Tell me if I am getting anywhere with:
Kp ≔ x knows p
Bp ≔ x believes p

1. Kp ⊨ Bp (premise)
2. ∀p: Kp (premise)
3. ∃p: ¬□p (premise)
Ignoring 4 for now.

1. Knowing and believing become synonymous in the case of omniscience.
2. If all knowledge is accessible then something (person/transhuman/computer) could become omniscient
3. There exists at least 1 example of omniscience that does not logically contradict itself or its own existence.

Tell me if I am getting anywhere with: — universeness

1. Knowledge entails belief. In other words, if I know p then I believe p.
2. For everything that is the case, this hypothetical person knows that it is the case. In other words, our hypothetical person is omniscient.
3. At least one thing that is the case is not necessarily the case. In other words, it is possibly not the case.
4. For at least one thing that is the case, this hypothetical person believes that it is the case and it is possibly not the case. In other words, he could be wrong.

The counterintuitive conclusion is that an omniscient being could be wrong about something.

1. Knowledge entails belief. In other words, if I know p then I believe p. — Michael

Why would it be wrong in modal logic to state knowledge subsumes belief?

2. For everything that is the case, this hypothetical person knows that it is the case. In other words, our hypothetical person is omniscient. — Michael

So, in general, line 2 is just a modal logic premise that it's possible for an omniscient to exist which is why you stated earlier

I am not saying that there is anything that satisfies the second premise — Michael

yes?

3. At least one thing that is the case is not necessarily the case. In other words, it is possibly not the case — Michael

Is this not equivalent to stating that the exception does not prove the rule? Yet in the case of an omniscient it would, to me it would be like demonstrating one system only that has mass but could still travel at light speed in normal space (ie not using a worm hole of some currently sci-fi warp tech etc). You only need to climb Everest once to show it is possible.

4. For at least one thing that is the case, this hypothetical person believes that it is the case and it is possibly not the case. In other words, he could be wrong — Michael

Ok, I agree this last one is much tougher but you describe it very well. I can see the underlying paradoxical logic in play. As you say if 1, 2 and 3 are true then 4 'An omniscient can,' exist can be true.
but if your modal logic ends up in paradox then that is not good logical evidence for the existence of the omniscient, or am I missing something? Do we not end up in the same place as stating that a logical approach to omniscience is an unsuccessful one and you are left with nothing but appeals to esoteric ideas such as the supernatural?
Are you hoping that others will offer an argument that provides stronger evidence for or against the existence of omniscience based on a modal logic approach?

I thought I might try to explain my thinking on the 'exception' issue.
I was thinking about stuff like 'all snow is white,' except for some bits of snow that are not white but in general the rule holds. Some system might seem omniscient but if it's impossible to ask all possible questions then how can omniscience ever be proved?

This might be an interesting description for those who can follow this kind of symbolic logic. I can't.
However, I have something to say about the subject of this topic.

This kind of paradoxes --"The paradox of omniscience", "The paradox of omnipotence", etc.-- are well known and have been and are being discussed quite a lot in here and elsewhere.
What they all have in common is that they are not actual paradoxes because they are based on arbitrary and inexistent elements and/or facts. First we assume that there is some entity, a being, e.g. "God" --not done in this topic, but it is implied, since "omniscience" has a meaning only if it is an attribute of some entity-- then we attribute imaginary features to it --"omnipotent", "omniscient", etc.-- and then we try to prove, and we actually do, that these are impossible to exist or happen and even maybe the entity itself. In this way we create a "pseudo-paradox", a paradox with unsound foundations. From the moment that we assume the existence of an "omniscient" being, or of a concept of "omniscience", which have no foundation whatsoever, and start "building" on them, what should we expect other than a construct that will fall apart on the first blow of air?

Again, it may be a good "exercise" in modal logic to prove that there's no such a thing as omniscience, but the same process can be described using plain logic: If I say "I know everything", it means that I know every thing, i.e. "all things". But "all things" cannot be defined. They indicate a quantity and this quantity is undefinable. They also indicate a quality, which is also undefinable. What are these things? What do these things consist of? Therefore, my statement "I know everything" is empty, just air. It's just nonsense.

But "all things" cannot be defined. They indicate a quantity and this quantity is undefinable. They also indicate a quality, which is also undefinable. What are these things? What do these things consist of? — Alkis Piskas

I am mostly with you here. Certainly, the only system that could prove a system was omniscient would have to be itself, omniscient. As only an omniscient would know all possible questions.
I am now positing a multiplicity of omniscients! What would be the point of two omniscients communicating? This is as bad as 1/0=infinity, your calculator just reports ERROR.

What they all have in common is that they are not actual paradoxes because they are based on arbitrary and inexistent elements and/or facts. — Alkis Piskas

Yes

1. Knowledge entails belief. In other words, if I know p then I believe p. — Michael

Prima facie, this seems ridiculous. I know Crowley's religion , Thelema, but I do not believe it. I must be missing the subtlety. I haven't read the entire thread; the symbolism seems alien which demonstrates the fact that mathematicians don't have to be logicians. We operate at a much more humble level.

Prima facie, this seems ridiculous. I know Crowley's religion , Thelema, but I do not believe it. — jgill

Sorry, I wasn’t precise. If I know that p is true then I believe that p is true.

What they all have in common is that they are not actual paradoxes because they are based on arbitrary and inexistent elements and/or facts.
— Alkis Piskas

Yes — jgill

But, is it not the same for the liars paradox or the barbers paradox? Why are they 'actual paradoxes?
I mean 'This statement is false,' is not well defined, is it? I would call this arbitrary, a self-referential statement that states it is false, but based on what?
If the barber only shaves those who do not shave themselves then who shaves the barber?
Also very arbitrary and inexistent as any real barber can shave themselves regardless of what propositional logic demands.

For those of you who can't seem to get past "omniscience", consider instead this simplified paradox of knowledge:

Kp ≔ x knows that p is true
Bp ≔ x believes that p is true

1. Kp ⊨ Bp (premise)
2. Kp (premise)
3. ¬□p (premise)
4. Bp ∧ ◇¬p (from 1, 2, and 3)

In ordinary language:

1. Knowing that something is true entails believing that this thing is true (premise)
2. I know that something is true (premise)
3. This thing is not necessarily true (premise)
4. I believe that this thing is true and it is possible that this thing is not true (from 1, 2, and 3)

The counterintuitive conclusion is that I could be wrong in believing that something is true even though I know that this thing is true.

The counterintuitive conclusion is that I could be wrong that p is true (even though I know that p is true). — Michael

I get that so if applied to omniscience this shows we land in paradox, does it not?
If it does then does this not suggest omniscience cannot be evidenced by this method so we are left with appeals to metalogic are we not?

It seems to be paradox whether it's about omniscience or our everyday, limited knowledge.

I know that my name is Michael but it's possible that I'm wrong.

Well does that not bring us back to the other thread about absolute truths and the proposal that there are no universal absolute truths but we can get pretty close with values we might declare 'constants' such as the mass of an electron but any declared constant will not be absolutely accurate. So we might get close to omniscience or the speed of light in a vacuum but we will never reach it.

I see nothing counterintuitive here. Here's my analysis...

2. Kp ⊬ □p

Kp is knowledge that p is true. It is not knowledge of whether p is metaphysically contingent or necessary.

4. ¬□p (premise)

...this entails p is metaphysically contingent. (but does not imply you know this to be the case)

7. Bp ∧ ◇¬p
By this stage: a) you believe p; b) p is true; and c) p is metaphysically contingent. d) you don't have a belief (or knowledge) about whether p is metaphysically necessary or contingent.

8. ∀p: Kp (premise)
OK, then you now are stating that you know the truth of all propositions, this includes the proposition "p is metaphysically contingent"

9. ∃p: ¬□p (premise)
Which means some propositions are metaphysically contingent.


10. ∃p: Bp ∧ ◇¬p (from 6, 8, and 9)
Which means you believe some propositions that are true, but are metaphysically contingent. Not really a problem.

So what does your analysis tell you about whether omniscience or/and absolute truths has/have existed, can exist or will exist?

So what does your analysis tell you about whether omniscience or/and absolute truths has/have existed, can exist or will exist? — universeness

My analysis of what? Of the argument? Premise 8 asserts someone is omniscient, and this contradicts no other premise.

Aside from the argument, I'd say that it seems logically possible (but not physically possible) for something to have knowledge of everything that is knowable. However, the outcome of a future metaphysically indeterminate outcome (like quantum indeterminacy) is unknowable. So it depends on how you define "omniscience".

I only value that which is physically possible. I also would not connect metaphysical indeterminacy (whatever that might mean) with quantum indeterminacy. As I suggested, omniscience belongs to musings on the supernatural but it remains true that through scientific endeavor, we will learn more in the future, than we know now.

. ∀p: Kp — Michael

I'm probably being slow as usual, but can all p be known? I'd have thought only the true ones could.

10. ∃p: Bp ∧ ◇¬p (from 6, 8, and 9)
Which means you believe some propositions that are true, but are metaphysically contingent. Not really a problem. — Relativist

Do you not think this means “I believe p but it’s possible that I’m wrong”?

Good point, maybe I should have had it as if p then Kp and the conclusion as if p then Bp ∧ ◇¬p.

Or consider the simplified argument here.

Edit: I’ve edited that in.

I feel for you, arguing in two directions by having to explain a fairly simple notation while trying to make a point with that notation to neophytes.

1. I believe p but I am wrong
2. I believe p but I could be wrong — Michael

The difference can be made explicit using possible worlds - but also more complex...

It's the difference between Moore's paradox and the simple admission that things may have been otherwise than they actually are.

1. I believe p but I am wrong
Suppose that in this possible world - the actual world, I believe p but I am wrong - hence ⊢~p.

1. In the actual world, (Bp ^ ~p)

This is of course Moore's paradox, since the speaker has contradicted themselves.

2. I believe p but I could be wrong
But suppose also that in the actual world, p, but in other possible worlds, ~p; then we have

2. (In the actual world, p and Bp) ^ (in other possible worlds, ~p)

Now I hope this parsing makes the difference clear to you , but doubtless from others there will be ensuing posts taking up irrelevant or trivial issues.

So in summary,

3. ∃p: Bp ∧ ◇¬p (from 1 and 2) — Michael

Might be read as "I believe p is true in the actual world, but in other possible worlds it is false" without contradiction.

2. I believe p but I could be wrong
But suppose also that in the actual world, p, but in other possible worlds, ~p; then we have

2. (In the actual world, p and Bp) ^ (in other possible worlds, ~p) — Banno

I don’t think that’s right because when I say “I believe she’s 30 years old but I might be wrong” I’m not saying that in some alternate world she might not be 30 years old, I’m saying that she might not be 30 years old in the actual world.

If it helps, think of it in the third person instead. John believes that p is true and it is possible that p is not true.

Prima facie, this seems ridiculous. I know Crowley's religion , Thelema, but I do not believe it. I must be missing the subtlety. I haven't read the entire thread; the symbolism seems alien which demonstrates the fact that mathematicians don't have to be logicians. We operate at a much more humble level. — jgill

Kp is read as "x knows p", where p is a proposition or a statement. So implicitly Kp is "x knows that the statement p is true". .

When you say you know Crowley's religion, you are presumably saying that you know Crowley made certain claims, but not that you know those claims to be true.

That to know something is to believe it follows from the Theaetetus portrayal of knowledge as true belief with an account. It would be hard to justify going against this without demonstrating much misunderstanding.

I don’t think that’s right because when I say “I believe she’s 30 years old but I might be wrong” I’m not saying that in some alternate world she might not be 30 years old l, I’m saying that she might not be 30 years old in the actual world. — Michael

Well, yes, you are saying that in some possible world here age might not be 30. The congenital problem with possible worlds is that folk over interpret them. A possible world is exactly a situation that might have been otherwise; it's just a way of talking about modality.

Note the change from "possible world" to "alternate world" in your text. Possible worlds are perhaps not alternate worlds... but thereby hang more than a few Doctoral dissertations...

If it helps, think of it in the third person instead. John believes that p is true and it is possible that p is not true. — Michael

Makes no difference. There is no paradox here.

The implication of this is that if p is not necessarily true then I can know that p is true even if it is possible that p is not true. — Michael

Well, you can. You know that her age is 30 even if it might have been that she was 29. There's no paradox here.

That to know something is to believe it follows from the Theaetetus portrayal of knowledge as true belief with an account — Banno

A priori underlying the arguments in the thread then.

Not at all a priori. Just terminological assumptions. The p in Kp is a proposition.

10. ∃p: Bp ∧ ◇¬p (from 6, 8, and 9)
Which means you believe some propositions that are true, but are metaphysically contingent. Not really a problem. — Relativist


Do you not think this means “I believe p but it’s possible that I’m wrong”? — Michael

No. That's because Premise 8 states: ∀p: Kp
Which means there is knowledge of p (knowledge = a belief that is true).

Consider Schroedinger's cat. You open the box, and observe the cat is alive. You KNOW the cat is alive. Therefore you BELIEVE the cat is alive.

But it could have turned out differently: the cat could have been dead. So the fact that the cat is alive is a contingent fact. This isn't a case of ignorance, it's a case of metaphysical possibility.

As the argument is written, the modality of the statements is a bit confusing. I think some people are confusing epistemic possibility ("AFAIK I could be wrong") with metaphysical possibility (x is the case, but ~x could have been the case). I previously gave my interpretation of the argument, applying what seemed the intended modality. But it could be cleared up by revising the argument and specifying the modality in each statement (epistemic, metaphysical, conceptual, logical...)

↪jgill
Not at all a priori. Just terminological assumptions. — Banno

When I see a philosophical thread with the words Omniscient or God, I wonder what a philosopher thinks when he sees a mathematical one with the word, infinity. :chin: