Define "omniscience" as the knowledge of the truth value of every well-formed proposition (i.e. every proposition that has a truth value). Now, consider an allegedly omniscient being. This being is supposed to know the truth value of every well-formed proposition. This surely includes the proposition P: "I am omniscient". It of course believes that P is true. If you assume P is true, then it believes correctly, and we may say that it is actually omniscient. But what if you assume P isn't true? Then the being believes falsely - and this false belief is sufficient to show that P isn't true. The falsehood is self-demonstrating. And if the being knows the truth value of every other proposition, there is no external way to adjudicate whether P is true or false: it is true if you accept that it is true, and it is false if you accept that it is false. Omniscience is only possible to the optimist.
But wait, there's more. Knowledge can be roughly defined as justified true belief. To say that this being knows the truth values of all well-formed propositions is to say that it believes truly and with justification. So what's its justification for all its knowledge? Well, if it knows that it's omniscient, then that would be credible justification: it knows that anything it believes must be true. But as we've just seen, whether or not it knows that it's omniscient is indeterminate. It may be believing that it is omniscient falsely, in which case all its justifications are based on a falsehood, and dissolve. Then, since knowledge requires sound justification, we might say that the being doesn't know anything, even if all its other beliefs happen to be true. It's like the man who looks at a watch stopped at noon when it is noon: he forms the belief that it is noon, and that belief is correct, but his justification is faulty, so we can't say he knows it's noon.
So we cannot tell whether the being is omniscient or not. And the being must know all this as well (or else it is out of the running for omniscience automatically). So it can't tell either; it has no way of knowing whether its belief that it is omniscient is true or false. Which leads us to the following exchange...
"Are you omniscient?"
"I don't know."
"Doesn't that mean you're not omniscient?"
"I don't know."
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Vive, vale. Siquid novisti rectius istis,
candidus inperti; si nil, his utere mecum.
The only problem I have with this is that although knowledge must be justified, we generally don't require that justification to rise to the level of proof. Good evidence is enough.
So here's an interview with an omniscient being:
Q: Are you omniscient?
A: Yes.
Q: Is that just a guess, or do you know that you're omniscient?
A: I know, obviously. If I didn't know, then that would be something that I didn't know, so I wouldn't be omniscient in that case.
Q: But knowledge requires justification. What is your justification for believing you are omniscient?
A: I believe I'm omniscient because I am not aware of anything I don't know.
Q: That's hardly a proof of your omniscience.
A: I didn't claim it was a proof. I just claimed it was a justification.
Q: But an man who has lived his whole life in a dark cave with no outside contacts could also be unaware of anything he doesn't know. In that case, doesn't he also have a justification for believing he is omniscient? And mightn't he also know that he is omniscient?
A: Sure, he would have a justification. But he wouldn't know, because knowledge is justified true belief. His belief is false.
Q: But that means you only know you're omniscient if it is in fact true that you are omniscient.
A: Good thing it's true, then.
Q: But what if you're wrong? How could I tell the difference?
A: I didn't claim to be able to prove to you that I was omniscient.
Q: Okay, then, how could you tell the difference?
A: I didn't claim to be able to prove to myself that I was omniscient either. I just claimed to know it. I believe it, it's true, and I have justified my belief. Therefore, I know I am omniscient.
Q: Sigh... But if you imagine a version of you that isn't omniscient, I could be having the exact same conversation with that version of you as I am with the real you.
A: Yes, but that version, when he said he knew he was omniscient, would be wrong.
So omniscience is circular? And in consequence a being who knows the truth-value of every well-formed proposition besides "I am omniscient" is omniscient if they believe they are and is not omniscient if they don't...?
I'm going to go ahead and assume this is wrong in some way. Will you tell me how?
Even if they believe you know something, the possibility exists that they are mistaken. Their belief that they are omniscient cannot make that proposition true. What would make the proposition true is... the fact that it is true (plus that everything else they believe is true, and nothing is true that they do not believe, but it's the recursive element that's the trouble).
I know this might be a stupid question, but why is this just true for the being's belief in his omniscience?
If he can't "just know" that he is omniscient, then why is he allowed to "just know" the rest of what he knows is true?
If a being knows that he is omniscient, this constitutes justification for anything else he "just knows". But if he tries to justify his knowledge that he is omniscient with his knowledge that he is omniscient, he runs into a slight circularity problem.
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Vive, vale. Siquid novisti rectius istis,
candidus inperti; si nil, his utere mecum.
If a being knows that he is omniscient, this constitutes justification for anything else he "just knows". But if he tries to justify his knowledge that he is omniscient with his knowledge that he is omniscient, he runs into a slight circularity problem.
So, everything would be resting on his omniscience, which--itself--isn't resting on anything?
Yeah, the problem here comes from the assumption. If
1. we define omniscience as having justified, true belief embedded within it, and
2. we have no pragmatic explanation for the being's omniscience besides our assumption, and corresponding suspension of belief to the contrary, that the being is omniscient,
THEN the omniscience evaporates, because we cannot justify it. The suspension of belief cannot hold. In this example, no matter how much we try to escape this by making it hypothetical, the being's omniscience comes from our assumption that it is omniscient. Real-life omniscience, if it existed, would have justified true belief embedded within it, but this justified true belief of each well-formed proposition would be the set of characteristics that lead us to call the being omniscient, not the other way around. Logic simply does not have the tools to let us work backwards like this.
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2UURRniv mizzet, the firemind endless fiery mind wheels of death 3WRjor kadeen, the prevailer weenies and extra combat forever 3RRzirilan of the claw dragons and damage doublers 4:SymRG::SymRG:wort, the raidmother burn is now EDH viable 2WUkangee, aerie keeper birds 1UBRjeleva, nephalia's scourge spellslinger/storm
here are the mana costs of generals i no longer play: 2BR3BB3UBG4UB:SymUB::SymUB:2URRG3WWU2UU2GGUUBGG
A bit off topic but hearing the word paradox reminded me of this one where (forget the specifics) Racer A starts ahead of racer B, and racer B is traveling at a faster pace than racer A. But racer B cannot pass racer A, because, in the time it took to catch up, Racer A will have covered x amount of distance. I guess a picture might help demonstrate this, but I'm sure someone here knows what I'm talking about.
Obviously this is wrong, but where exactly does the logic become faulty?
A bit off topic but hearing the word paradox reminded me of this one where (forget the specifics) Racer A starts ahead of racer B, and racer B is traveling at a faster pace than racer A. But racer B cannot pass racer A, because, in the time it took to catch up, Racer A will have covered x amount of distance. I guess a picture might help demonstrate this, but I'm sure someone here knows what I'm talking about.
Obviously this is wrong, but where exactly does the logic become faulty?
It's some variant of Xeno's Paradox (I'm not sure if it's formalized as a different paradox or not), but it's faulty because in the time it took B to get to A's initial position, A moved a smaller distance... and the smaller time it takes B to reach A's next position, A moves a smaller distance (if the speed discrepancy is very large, a very small distance indeed!). As the times go to the limit of 0 seconds, practically no movement is occurring... but time doesn't actually stop. It's just that the time it takes B to pass A is not time with the characteristic "time it takes B to get from A's previous position to A's next position", so it falls outside the scope of the example.
Essentially the 'paradox' is saying nothing more interesting than that B can't pass A in less than the time it takes B to pass A, it just does so in an intuition-warping fashion for certain types of people.
Define "omniscience" as the knowledge of the truth value of every well-formed proposition (i.e. every proposition that has a truth value). Now, consider an allegedly omniscient being. This being is supposed to know the truth value of every well-formed proposition. This surely includes the proposition P: "I am omniscient". It of course believes that P is true. If you assume P is true, then it believes correctly, and we may say that it is actually omniscient. But what if you assume P isn't true? Then the being believes falsely - and this false belief is sufficient to show that P isn't true. The falsehood is self-demonstrating. And if the being knows the truth value of every other proposition, there is no external way to adjudicate whether P is true or false: it is true if you accept that it is true, and it is false if you accept that it is false. Omniscience is only possible to the optimist.
But wait, there's more. Knowledge can be roughly defined as justified true belief. To say that this being knows the truth values of all well-formed propositions is to say that it believes truly and with justification. So what's its justification for all its knowledge? Well, if it knows that it's omniscient, then that would be credible justification: it knows that anything it believes must be true. But as we've just seen, whether or not it knows that it's omniscient is indeterminate. It may be believing that it is omniscient falsely, in which case all its justifications are based on a falsehood, and dissolve. Then, since knowledge requires sound justification, we might say that the being doesn't know anything, even if all its other beliefs happen to be true. It's like the man who looks at a watch stopped at noon when it is noon: he forms the belief that it is noon, and that belief is correct, but his justification is faulty, so we can't say he knows it's noon.
So we cannot tell whether the being is omniscient or not. And the being must know all this as well (or else it is out of the running for omniscience automatically). So it can't tell either; it has no way of knowing whether its belief that it is omniscient is true or false. Which leads us to the following exchange...
"Are you omniscient?"
"I don't know."
"Doesn't that mean you're not omniscient?"
"I don't know."
There are indeed ways to define "omniscience" and "knowledge" such that omniscience, so defined, is logically impossible.
A bit off topic but hearing the word paradox reminded me of this one where (forget the specifics) Racer A starts ahead of racer B, and racer B is traveling at a faster pace than racer A. But racer B cannot pass racer A, because, in the time it took to catch up, Racer A will have covered x amount of distance. I guess a picture might help demonstrate this, but I'm sure someone here knows what I'm talking about.
Obviously this is wrong, but where exactly does the logic become faulty?
An infinite sum of values does not necessarily fail to be finite. With calculus and some values for the racer's speeds, you can show that the terms of the sum of "the time it takes" become vanishingly small. The sum in total being finite, it is no hurdle to claim racer B overtakes racer A.
Now with that awkward vector out of the way,
B_S, it would appear you've done this:
P: C believes that P. Q: C believes everything that is true. R: P & Q.
If P is true then C is correct. If P is false, then C does not believe it.
Thus, being able to show that Q is sustained no matter the condition of P, Q is logically equivalent to R Well that's not true. Excuse me. I forget the name of this principle, I learned it in an electronics course. Basically you could say "S ^ (S^P V S^-P)" can derive and be derived from S.
You've found one of I assume an infinite class of interesting-looking statements that take the role of P, but which are irrelevant in a sense that seems to require a more technical term than irrelevant.
But wait, there's more. Knowledge can be roughly defined as justified true belief. To say that this being knows the truth values of all well-formed propositions is to say that it believes truly and with justification. So what's its justification for all its knowledge? Well, if it knows that it's omniscient, then that would be credible justification: it knows that anything it believes must be true. But as we've just seen, whether or not it knows that it's omniscient is indeterminate. It may be believing that it is omniscient falsely, in which case all its justifications are based on a falsehood, and dissolve. Then, since knowledge requires sound justification, we might say that the being doesn't know anything, even if all its other beliefs happen to be true. It's like the man who looks at a watch stopped at noon when it is noon: he forms the belief that it is noon, and that belief is correct, but his justification is faulty, so we can't say he knows it's noon.
So we cannot tell whether the being is omniscient or not. And the being must know all this as well (or else it is out of the running for omniscience automatically). So it can't tell either; it has no way of knowing whether its belief that it is omniscient is true or false. Which leads us to the following exchange...
"Are you omniscient?"
"I don't know."
"Doesn't that mean you're not omniscient?"
"I don't know."
candidus inperti; si nil, his utere mecum.
So here's an interview with an omniscient being:
Q: Are you omniscient?
A: Yes.
Q: Is that just a guess, or do you know that you're omniscient?
A: I know, obviously. If I didn't know, then that would be something that I didn't know, so I wouldn't be omniscient in that case.
Q: But knowledge requires justification. What is your justification for believing you are omniscient?
A: I believe I'm omniscient because I am not aware of anything I don't know.
Q: That's hardly a proof of your omniscience.
A: I didn't claim it was a proof. I just claimed it was a justification.
Q: But an man who has lived his whole life in a dark cave with no outside contacts could also be unaware of anything he doesn't know. In that case, doesn't he also have a justification for believing he is omniscient? And mightn't he also know that he is omniscient?
A: Sure, he would have a justification. But he wouldn't know, because knowledge is justified true belief. His belief is false.
Q: But that means you only know you're omniscient if it is in fact true that you are omniscient.
A: Good thing it's true, then.
Q: But what if you're wrong? How could I tell the difference?
A: I didn't claim to be able to prove to you that I was omniscient.
Q: Okay, then, how could you tell the difference?
A: I didn't claim to be able to prove to myself that I was omniscient either. I just claimed to know it. I believe it, it's true, and I have justified my belief. Therefore, I know I am omniscient.
Q: Sigh... But if you imagine a version of you that isn't omniscient, I could be having the exact same conversation with that version of you as I am with the real you.
A: Yes, but that version, when he said he knew he was omniscient, would be wrong.
If he can't "just know" that he is omniscient, then why is he allowed to "just know" the rest of what he knows is true?
If a being knows that he is omniscient, this constitutes justification for anything else he "just knows". But if he tries to justify his knowledge that he is omniscient with his knowledge that he is omniscient, he runs into a slight circularity problem.
candidus inperti; si nil, his utere mecum.
So, everything would be resting on his omniscience, which--itself--isn't resting on anything?
Alright, got it.
1. we define omniscience as having justified, true belief embedded within it, and
2. we have no pragmatic explanation for the being's omniscience besides our assumption, and corresponding suspension of belief to the contrary, that the being is omniscient,
THEN the omniscience evaporates, because we cannot justify it. The suspension of belief cannot hold. In this example, no matter how much we try to escape this by making it hypothetical, the being's omniscience comes from our assumption that it is omniscient. Real-life omniscience, if it existed, would have justified true belief embedded within it, but this justified true belief of each well-formed proposition would be the set of characteristics that lead us to call the being omniscient, not the other way around. Logic simply does not have the tools to let us work backwards like this.
3WR jor kadeen, the prevailer weenies and extra combat forever
3RR zirilan of the claw dragons and damage doublers
4:SymRG::SymRG: wort, the raidmother burn is now EDH viable
2WU kangee, aerie keeper birds
1UBR jeleva, nephalia's scourge spellslinger/storm
here are the mana costs of generals i no longer play: 2BR3BB3UBG4UB:SymUB::SymUB:2URRG3WWU2UU2GGUUBGG
3UWR numot, the devastator of [the spirit of edh]
Obviously this is wrong, but where exactly does the logic become faulty?
It's some variant of Xeno's Paradox (I'm not sure if it's formalized as a different paradox or not), but it's faulty because in the time it took B to get to A's initial position, A moved a smaller distance... and the smaller time it takes B to reach A's next position, A moves a smaller distance (if the speed discrepancy is very large, a very small distance indeed!). As the times go to the limit of 0 seconds, practically no movement is occurring... but time doesn't actually stop. It's just that the time it takes B to pass A is not time with the characteristic "time it takes B to get from A's previous position to A's next position", so it falls outside the scope of the example.
Essentially the 'paradox' is saying nothing more interesting than that B can't pass A in less than the time it takes B to pass A, it just does so in an intuition-warping fashion for certain types of people.
There are indeed ways to define "omniscience" and "knowledge" such that omniscience, so defined, is logically impossible.
An infinite sum of values does not necessarily fail to be finite. With calculus and some values for the racer's speeds, you can show that the terms of the sum of "the time it takes" become vanishingly small. The sum in total being finite, it is no hurdle to claim racer B overtakes racer A.
Now with that awkward vector out of the way,
B_S, it would appear you've done this:
P: C believes that P.
Q: C believes everything that is true.
R: P & Q.
If P is true then C is correct. If P is false, then C does not believe it.
Thus, being able to show that Q is sustained no matter the condition of P,
Q is logically equivalent to RWell that's not true. Excuse me. I forget the name of this principle, I learned it in an electronics course. Basically you could say "S ^ (S^P V S^-P)" can derive and be derived from S.You've found one of I assume an infinite class of interesting-looking statements that take the role of P, but which are irrelevant in a sense that seems to require a more technical term than irrelevant.
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