 22:14
  • The terms "prove" and "proposition" are widely used in non-mathematical discourse. – Rob Arthan Aug 01 '16 at 22:31
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    Please, give me a moment. I took the example of the sun because I thought it would pave the easiest way to understand my query. My question, as I'd like it to be, in context of mathematics. @RobArthan – Vibhu Aug 02 '16 at 20:59