When I see a philosophical thread with the words Omniscient or God, I think "that's not philosophy".

What we have here is a failure to parse stuff well. It's not too unlike those proofs that 0.9999... is not equal to 1.

Sadly perhaps, philosophy often involves considering the implications of concepts that nothing to do with the real world.

Medieval scholasticism seems to creep back into the present, where it rubs shoulders with quantum mysticism.

The problem of scope is within your English phrasing (the worlds which we read the word 'possibly' to quantify over), not the trivial inference (Kp ∧ ¬□p) ⊢ (Kp ∧ ◇¬p), which really is just an instance of a replacement rule.

We could not be wrong about p given that we know that p when we already restricted 'could've been' on the condition of whatever is accessible from our epistemic modality (meaning that Kp is true in all accessible worlds). Kp entails p in all worlds where Kp, so p is true in all epistemically possible worlds. Had we not known that p, it could've been that ¬p, which is a trivial and uninteresting result.

If 'could've been' is just our broadest alethic modality, then Kp does not entail □p for all contingent truths. This just means that we could've not known that p (where not-p could've been the case), not that we could've known p and been wrong about p

As you yourself suspect,

Note that I'm not saying: ◇(Kp ∧ ¬p)

From here:

1. It is possible that I know everything and am wrong about something
2. I know everything and it is possible that I am wrong about something

The former is false but the latter seems possible as the argument above shows. — Michael

(1) and (2) are both false when the modality of 'possible' is epistemic, (2) is only true when the modality of 'possible' is alethic, or if we know that p fallibly. There's really not much more to it.

Well, yes, you are saying that in some possible world here age might not be 30. — Banno

I'm not just saying that. I'm saying that she might not be 30 in the actual world. When we admit to the possibility of being wrong we're not saying "I'm actually right, but in some possible world I'm wrong". We're saying "I might actually be wrong." That's fallibilism.

(2) is only true when the modality of 'possible' is alethic, or if we know that p fallibly. — Kuro

Yes, that's what I said here:

I suppose the latter is the implication of fallibilism. If knowledge does not require certainty then I can know everything even if I am not certain about anything. — Michael

Omniscience aside, if we're fallibilists then we could be wrong even if we have knowledge.

I'm saying that she might not be 30 in the actual world. — Michael

But the actual world is a possible world...

So when you say that she might be 30 in the actual world, you are saying she might be 30 in some possible world...

In any case, it's clear this line of thought is not going to help.

But the actual world is a possible world...

So when you say that she might be 30 in the actual world, you are saying she might be 30 in some possible world... — Banno

That's why I said "I'm not just saying that."

The point is that when I say "I believe p but I could be wrong" I'm not saying "I believe p and in the actual world I'm right and in some other possible world I'm wrong". I'm saying "I believe p and in the actual world I could be wrong".

When I see a philosophical thread with the words Omniscient or God, I think "that's not philosophy". — Banno

I regret using that term now. Too many can't seem to get past it. That's why I offered this as an alternative.

Better.

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

That's not a good rendering of 4.

A better would be something like "I believe that in the actual worlds, p, although in other possible worlds, ~p"

And note that since your premise is now that that you know the woman is 30, you cannot conclude that you might be wrong.

A better would be something like "I believe that in the actual worlds, p, although in other possible worlds, ~p" — Banno

The literal translation of 4 is "I believe that this thing is true and it is possible that this thing is not true" which is just what I understand "I could be wrong" means.

Because, as above, when I say "I believe p but I could be wrong" I'm not saying "I believe p and in the actual world I'm right and in some other possible world I'm wrong". I'm saying "I believe p and in the actual world I could be wrong".

I think the mistake is in conflating modal logic with possible world semantics. You can have the former without the latter. I'm just using modal logic here.

It is possible that I am wrong about what is the case in the actual world. That seems to me to be the ordinary meaning of "I could be wrong".

by having to explain a fairly simple notation while trying to make a point with that notation to neophytes. — Banno

Who you calling a neophyte? Ya neophyte!

It is possible that I am wrong about what is the case in the actual world. — Michael

Not about things you know - they are true.

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 — Banno

She cant be 30 and not 30 at the same time. That's the paradox and if she is 29 then she's not 30 so how can you know she is 30 if she is 29? REAL world.

Not about things you know - they are true. — Banno

Some of the things I believe but don't know are true as well. When I say "I believe p but I could be wrong" I'm not saying "I believe p but I am wrong". And when I say "I know p but I could be wrong" I'm not saying "I know p but I am wrong".

"I could be wrong" can be true even if I'm not wrong. So that I'm not wrong if I know isn't that I can't be wrong if I know.

"I could be wrong" can be true even if I'm not wrong. — Michael

Can you give a real-life example or does this have to firmly stay in propositional logic?
Say about your own attributes for example. I know I am male, could I be wrong? I know I am caucasian, could I be wrong? etc.

Can you give a real-life example or does this have to firmly stay in propositional logic?
Say about your own attributes for example. I know I am male, could I be wrong? I know I am caucasian, could I be wrong? etc. — universeness

It's better understood with belief.

1. I believe that you are a mathematician, but I could be wrong.

Even if my belief is true it is still the case that it could be false. There is a difference between "I am wrong" and "I could be wrong". I could be wrong even if I'm not wrong.

If 1 is true of a true belief then it's also true of knowledge because knowledge just is true belief (with sufficient justification).

I think the reason that this conclusion seems counterintuitive is that even if we claim to be fallibilists there is this intuitive sense that knowledge entails certainty.

I think the reason that this conclusion seems counterintuitive is that even if we claim to be fallibilists there is this intuitive sense that knowledge entails certainty — Michael

I like your introduction of 'fallibility' into the discussion. Fallible, described as 'capable of making mistakes or being wrong.' Fallibility could imo be a candidate for an absolute or objective truth about humans.
I believe @jgill is a mathematician but is he the best mathematician that has ever existed in human history? I think this is the direction of play with concepts such as the omnis. I am currently convinced that we can only ever display an asymptotic approach to the omnis. If jgill was the best ever then would that mean he is not fallible in the subject of maths, he cannot make a mistake. It seems intuitive to me that for him to be omnimathematical, he must demonstrate infallibility in maths, do you agree?

Based on the discussion so far, has your opinion altered in anyway?
Your statement in the OP was:

This strikes me as a somewhat counterintuitive conclusion. — Michael

I assumed this meant that you were undecided whether or not there was a modal logic pathway to strong/good/respectable evidence to support or refute the possibility of an existent omniscient, past, present or future. Is this accurate? and has your position altered any based on the discussion so far?

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. — universeness

Exactly. Good point!
So, we have a closed, self-referential system, about and from which no knowledge can be obtained. As far as we are concrened, it represents an absolute "unknown". Yet, we are giving it a name it and even we define it. How can we do that, if we don't know what's in it? Well, I guess, in order to be able then to refute it! :smile: It's one of these games/tests that we create --knowingly or not-- so that we can challenge our logic. This is good. We must test and challenge our logic on a constant base!

I agree.
I like your

This is good. We must test and challenge our logic on a constant base! — Alkis Piskas

Do you have any philosophical musings about why we are compelled to do this, without going off thread!
You don't want to invoke the wrath of the mighty titans, labeled 'mods,' and their terrifying siren call!

We arra mods, We arra mods, We are, We are, We arra mods! :scream:

If you don't feel you can comment without going off thread then you could PM me a response if you want.

Even if my belief is true it is still the case that it could be false. — Michael

I just don't see how you're getting here.

Here's Oxford Reference...

A necessary truth is one that could not have been otherwise. It would have been true under all circumstances. A contingent truth is one that is true, but could have been false.

The point is all about tense. The definition given here is in the past tense (modal) 'could have been'. Not 'still could be'.

So if p is not a necessary truth it still only means that p could have been otherwise, not that p still could be otherwise. As such one's belief that p is true, p not being a necessary truth doesn't seem to imply that p could be false, only that it could have been.

When I say "I believe that you are American but I could be wrong" I'm not saying "I believe that you are American but I am wrong" and I'm not just saying "I believe that you are American but in some other possible world I am wrong".

"I believe that you are American but I could be wrong" is true even if you are in fact an American.

There seems to be this assumption made by some that if I'm not wrong then I can't be wrong, but then via modus tollens it then follows that if I can be wrong then I am wrong. This obviously isn't right.

When I say "I believe that you are American but I could be wrong" I'm not saying "I believe that you are American but I am wrong" and I'm not saying "I believe that you are American but in some other possible world I am wrong". — Michael

Yes. We all agree that beliefs could be wrong.

"I believe that you are American but I could be wrong" is sensible, and possibly true, even if you are in fact an American. — Michael

Again, it's just not clear how you're getting here. The "...I could be wrong" cannot be true if the proposition is it's referring to is true.

I'm not wrong then I can't be wrong, but then via modus tollens it then follows that if I can be wrong then I am wrong. — Michael

Draw that out for me...?

Draw that out for me...? — Isaac

p ≔ I am wrong
q ≔ I can be wrong

¬p → ¬q
∴ q → p

If I'm not wrong then I can't be wrong
Therefore, if I can be wrong then I am wrong

I think the conclusion is false, therefore I think the premise is false.

Again, it's just not clear how you're getting here. The "...I could be wrong" cannot be true if the proposition is it's referring to is true. — Isaac

Are you saying that "I believe this but I could be wrong" is only true if I am wrong (as per the above conclusion)? Then what's the difference between "I am wrong" and "I could be wrong"?

If I know that p is true then I believe that p is true. — Michael

I just wonder ... If you know that 'p' is true, why do you have also to believe it is true? What does "belief" add to it? Anyway, knowledge is stronger than belief. Beliefs are not knowledge. A belief may be an opinion, a certainly and at most a conviction. But all these are relative. They lie on a scale of of certainty: from very low to very high. Knlowledge on the other hand is not relative. It refers to something absolute or definite. "I know that this is true." That's all. (If it is true or not for you or most people, it's another story.) On the other hand, saying "I believe this is true" is quite different. I leave a window of uncertainly open, however small.

In fact, I could say that belief precedes knowledge, rather than follows it. I say, "I believe he is innocent. There's some good evidence about this." Then, when if the person is proved indeed innocent, I will know that he is.

What do you think?

Do you have any philosophical musings about why we are compelled to do this, without going off thread! — universeness

(I had to look up "musings". :smile:) I don't think we can discuss this without going off thread! :grin:
Anyway, I don't know if we are compelled to do that or just "fall" into that, like in a trap. And our mind can set up a lot of traps. Esp. if we let it do that. A fallacy, a self-contradition and all kinds of mistakes of logic, occur when we give our logic a "break". When, e.g. we are too cool, or inattentive or not interested enough, etc. Not a big deal, though. As long, of course, as we are able to detect and correct these mistakes ...
Well, all these are offhand thoughts ... I was never asked such a question! :grin:

You don't want to invoke the wrath of the mighty titans, labeled 'mods,' and their terrifying siren call! — universeness

"Mods" ... It seems it's my vocabulary day ... But I wan't so lucky with this one! :smile:

If you don't feel you can comment without going off thread then you could PM me a response if you want. — universeness

I think this is a wise idea. Well, it was, because I already replied in here! :grin:)

I will PM you.

If I'm not wrong then I can't be wrong
Therefore, if I can be wrong then I am wrong — Michael

If I'm not a cat then I can't be a cat
Therefore, if I can be a cat then I am a cat.

It doesn't work no matter what.

So either you've broken modus tollens, or you've rendered it wrong. It doesn't seem to say anything specific about knowledge.

I think there's something wrong with using 'can be' as proposition to be negated...?

I understand "I can be wrong" as "it is possible that I am wrong". So:

p ≔ I am wrong

¬p → ¬◇p
∴ ◇p → p

If I am not wrong then it is not possible that I am wrong
Therefore, if it is possible that I am wrong then I am wrong

I think the conclusion is false, therefore I think the premise is false.

Which part do you disagree with? Do you think that the conclusion is true or do you think that the premise is true even though the conclusion is false?