My take would be if there is no proof of p, there's no reason to think p exists. Depending on what p is imagined to be, however, we may make inferences regarding the likelihood of p's existence. That we have no proof there is a planet-sized turtle orbiting the sun doesn't mean there could be one. — Ciceronianus

Yes but that there is no reason to think p exists isn't a reason to think p does not exist. That's how things stand as of now I'm told. As one poster remarked, absence of proof is not proof of absence. To say p does not exist, we need proof that p does not exist.

Russell's celestial teapot is a response to some theists who take the stance that because science can't disprove god, god exists. It's basically an argumentum ad ignorantiam (argument from ignorance) fallacy to think this way. Note Russell stops short of claiming there is no celestial teapot in orbit around the sun. The reason: absence of proof is not proof of absence.

To make the long story short,

1. Despite claims that you can't prove a negative, you have to. You know what, this idea is probably a derivative of the legal principles innocent until proven guilty and onus probandi (burden of proof). I'm not sure about this though. However, sometimes it makes more sense to assume truth rather than falsity (Pascal's Wager). Like one poster remarked, it depends on the risk involved.

2. As for the main thrust of my OP (p can't be proven implies impossible that p is true i.e. in bivalent logic, p has to be false), it appears that, here too, the legal principle innocent until proven guilty applies but the argument for falsity/innocence is, to my reckoning, much, much stronger; after all, guilt/truth can't be proved (no proof exists).

What are you trying to get to in all this? I don't do math or logic, sorry. But I am curious what you have in mind.

However, sometimes it makes more sense to assume truth rather than falsity (Pascal's Wager). Like one poster remarked, it depends on the risk involved. — Agent Smith

Pascal's Wager is nuts. What if you 'decide to believe' in the wrong God? It leads you no where. Also there's the problem, as I see it, that beliefs can't just be faked like this. You either believe something or you don't.

Main quest: If it's impossible to prove the proposition p is true , what truth value do we assign that proposition, p?

Side quest: Why can't, rather shouldn't, we prove a negative?

Ok. Unknown?

Ok. Unknown? — Tom Storm

But why?

Given proposition p, p is either

1. True

or

2. False

For me to know about p, I need

3. Proof/justification for p

Suppose I looked thoroughly in The Book (of all proofs) and found p has no proof. Is this equivalent to it is impossible to prove p? So what if it is? More importantly, from this alone, what truth value should I assign p?

Note that there's a difference between I have no proof and there is no proof. If from the former, I say p is false (~p), I commit the argument from ignorance fallacy. From the latter, since it's not the same as the former, I reckon the aforementioned fallacy is not committed. Why? How? My questions.

I have a feeling that I'm getting mixed up between knowledge (JTB) and logic/truth. I'm still in a fog.

I have a feeling that I'm getting mixed up between knowledge (JTB) and logic/truth. I'm still in a fog. — Agent Smith

I'm not the best person to talk to as logic doesn't engage me. The logical absolutes are tautologies.

No one denies that we can say that God either exists or does not exist. There's your Aristotle. But in the real world it is not always possible to know what is the case.

Justified true belief is not logic. It simply means that a belief held is resting on good evidence. And of course justified true belief is not always correct.

My own view is that is no such thing as absolute truth and that truth is not found outside us but made by humans. No doubt many will disagree.

:ok:

Update

Tom Storm has stated something very important by proferring a "truth value" over & above the classical Aristotelian true and false viz. unknown.

This is interesting, very interesting in fact, because unknown pertains to, in my universe, knowledge and it's about us, our ignorance to be precise and not, presumably, about a/the propsoition and the state of affairs it talks about. In short, unknown isn't/can't be (?) be a truth value.

As to how all this relates to my OP, all I can say is unknown, if posited as a truth value. is N/A.

I have a feeling that I'm getting mixed up between knowledge (JTB) and logic/truth. — Agent Smith

Let p = Shakespeare drank a cup of coffee on the morning of 15 July 1584. P is either true or false, as a proposition is either true or false by definition.

unknown isn't/can't be (?) be a truth value. — Agent Smith

P has a truth value, even though we may never know what it is.

To say p does not exist, we need proof that p does not exist. — Agent Smith

No statement can ever be proved

The phrase "to say p does not exist" is incorrect. The question is whether the proposition is true or false, not whether the proposition exists or doesn't exist.

If I say that p is false, then I am saying that Shakespeare did not drink a cup of coffee on the morning of 15 July 1584. If I make such a statement, then it is only reasonable that I justify myself, giving sensible reasons why I believe that my statement is true. But I can never prove my statement. Even science only deals in probabilities, not proofs. No statement can ever be proved with 100% certainty.

Consider an example of deductive reasoning: “All men are mortal. Harold is a man. Therefore, Harold is mortal". It is true that there is a sequence of logical statements, one implying another, and giving an explanation of why a given statement is true. But the starting point is always axioms, accepted "rules", statements or propositions which are regarded as being established, accepted, or self-evidently true, ie, unproved.

IE, for me to say that Shakespeare did not drink a cup of coffee on the morning of 15 July 1584, it is reasonable that I am required to justify my belief giving sensible reasons, but it would be impossible for me to prove my statement, in that it is impossible to prove any statement.

No statement can ever be proved with 100% certainty. — RussellA

As a wannabe skeptic, that's music to my ears but then deduction?

it is impossible to prove any statement. — RussellA

Nice!

Is there is no proof equivalent to impossible to prove p?

In a legal context, the concept of burden of proof would be more applicable than innocent until proven guilty, I think, as the latter may be said to represent a high burden of proof. Generally, those making a claim have the burden to establish it, or at least make what we love to call a prima facie case, in legal Latin. Then, the burden may shift to the opponent of the claim, rather than the proponent. In some instances, the significance of the public policy behind a law is said to place the burden on the opponent rather than the proponent of a claim.

If the analogy holds, those making the claim that there is a God would have the burden to establish that's the case.

Is there is no proof equivalent to impossible to prove p? — Agent Smith

How to prove that the statement "no true statement can ever be proved true" is true.

Analytic statements
Quine in The Two Dogmas of Empiricism distinguished between two kinds of analytic claims - i) logical truths, true no matter how we interpret the non-logical parts in the statement, such as "no not-X is X" - and ii) synonymous truths, such as comparing bachelor with unmarried man.

As regards logical truths, logical truths are necessary truths, and necessary truths are beyond proof, in that no argument can be found to establish the truth of a necessary truth, as the nature of a necessary truth is to be true.

As regards synonymous truths, consider the statement "the sun rises in the east". By definition, the "sun" is something that rises in the east. So, the statement "the sun rises in the east" is analytically true. IE, if one morning something rose in the west, rather than the east, then that something wouldn't be the "sun", it would be something else.

IE, logical truths are beyond proof, and synonymous truths don't need to be proved as they are true by definition.

Synthetic statements
As regards whether the statement "the sun rises in the east" is synthetically true, one needs to prove that the sun rises in the east rather than the west.

If Idealism is true, and there is no external reality and the world only consists of ideas, then there is no sun, and the question as to where it rises becomes irrelevant.
If Indirect Realism is true, and our conscious experience is not of the real world itself but of an internal representation, then impossible to prove something outside that which we have direct knowledge of.
If Direct Realism is true, and the senses provide us with direct awareness of the external world, then it would be possible to prove that the sun rises in the east.

But how do we prove which of Idealism, Indirect Realism or Direct Realism is true, something philosophers have debated for thousands of years. As my perception of the sun would be identical, regardless of whether Idealism, Indirect Realism or Realism was true, it would therefore be impossible to prove which of these is in fact the case.

IE, synthetic statements cannot be proved.

Summary
Given the statement S "no true statement can ever be proved true", if S is true, proving it true would result in a contradiction.

IE, some true statements cannot be proved true. Though, in practice, proof is secondary to what pragmatically works.

Positive and Negative statements:

1. Positive statement: G = God exists. To assert that something is a fact (a part of reality). It is possible that one believes something to be a fact when it isn't.

2. Negative statement: ~G = God doesn't exist. To aver that something is not a fact (not a part of reality). It is possible that one believes something to be not a fact when it is

If one believes something to be a fact, all one needs to do is show it is so in a particular corner of reality while to prove that something is not a fact, one has to exhaust all reality itself (a corner just won't do). It's quite obvious that the former is orders of magnitude easier than the latter. Ergo, one can't prove a negative.