I have a true belief and yet "I might actually be wrong" is true. — Michael

Not if it is a true belief.

My claim that I might actually be wrong is true even if aliens happen to exist and I'm not wrong. — Michael

See the word "actually"? It sneaks in a misplaced modal quantifier.

Not if it is a true belief. — Banno

1. I believe that aliens exist but I don't know that aliens exist.
2. I claim that I might actually be wrong.
3. My claim that I might actually be wrong is true even if aliens happen to exist and I'm not wrong.

Are you now saying that the third claim is false?

See the word "actually"? It sneaks in a misplaced modal quantifier. — Banno

I am using that word to account for what I explained here:

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

@Banno

Incidentally your reasoning entails that everyone lies when they admit that they might be wrong. If they’re not wrong then they can’t be wrong (according to your reasoning). They believe that they’re not wrong. Therefore they believe that they can’t be wrong (if they accept your reasoning). When they admit that they might be wrong they’re saying something they believe to be false.

:up:

Do you think there is credence in the proposal that there WILL BE a time in the future when a human, a transhuman or a non-human system will be declared as omnimathematical and have some accompanying proof from first principles based on modal logic?

3. Kp (premise)
4. ¬□p (premise)
5. Kp ∧ ◇¬p (from 3 and 4) — Michael

What about

3. p (premise)
4. ¬□p (premise)
5. p ∧ ◇¬p (from 3 and 4)

Isn't that unintuitive enough?

I mean, do you want it to be true, even if you're not wrong that p?

And what I say is true even if I am not wrong. — Michael

Green zone is p ∧ ◇¬p. Imagine the complement-donuts on the left are un-wound to make their own circles on the right.

5. Kp ∧ ◇¬p (from 3 and 4) — Michael

We then conclude that I could be wrong even if I know everything (and assuming that some p is not necessarily true): — Michael

Is it relevant that your K is an ordinary predicate not a modal operator? Sending the modal operator (or rather a corresponding extension) on the same adventure of enlargement would push back the territory of ◇¬p

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. — Michael

Try this: there are possible worlds in which she is 30, and possible worlds in which she is not; if you do not know whether she's 30, you do not know which bucket this world goes in.

To say, I know she's 30, is to say, I know this is one of the worlds in which she's 30. You can happily say I think she's 30 but maybe I'm wrong, because that's just saying, I think this is one of the worlds in which she's 30 but it might be one of the others.

But if you want to say, I know she's 30 but I might be wrong, then you're trying to say, I know this is one of the worlds in which she's 30 but I don't know if this is one of the worlds in which she's 30. That's a tough sell.

I'm not talking about knowledge at this point, just belief.

1. I believe that aliens exist but I don't know that aliens exist.
2. I claim that I might actually be wrong.
3. My claim that I might actually be wrong is true even if aliens happen to exist and I'm not wrong.

1. I believe that aliens exist but I don't know that aliens exist.
2. I claim that I might actually be wrong.
3. My claim that I might actually be wrong is true even if aliens happen to exist and I'm not wrong — Michael

This just seems to confirm that belief does not require irrefutable evidence but knowledge does.

There are possible worlds in which aliens exist and possible worlds in which they don't.

1. I believe that aliens exist but I don't know that aliens exist.

I believe this world is one in which they do, but I don't know it for a fact.

2. I claim that I might actually be wrong.

This world might be one of the ones in which aliens don't exist.

3. My claim that I might actually be wrong is true even if aliens happen to exist and I'm not wrong.

This world might, for all I know, be one of the ones in which aliens don't exist, even if it is one of the ones in which aliens exist.

The problem here is that we still have subjunctives, because we're layering the epistemic issue on top of the, let's say, metaphysical one. What we want is all indicatives, so we can quantify properly:

There are possible worlds in which I think my world is one of the worlds in which aliens exist.

That looks fine. Can we also say this?

There are possible worlds in which I think my world is one of the worlds in which aliens exist, but I am wrong and there are no aliens in that world.

Of course. What about this?

There are possible worlds in which I think my world is one of the worlds in which aliens exist, and they do, I'm right, and (new scope here) there are possible worlds in which they don't.

Yes.

There are possible worlds in which I think my world is one of the worlds in which aliens exist, and they do, and I think, in that world, there are possible worlds in which they don't exist, and I'm right (or wrong, whatever).

If the sorts of available possible worlds are clear enough, but the problem is knowing which sort this one is, then you can still analyze it as sets of possible worlds in which you're right about which sort of possible world this one is, and possible worlds in which you're wrong.

This world might ... be one of the ones in which aliens don't exist, even if it is one of the ones in which aliens exist. — Srap Tasmaner

Yes, and do you accept that this is true?

To make it simpler to understand, if necessary:

1. This world might be one of the ones in which aliens don't exist

Is this true? If yes then does it entail that aliens don't exist? If no then it is true even if aliens exist

The problem here is that we still have subjunctives, because we're layering the epistemic issue on top of the, let's say, metaphysical one. — Srap Tasmaner

I'm not sure what's problematic about the sentence above. It seems straightforward to me?

If aliens don't exist in this world, then this is one of the worlds in which they don't exist, whether you know it or not. If aliens are possible, then they are possible whether you happen to be in a world where they are actual or not, and whether you know they're possible or not.

I think you're trying to ask if this world could conceivably be a different world, but that's already baked in. All the possible worlds are already there; the question for you, the epistemic question, is which one you're in.

All the possible worlds are already there; the question for you, the epistemic question, is which one you're in. — Srap Tasmaner

Sure, so is the below true?

This world might be one of the ones in which aliens don't exist

is the below true?

This world might be one of the ones in which aliens don't exist — Michael

If that's a way of saying, I don't know which sort of world I'm in, sure.

But if you know aliens are actual, then they are actual, and you know this is not a world where they're not.

If that's a way of saying, I don't know which sort of world I'm in, sure. — Srap Tasmaner

So returning to my original wording, my claim that I might actually be wrong is true even if aliens happen to exist and I'm not wrong.

So I might actually be wrong even if I have a true belief.

Are you trying to equivocate? Why not distinguish the issue of what sort of world you happen to be in from the issue of what sort of world you think you're in?

I don’t understand what you’re asking. I’m simply saying that I believe that aliens exist but I might actually be wrong. Do you agree?

Except you seem also to want to say that your true belief "might be" false.

Do you think there is credence in the proposal that there WILL BE a time in the future when a human, a transhuman or a non-human system will be declared as omnimathematical and have some accompanying proof from first principles based on modal logic? — universeness

The words "omniscience" and "God" are not in my vocabulary normally. Initially I would have thought that modal logic had no place in the mathematics I have known, but I see according to page views on Wiki it is popular and there is something called "multiverse set theory", so I guess I was wrong. There are now so many mathematical topics I can no longer even pretend to define what "mathematics" is.

Except you seem also to want to say that your true belief "might be" false. — Srap Tasmaner

Yes.

1. I believe that aliens exist
2. I might be wrong

These are both true.

Can I derive “aliens don’t exist” from 2? No, because aliens can still exist even though 2 is true. It then follows that if aliens exist then I have a true belief and I might be wrong.

To deny the conclusion you must argue that you can derive “aliens don’t exist” from the truth of 2. You must argue that if I might be wrong then I am wrong.

If aliens exist, then it follows that you are not wrong to think they do.

Not being wrong won't stop you from thinking you might be, but it quite definitely stops you from actually being wrong.

If aliens exist, neither you nor anyone else can be mistaken in thinking they exist. That possibility is blocked by them existing.

You can still think they might not all you like. You'll always be wrong, though.

I'm not saying that if aliens exist then I have a true belief and am wrong. I'm saying that if aliens exist then I have a true belief and I might be wrong.

Again, these two are true:

1. I believe that aliens exist
2. I might be wrong

One of these is true:

3. Aliens exist
4. Aliens do not exist

Therefore, either these 3 are true:

1. I believe that aliens exist
2. I might be wrong
3. Aliens exist

Or these 3 are true:

1. I believe that aliens exist
2. I might be wrong
4. Aliens do not exist

If the first set is true then I have a true belief that might be wrong.

Continuing on from the above, what is the difference between a true belief and knowledge? If we're fallibilists and agree with the JTB definition of knowledge then knowledge is justified true belief, so let's amend the above:

These two are true:

1. I believe with justification that aliens exist
2. I might be wrong

One of these is true:

3. Aliens exist
4. Aliens do not exist

Therefore, either these 3 are true:

1. I believe with justification that aliens exist
2. I might be wrong
3. Aliens exist

Or these 3 are true:

1. I believe with justification that aliens exist
2. I might be wrong
4. Aliens do not exist

If the first set is true then I have knowledge that might be wrong.

To avoid the conclusion you must be an infallibilist and claim that knowledge requires certainty. In which case if I am certain that aliens exist then "I might be wrong" is false, and so I can't have knowledge that might be wrong.

I think a mistake that some fallibilists here are making is that they switch to infallibilism. They say that we don't require certainty to have knowledge but then imply that if we have knowledge then we're certain.

:smile: I know what you mean. It's hard to keep up with developments in computing science as well, which is my field of expertise.
But projecting my knowledge of computing science I cannot conceive of any system, human, transhuman or machine based that could be omniscient, regardless of any mathematical system that suggests it may be or is possible.
So if, a mathematician like you who achieved the title of professor does not think there is any mathematical pathway to omniscience, based on:

The words "omniscience" and "God" are not in my vocabulary normally. — jgill

Then this adds to my skepticism regarding a modal logic path towards it.

1. I believe that aliens exist
2. I might be wrong

One of these is true:

3. Aliens exist
4. Aliens do not exist — Michael

These premises are not independent. The truth-value of (2) depends on which of (3) and (4) is true. If aliens exist, you cannot be wrong to think they do; if they don't, you cannot be wrong to think they don't.

The actual world cannot "possibly" be a different one. It simply is whichever one it is, just as all the others are. It is not a matter of which name goes on which world, because "actual" is not a name but an indexical.

Again, these two are true:

1. I believe that aliens exist
2. I might be wrong

One of these is true:

3. Aliens exist
4. Aliens do not exist — Michael

These premises are not independent. The truth-value of (2) depends on which of (3) and (4) is true. If aliens exist, you cannot be wrong to think they do; if they don't, you cannot be wrong to think they don't. — Srap Tasmaner

So if I might be wrong then I am wrong?

Then this adds to my skepticism regarding a modal logic path towards it. — universeness

"Possibly" in current math would be a conjecture. It would go nowhere logically by itself. "Believe" really would be nonsense in mathematics.

I think that's the converse of what I was at least trying to say.

Let's suppose aliens might not exist. Then there is at least one possible world in which they don't.

Can the possible world in which aliens do not exist be a world in which aliens exist? No.

Can it be true, in a world in which aliens do exist, that there are other worlds in which they don't?

That's the same as asking if there are worlds in which aliens don't exist. We are supposing that aliens might not exist, so by stipulation, yes, there is at least one world in which they don't, and that continues to be true even in a world in which they do.

I think that's the converse of what I was at least trying to say. — Srap Tasmaner

Yes, and I think it's false, and so I think your premise is false.

When I say "I believe this but I might be wrong" I'm not saying, in a roundabout way, "I believe this but I am wrong". That I might be wrong has nothing to do with whether or not I am wrong. I'm just admitting to the possibility of being wrong (and not in Banno's "I'm actually right, but there's a possible world where I'm wrong" sense but in the "I might actually be wrong" sense). ◇¬p does not entail ¬p.

I think what's happening is that you're misinterpreting "it's possible that I have a true belief that might be wrong" as something like ◇(p ∧ Bp ∧ ¬p), but this symbolism actually say "it's possible that I have a true belief that is wrong" and is, of course, false. You should be interpreting it as "◇(p ∧ Bp ∧ ◇¬p)" which is true if ¬□p.

Nice try.

Michael might think he is making a modal point when he is making an epistemic one. On that basis we might interpret

"I believe this but I might be wrong" — Michael

as just saying that one is not certain of one's beliefs.

Trying to be charitable, that might explain why he does not recognise the modal answers to his question - it's not really a modal question.

Just a suggestion. The alternative is that he is infatuated with the muddle he has created, and doesn't want to see the answer. It happens.

I believe this but I might be wrong — Michael

as just saying that one is not certain of one's beliefs. — Banno

That's exactly what I'm saying as I made clear here:

To avoid the conclusion you must be an infallibilist and claim that knowledge requires certainty. In which case if I am certain that aliens exist then "I might be wrong" is false, and so I can't have knowledge that might be wrong.

I think a mistake that some fallibilists here are making is that they switch to infallibilism. They say that we don't require certainty to have knowledge but then imply that if we have knowledge then we're certain.