I think I do. — Srap Tasmaner

Then using that understanding, we have three options:

Option 1
It is possible that Jane's belief is actually false
Therefore, if Jane's belief is true then it is possible that Jane's belief is actually false

Option 2
It is possible that Jane's belief is actually false
Therefore, Jane's belief is actually false

Option 3
It is not possible that Jane's belief is actually false

Assume, for the sake of argument, that Jane's belief is "John is a bachelor".

Which option is correct? It must be one of them.

Based on what I understand of your position, you're saying that if the actual world is as Jane believes it to be then it isn't possible for the actual world to be other than Jane believes it to be?

If so then via modus tollens if it is possible for the actual world to be other than Jane believes it to be then the actual world isn't as Jane believes it to be, and so Option 2 is correct.

You are right that Jane's belief, which is true in the actual world, isn't false in the actual world, but what do you mean by saying that it can't be false in the actual world? — Michael

Only the law of of noncontradiction. It's a matter of, let's say, "logical" necessity. What is true cannot be false, even if it might have been false. No such law prevents a sentence from being both true and English, say. But by the same token an English sentence cannot be a Russian sentence.

Option 1
It is possible that Jane's belief is actually false
Therefore, if Jane's belief is true then it is possible that Jane's belief is actually false — Michael

No. If Jane's belief is actually true, it can only be counterfactually false, not actually false. It's what "counterfactual" means.

Option 2
It is possible that Jane's belief is actually false
Therefore, Jane's belief is actually false — Michael

No. We do not know that Jane's belief is necessarily false, so we do not know that it is not possible that Jane's belief is actually true.

This is tricky though, and can be put a little more clearly the other way around. There are worlds in which Jane's belief is false; the actual world might be one of those worlds. "Might be" here is epistemic; it's about our knowledge of what sort of world this one is, not about what sort of world it is. There are two steps: determining what sorts of possible worlds there are, and then determining which of those we happen to live in. We simply do not have enough here to conclude that this world is the sort of world in which Jane's belief is false. The other kind of world may be possible, and this might be one of those.

(For what it's worth, the epistemic issue is forcing me to talk about possibilities that might or might not be, which is terribly uncomfortable, but I'm not sure how to get around it. I wondered aloud once before whether we could just capture the epistemic options in more sets of possible worlds, but that's more work than I feel like doing unless I have to, and I don't even know that it works. Again, very likely reinventing the wheel here, as this sort of stuff is a very hot topic today in epistemology.)

Option 3
It is not possible that Jane's belief is actually false — Michael

So this is not freestanding but the fallback if I reject 1 and 2.

Option 1 I have a problem with because even if JB is only contingently actually true, it cannot be actually but only counterfactually false. I am not committed to JB being necessarily true.

Option 2 tries to take the logical necessity of Option 1 and turn it into metaphysical necessity. That is, there are conditions under which I have said JB cannot be actually false (namely JB being actually true), therefore if JB can be actually false, those conditions must not be met (JB is not actually true). But JB need only be contingently actually true to block JB being actually (rather than counterfactually) false, so I am still not committed to JB being necessarily true.

If JB were necessarily true, it would not be possible for it to be actually false. Since I am not committed to JB being necessarily true, am I committed to anything else that would make it impossible for JB to be actually false? Nothing I can think of, so it's "no" to option 3 as well.

Which option is correct? It must be one of them. — Michael

Also no.

I understand your quandary here, I think, but it's mainly down to use of modal sounding terms in different senses. The fact that whenever P is true in W, it cannot be false in W, comes right out of the definition of W, which includes P carrying a value of "true". That's just not the same thing as saying that P is true in all possible worlds; it's only saying that the worlds in which it is true are defined by its being true there, and if you need it to be false then you're in another set of worlds.

Is anyone besides Michael checking my math?

I am barely qualified to be explaining this stuff and feel like I'm on the verge of making a hash of it.

Would really appreciate it if someone more knowledgeable chimed in.

No. If Jane's belief is actually true, it can only be counterfactually false, not actually false. It's what "counterfactual" means. — Srap Tasmaner

You accepted here that "Jane's belief might be false" and "Jane's belief is true" can both be true, so I don't understand your objection. Do the below two phrasings mean something fundamentally different to you? Obviously the first phrasing has as a premise the antecedent of the second phrasing's conclusion, and the second phrasing combines the second premise and the conclusion of the first phrasing into a material conditional, but the meaning of "Jane's belief might be false" is identical in every occurrence, and if one is valid then so is the other.

Phrasing 1:
Jane's belief might be false
Jane's belief is true
Therefore, Jane's belief might be false

Phrasing 2:
Jane's belief might be false
Therefore, if Jane's belief is true then Jane's belief might be false

Maybe I need to be even simpler.

World 1
Jane's belief might be false
Jane's belief is true

World 2
Jane's belief might be false
Jane's belief is false

No. If Jane's belief is actually true, it can only be counterfactually false, not actually false. It's what "counterfactual" means.
— Srap Tasmaner

You accepted here that "Jane's belief might be false" and "Jane's belief is true" can both be true, so I don't understand your objection. Do these two phrasings mean something fundamentally different to you? Obviously the second phrasing combines the premise and conclusion of the first phrasing into a material conditional, but the meaning of "Jane's belief might be false" is identical in every occurrence. — Michael

I doubt either of us has been perfectly consistent about this, but I can explain what I'm thinking.

When we say "P is true," for instance, I am taking that as "P is true in the actual world." I think that's generally what's intended with an unadorned "true" or "false." When it matters, I'm saying "actually true" or "true in W," something like that.

When we say "P is possibly true," with no other restriction, I am taking that as "true in some possible world," and the set of possible worlds in which P is true may or may not include the actual world.

So, in our earlier exchange, I took "Jane's belief is true" to mean true in the actual world, and "Jane's belief might be false" to mean false in some (other) possible world. They can both clearly be true, on this reading, even though JB being actually true means the actual world is not one of the worlds in which JB is false.

Phrasing 1:
Jane's belief might be false
Jane's belief is true
Therefore, Jane's belief might be false

Phrasing 2:
Jane's belief might be false
Therefore, if Jane's belief is true then Jane's belief might be false — Michael

Phrasing 2 is a degenerate argument in which JB's truth plays no role.

Phrasing 1 is fine because I'm reading "might be" as counterfactual. If you change that to "Jane's belief might actually be false," I'll say no.

Maybe I need to be even simpler. — Michael

I'd rather you be more explicit. It is absurd that in this sort of conversation we are not in every case saying "true in W" instead of unadorned "true." It would make many points much clearer.

Do you accept that both of these are possible worlds?

World 1
Jane's belief might be false
Jane's belief is true

World 2
Jane's belief might be false
Jane's belief is false — Michael

Worlds? Not sets of worlds?

For the first, if JB is true in W1, no it cannot be false in W1. How could it be?

For the second, I'll address the bit I skipped over with W1. What exactly do you mean by "might be"? I cheated a little in W1, because the second premise allowed me to construe it as "not violating the law of noncontradiction." But really what is "might be" supposed to mean within a given world?

If you want these claims to be true within a single world, I think we have to take "might be" as indicating our epistemic position, because it makes no sense at all to count a world as its own counterfactual. Worlds are collections of facts, not possibilities. Sets of worlds represent possibilities, depending on how the facts are distributed among them.

I can continue to read W2 as I read W1, that it does not violate the law of noncontradiction for JB to be false. But these "worlds" are specified using unnecessarily ambiguous language. I wish this language were just nonsensical, but it happens that there are things we want to say that can be made to fit (the logical and epistemic issues), and that are sometimes what people mean when they talk this way. Since we know there are several different types issues in play, there's no reason for us not to be much clearer.

What exactly do you mean by "might be"? I cheated a little in W1, because the second premise allowed me to construe it as "not violating the law of noncontradiction." But really what is "might be" supposed to mean within a given world? — Srap Tasmaner

The skeptic claims that we might be brains in a vat. There are two different ways to interpret this claim:

1. There is a possible world where "we are brains in a vat" is true
2. It is possible that "we are brains in a vat" is true in the actual world

I think it obvious in context that they are making a claim such as 2). So with that in mind, I will rephrase the above:

World 1
a) This world might be other than Jane believes it to be
b) This world is as Jane believes it to be

World 2
a) This world might be other than Jane believes it to be
b) This world isn't as Jane believes it to be

---

Option 1
Both World 1 and World 2 are possible

Option 2
Only World 2 is possible

Option 3
Neither World 1 nor World 2 is possible

(only World 1 being possible is technically an option but I suspect we can dismiss that option outright).

The skeptic claims that we might be brains in a vat. There are two different ways to interpret this claim:

1. There is a possible world where "we are brains in a vat" is true
2. It is possible that "we are brains in a vat" is true in the actual world

I think it obvious in context that they are making a claim such as 2). So with that in mind, I will rephrase the above: — Michael

3. There are possible worlds in which we are brains in vats, and we do not (or, perhaps, "cannot") know that this world is not one of those.

See how that separates the concerns you have mashed together in (2)? You must first argue that brains-in-vats worlds are possible (that it is coherent, and maybe a bunch of other stuff), and then further argue that we have and possibly can have no knowledge that our world is not one of those.

"Possibly true in W" is not nearly clear enough, not for this kind of discussion, and I'm tired of disentangling the various strands of meaning.

Not gonna address your worlds that "might be" this or that. Clarify your terminology or I'm done.

3. There are possible worlds in which we are brains in vats, and we do not (or, perhaps, "cannot") know that this world is not one of those. — Srap Tasmaner

This is acceptable except your use of the word “know”. I think “we are not certain” is a better phrasing.

This is the approach I took back here:

My belief is justified
I am not certain
John is a bachelor
Therefore, I have knowledge that is not certain

—

And here:

The paradox, then, is that:

1. "might be wrong" means either "is not certain" or "is not necessarily true"
2. It is acceptable to say that we can have knowledge that is not certain (if we're fallibilists)
3. It is acceptable to say that we can have knowledge that is not necessarily true
4. It is unacceptable to say that we can have knowledge that might be wrong

—

And back on page 1:

Maybe the problem is with the interpretation of the English sentence. These two don’t mean the same thing:

It is possible that I know everything and am wrong about something

I know everything and it is possible that I am wrong about something

The former is false but the latter seems possible.

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.

—

If this is the approach you want to take now then I don’t understand what you’ve been arguing against. Perhaps a misrepresentation of my position? I made it clear, again back on page 1 that I wasn’t suggesting anything like ◇(Kp ∧ ¬p).

3. There are possible worlds in which we are brains in vats, and we do not (or, perhaps, "cannot") know that this world is not one of those.
— Srap Tasmaner

This is acceptable except your use of the word “know”. — Michael

And I think you should have raised an eyebrow at "cannot". That would pretty much force us to start sorting worlds by our epistemic condition and then determining (1) whether worlds in which we know we're brains in vats are possible, and (2) whether worlds in which we are brains in vats but don't know it are possible, and so on and on. There is, I understand, quite a bit of literature along exactly these lines, none of which I've spent any time with.

I don’t understand what you’ve been arguing against. — Michael

I think that's true.

Some of what I've been saying has become clearer to me, but not to you, as we went along. Some of it is just subtle enough that I think I've expressed myself poorly at least a couple times, but I have been trying to be more precise with each post.

I think I have been mistaken, even in that last post, in how I imagined we would keep the issues of how the world is and our knowledge of it separate. I still think we should keep them separate, but it's becoming clearer to me that they are both ways of sorting and partitioning sets of worlds -- different ways, yes, but the only way forward is to treat them similarly if separately. Otherwise there are too many things I can't say without serious cheating.

As far as I can tell, and you'll correct me if I'm wrong, your position and your understanding of the issues involved has changed not at all since the OP, despite everything I and others have posted. You still appear to be baffled that anyone would disagree with anything you've posted and just post it again, as here.

If you have something new to say, I'll listen, but for now I've put as much work into this as I intend to, and I'm not going down the rabbit hole of modal conditions on knowledge without very good reason.

As far as I can tell, and you'll correct me if I'm wrong, your position and your understanding of the issues involved has changed not at all since the OP, despite everything I and others have posted. You still appear to be baffled that anyone would disagree with anything you've posted and just post it again, as here. — Srap Tasmaner

You and others have said things like “if p is true then p cannot be false” which I’m not sure how to understand.

If it’s something like p → ¬◇¬p then it’s false.

If it’s something like “if p is true then we are certain that p is true” then it’s false.

If it’s something like ¬◇(p → ¬p) then it’s not addressing anything I’m saying.

If it’s something else then please tell because then I honestly have no idea what you’re trying to say.

It's something like ~(P & ~P). It's really that simple.

It's something like ~(P & ~P). It's really that simple. — Srap Tasmaner

I’ve never said anything to suggest otherwise which is why I don’t understand the objections.

Again from the first page, but amended to not be about omniscience:

Maybe the problem is with the interpretation of the English sentence. These two don’t mean the same thing:

a) It is possible that I know something and am wrong about that thing
b) I know something and it is possible that I am wrong about that thing

The former is false but the latter seems possible as the arguments show.

I suppose the latter is the implication of fallibilism. If knowledge does not require certainty then I can know and not be certain.

It seems you and others are misinterpreting my position as saying something like a).

Maybe the problem is with the interpretation of the English sentence. These two don’t mean the same thing:

a) It is possible that I know something and am wrong about that thing
b) I know something and it is possible that I am wrong about that thing

The former is false but the latter seems possible as the arguments show. — Michael

I think on ordinary usage, b) is also false.

If I know it's raining outside then I can't be wrong that it's raining outside. Knowledge entails truth.

So consider instead a scenario where something isn't known. Suppose Alice flips a coin, observes the outcome (i.e., knows what it is) and then places her hand over the coin. She then asks Bob, who hasn't observed the outcome, whether it is heads or tails. He could reasonably say, "I don't know. It could be heads or it could be tails, with equal likelihood of either."

Bob's second sentence is true since he's not in a position to rule out either possibility or prefer one possibility to the other. One of the possibilities is the actual outcome, he just doesn't know which one that is.

Whereas for Alice, there is only one possibility - the outcome she observed.

. If Jane's belief might be wrong and if Jane's belief is true then Jane's belief is true and might be wrong. — Michael

Jane's belief might be wrong or it might be right. If Jane' s belief is right then it's not true that it might be wrong. If Jane's belief is wrong, then it is not that it merely might be wrong, but that it is in fact wrong. It is only appropriate to say a belief might be wrong, if we don't know whether it is right or wrong.

You said that if we have a justified belief it might be wrong, which is true; but a justified belief is not knowledge, since knowledge is defined as a justified true belief. It is not the case that a justified true belief might be wrong. So many pages on this thread on account of a very basic confusion; it's puzzling!

If I know it's raining outside then I can't be wrong that it's raining outside. Knowledge entails truth. — Andrew M

A true belief entails truth as well. I guess that it's raining outside and I might be wrong. Even if I guessed correctly. Otherwise "I might be wrong" is only true if I guessed incorrectly, i.e. if I am wrong.

So, either "I might be wrong" can be true even if I have a true belief or "I might be wrong" is only true if I have a false belief.

You said that if we have a justified belief it might be wrong, which is true; but a justified belief is not knowledge, since knowledge is defined as a justified true belief. It is not the case that a justified true belief might be wrong. — Janus

See above.

So, either "I might be wrong" can be true even if I have a true belief or "I might be wrong" is only true if I have a false belief. — Michael

The first option is fine when understood as an expression of uncertainty as in, "I believe it is raining but I'm not certain". But not in the sense of, "My true beliefs could be false".

The first option is fine when understood as an expression of uncertainty as in, "I believe it is raining but I'm not certain". But not in the sense of, "My true beliefs could be false". — Andrew M

What does "could be false" mean? Either "there is a possible world where it is false" or "I am not certain that it is true". In both cases "My true belief could be false" can be true.

If you just mean that my true belief could not be both true and false then I agree, but "my true belief could be false" doesn't mean "my true belief could be both true and false".

The "could be false" in "my true belief could be false" means the same thing it does in "my belief could be false". Therefore, if "my belief could be false" being true does not entail that my belief is false then "my true belief could be false" could be true.

What does "could be false" mean? Either "there is a possible world where it is false" or "I am not certain that it is true". In both cases "My true belief could be false" can be true. — Michael

Either of those two senses are fine. But "My true belief could be false" is a conceptual claim. Compare "John could be married" to "Bachelor John could be married". There are possible worlds where John is married (and others where he is not). But there are no possible worlds where John is a bachelor and married.

But there are no possible worlds where John is a bachelor and married. — Andrew M

I know, but "bachelor John could be married" doesn't mean "there is a possible world where John is a bachelor and married".

There is a difference between:

a) John is a bachelor and could be married
b) John could be a bachelor and married

There is a ball hidden in a box. That ball is either red or blue.

a) The ball might be red.

This proposition is true whatever the colour of the ball in the box. It is true if the ball is red and it is true if the ball is blue.

If you want to say that a) is false if the ball is blue then you are saying that if the ball might be red then it is red.

If you accept that a) is true even if the ball is blue then you accept that there is a possible world where the ball is blue and a) is true; you accept that there is a possible world where the ball is blue and the ball might be red.

There is a ball hidden in a box. That ball is either red or blue.

a) The ball might be red.

This proposition is true whatever the colour of the ball in the box. It is true if the ball is red and it is true if the ball is blue. — Michael

Yes.

If you want to say that a) is false if the ball is blue — Michael

No, I'm saying that it is false that "the blue ball might be red", just as it is false that "The number 2 might be odd". There's a difference between conceptual and empirical claims.

No, I'm saying that it is false that "the blue ball might be red", just as it is false that "The number 2 might be odd". There's a difference between conceptual and empirical claims. — Andrew M

a) The ball might be red

If you accept that a) is true even if the ball is blue then you accept that there is a possible world where the ball is blue and a) is true.

And then I don't see a difference between these phrasings:

1. The ball is blue and a) is true
2. The ball is blue and the ball might be red
3. The ball is blue and might be red
4. The blue ball might be red

Do these mean different things to you, and so have different truth-conditions?

No, I'm saying that it is false that "the blue ball might be red", just as it is false that "The number 2 might be odd". There's a difference between conceptual and empirical claims. — Andrew M

Do you think that the number 2 might be (or could be) odd?

a) The ball might be red

If you accept that a) is true even if the ball is blue then you accept that there is a possible world where the ball is blue and a) is true.

And then I don't see a difference between these phrasings:

1. The ball is blue and a) is true
2. The ball is blue and the ball might be red
3. The ball is blue and might be red
4. The blue ball might be red

Do these mean different things to you, and so have different truth-conditions? — Michael

They mean the same thing to me. But I (and I suspect most people) would interpret them as asserting a contradiction (in the conceptual sense I mentioned above). Whereas you seem to be interpreting them in a Moorean sentence sense. While such sentences can be true, no-one would ever assert them. People would either say the ball is blue (when they knew it was blue) OR say the ball might be red (when they didn't know it was blue), but not both together.

Whereas you seem to be interpreting them in a Moorean sentence sense. While such sentences can be true, no-one would ever assert them. People would either say the ball is blue (when they knew it was blue) OR say the ball might be red (when they didn't know it was blue), but not both together. — Andrew M

Yes, exactly that. Moore's paradox was the inspiration for this discussion.

"I believe it is raining and it is not raining" is logically consistent and possibly true, but not something we would ever assert.

"Jane's knowledge might be wrong" is logically consistent (unless knowledge requires certainty) and possibly true, but not something we would ever assert.

Do you think that the number 2 might be (or could be) odd? — Andrew M

No, because the number 2 is necessarily even. My examples are only ever where the truth of the claim is not necessarily true. Hence in the OP:

1. Kp (premise)
2. ¬□p (premise)
3. Kp ∧ ◇¬p (from 1 and 2)

Yes, exactly that. Moore's paradox was the inspiration for this discussion. — Michael

:up:

No, because the number 2 is necessarily even. My examples are only ever where the truth of the claim is not necessarily true. — Michael

OK, so there is a number written on a piece of paper hidden in a box. That number is either 1 or 2.

a) The number in the box might be odd.

This proposition is true whatever the number in the box. It is true if the number is 1 and it is true if the number is 2.

Would you agree or disagree with that?

My first impression of your original post, is that you are implying ignorance as to whether you occupy your actual world versus a possible world occupied by someone else. In which case there is a contradiction.

But if by definition you take p, Kp and Bp to correspond to your actual world, then no contradiction arises with respect to the discrepancies with a possible world you talk about.

"I believe it is raining and it is not raining" is logically consistent and possibly true, but not something we would ever assert. — Michael

Not according to many people's grammar of "belief" including mine, although you appear to have company with a certain group of subjective Bayesians, who when designing an experiment insist on talking about their mental states rather than the experiment itself, much to the bemusement of any non-Bayesians present who merely wish to discuss reality.

Personally, if I am prepared to say "I believe X", then i am also prepared to assert "X" and "X is true". So according to my prescriptive usage, Moore's sentence is inconsistent. Only in the past or future tense would i invoke belief concepts.

Very clever, there is a possible world where both of these are true:

a) the number in the box might be odd
b) the number in the box is 2

Which contracts to:

c) the number in the box is 2 and might be odd

And even:

d) the number 2 might be odd

However, 2 is necessarily even and so we have a contradiction. And even if we understand "might be" in terms of one's own certainty rather than logical possibility we have:

e) I am not certain that the number 2 is even

Which may, in fact, be false.

How do you think this is resolved? I wonder if perhaps these don't mean the same thing?

1. The ball is blue and might be red
2. The blue ball might be red

3. The number in the box is 2 and might be odd
4. The number 2 might be odd

I'm happy to reject 2 and 4.

So in terms of my original argument, I'll still commit to 5 but reject 6:

5. My belief is true and might be wrong
6. My true belief might be wrong