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.

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.

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).

anyone, even if they aren't a transgender woman, who identifies as a woman should — ToothyMaw

This is why your wording is ambiguous. A biological male who identifies as a women is a transgender woman, as I understand the word “identify”. What do you mean by the word?

Should men who identify as women be allowed to use women's restrooms? Should male fighters get to smash female fighters because they identify as women? — ToothyMaw

Your wording is ambiguous. When you say “men who identify as women” do you just mean men who say they’re women, or do you mean transgender women?

When I say that I mostly mean that each man and woman possess characteristics that are - to them at least - necessary to their gender expression — ToothyMaw

I don’t think so. I can’t think of any such characteristics in my case. I just am a man.

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).

there are distinct qualities that are necessary to the identities of men and women — ToothyMaw

I question that assertion. I'm skeptical of essentialism.

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 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

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.

Because you have been very clear that you mean Jane's belief, which is true in the actual world, might be false in the actual world, and that's not an option. If it's true in ℋ, it cannot be false in ℋ; if it's possibly false, in addition to being true in ℋ, it's false counterfactually in some ℳ where ℳ ≠ ℋ. — Srap Tasmaner

I don't understand your modality.

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? That something isn't false isn't that it can't be false (unless it's necessarily true).

You seem to have this notion of modality that sits somewhere between p and ◇p?

Your reasoning here seems to be that "Jane's belief might be wrong" and "Jane's belief is true" cannot both be true — Michael

They can both be true, yes — Srap Tasmaner

Then I don't understand the issue you have. If Jane's belief might be wrong and if Jane's belief is true then Jane's belief is true and might be wrong. Why do you disagree so much with this contraction?

And this gets us no closer to your goal of fallibilist knowledge, so far as I can tell. — Srap Tasmaner

Throw in justification to the above.

If Jane's belief is justified and if Jane's belief might be wrong and if Jane's belief is true then Jane's belief is justified and true and might be wrong. Again, why do you disagree so much with this contraction?

No, it doesn't "mean the same thing," but it might or might not be different from how I think it is implies that I do not know whether it is how I think it is. — Srap Tasmaner

It's the meaning of the phrase that matters here, not what its assertion implies about the speaker.

Do you at least understand the difference between "the actual world might be other than I believe it to be" and "there is a possible world that is other than how I believe the actual world to be"?

I get what you're going for, I do. But if Jane's belief is true, Jane's belief can only be false counterfactually. We already know how to say that, and it's "Jane's belief might have been false," or "could have been false." — Srap Tasmaner

Yet something like this seems to be what we are saying when we say "I believe this but I might be wrong". We are claiming that the actual world might be other than how we believe it to be.

So if Argument 3 is invalid then either "if I might be wrong then I am wrong" is true or "I might be wrong" is necessarily false.

I was still (am still!) trying to figure out what's going on here. — Srap Tasmaner

I am trying to make sense of this:

Argument 1
Jane's belief might be wrong
Therefore, if Jane's belief is true then Jane's belief might be wrong

The argument is valid but the conclusion is counterintuitive despite its consequent simply being a restatement of the premise.

To understand the issue we need to understand what the premise is saying. One interpretation is:

Argument 2
There is a possible world where Jane's belief is false
Therefore, if Jane's belief is true then there is a possible world where Jane's belief is false

The conclusion is acceptable, but I think that the premise is an inaccurate interpretation of the original. The premise "Jane's belief might be wrong" isn't just saying that there is a possible world where Jane's belief is false; it's suggesting that the actual world might be such world, and so we need something like:

Argument 3
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

I understand that this might be adding a second layer of possible world semantics, but I don't know how else to phrase it. I know that there have been attempts to make sense of modal logic without possible world semantics, so maybe that is what is needed for claims like these.

Perhaps, as you suggested, this is now an epistemic matter, and the interpretation is something like:

Argument 4
Jane's belief is not certain
Therefore, if Jane's belief is true then Jane's belief is not certain

This, at least, appears to have an acceptable premise and conclusion, although I'm not sure if it's an accurate interpretation of Argument 3.

The paradox, though, is that whereas we may be willing to accept Arguments 2 and 4, we appear unwilling to accept Argument 1 (as shown by the resistance I am getting). Why is that?

And no, it isn't, and it isn't. — Srap Tasmaner

That first argument was a reference to your claim here. When I asked you if that meant that "I might be wrong" entails "I am wrong" you responded with "I think that's the converse of what I was at least trying to say."

If you don't like me referring to it as an "argument" then I'll call it a "sentence" and say that either it is true or Argument 2 is valid.

In a context like this, "might be" is deliberately misleading. — Srap Tasmaner

It's not "deliberately misleading" if I explicitly say "the meaning of 'Jane's belief might actually be wrong' in the conclusion means whatever it means in the premise".

If you want to say "Jane's belief might actually be wrong" in the conclusion means something like "Jane's belief might have been wrong had things been different" then you are saying that this is what it means in the premise, which I think is false.

But see this is not an argument. — Srap Tasmaner

It is. It’s like saying “I’m a bachelor, therefore I’m not married”.

So which of arguments 1 and 2 is valid?

(b) is a misuse of "possible" in this context, because of the "actually" there.

There are no leftover possibilities in the actual world. It is defined by which possibilities it actualizes and which it doesn't. A statement that has a different truth value from the one it has in the actual world, is a statement that belongs to and partly defines a different possible world.

I tried to work around this issue by suggesting that the epistemic dilemma can be cast as trying to figure out which sort of world the actual world is. That might work, for all I know, but I suspect it's reinventing the wheel. @Kuro seems to be much more knowledgeable about this stuff than me. — Srap Tasmaner

Then there's an issue with the claim "I believe p but I might be wrong".

It shouldn't be interpreted simply as "I believe p and I'm not wrong but there is some other possible world that isn't the actual world where I'm wrong".

And I don't think it should be interpreted simply as "I believe p but I'm not certain" as the claim prima facie says something about the subject matter of the belief rather than one's reasons for holding it.

So how do we make sense of such a claim?

I agree, although I don't see how that is relevant to what I am arguing. In a stripped down form:

Argument 1
Jane's belief might actually be wrong
Therefore, Jane's belief is false

If Argument 1 is invalid then Argument 2 is valid

Argument 2
Jane's belief might actually be wrong
Therefore, if Jane's belief is true then Jane's belief might actually be wrong and Jane's belief is true

The consequent of the conclusion of Argument 2 is simply the conjunction of the premise and the antecedent, and so the meaning of "Jane's belief might actually be wrong" in the conclusion means whatever it means in the premise.

So which is valid?

The only way in which they can both be invalid is if "Jane's belief might actually be wrong" is necessarily false.

It's just that the argument is vacuous, so I see no sense in posting it on a philosophy forum. — Relativist

And yet the conclusion has been met with such resistance. Why is that? Perhaps others are equivocating and reading something into the "might be wrong" in the conclusion that isn't read into the "might be wrong" in the premise.

And if it's "vacuously" true that Jane's knowledge might be wrong then it seems worthy of posting to me, but to each their own.

The issue is what to make of arguments that go like this:

1. P.
2. (1) might be wrong.
... — Srap Tasmaner

Not quite, as it's not asserting p and then asserting that p might be wrong. It's asserting that there is this belief and then asserting that this belief might be wrong (which is not the same as saying that that there is this belief might be wrong, as would be suggested by your wording above).

But I do think it has to do with the meaning of "might be wrong", which is why here I addressed various interpretations and here set out the paradox in that when we translate "might be wrong" using one of these interpretations there is no issue, but when we consider it untranslated it is rejected. Why is that? As above, perhaps others are equivocating and reading something into the "might be wrong" in the conclusion that isn't read into the "might be wrong" in the premise.

The part in bold makes no sense. (a) is the premise "Jane believes with justification that John is a bachelor". A premise is treated as true, so why make it the consequent of a conditional?

Then the other part of the conclusion is vacuous - it just repeats the antecedent of the conditional (If John is a bachelor, then John is a bachelor). — Relativist

a) I am a man
b) I am British
c) Therefore, if I am 30 years old then I am a man and I am British and I am 30 years old

It makes perfect sense. It's valid.

So in this case:

a. Jane's belief that John is a bachelor is justified
b. Jane's belief might be wrong
c. Therefore, if John is a bachelor then Jane's belief that John is a bachelor is justified and Jane's belief might be wrong and John is a bachelor

We can combine "Jane's belief that John is a bachelor is justified" and "John is a bachelor" to make "Jane's belief is justified and true":

a. Jane's belief that John is a bachelor is justified
b. Jane's belief might be wrong
c. Therefore, if John is a bachelor then Jane's belief is justified and true and Jane's belief might be wrong

We can combine "Jane's belief is justified and true" and "Jane's belief might be wrong" to make "Jane's belief is justified and true and might be wrong":

a. Jane's belief that John is a bachelor is justified
b. Jane's belief might be wrong
c. Therefore, if John is a bachelor then Jane's belief is justified and true and might be wrong

Argument 2 is invalid. If John is a bachelor, then her justified belief is true - it is not metaphysically possible for it to be false. — Relativist

Argument 2
a. Jane believes with justification that John is a bachelor
b. Jane's belief might be wrong
c. Therefore, if John is a bachelor then Jane's justified true belief might be wrong

I'll rephrase it:

a. Jane believes with justification that John is a bachelor
b. Jane's belief might be wrong
c. Therefore, if John is a bachelor then a) is true and b) is true and John is a bachelor

Does the conclusion follow?

I'm not presenting any of those premises as something like personal belief-assertions. They are intended as statements of fact, as is usually the case when we make arguments like these. Would it be clearer in the third-person, and perhaps with a material implication?

One of these is valid:

Argument 1
a. Jane believes with justification that John is a bachelor
b. Jane's belief might be wrong
c. Therefore, John is not a bachelor

Argument 2
a. Jane believes with justification that John is a bachelor
b. Jane's belief might be wrong
c. Therefore, if John is a bachelor then Jane's justified true belief might be wrong

But no one ever says "I know for a fact there are three left, but I could be wrong."

Why not? — Srap Tasmaner

Also on this, the entire point of Moore's paradox is that there is something that we would never say as prima facie it would be absurd even though it isn't a logical contradiction and is sometimes even true. Perhaps the same sort of thing is going on here.

But no one ever says "I know for a fact there are three left, but I could be wrong."

Why not? — Srap Tasmaner

I suspect because of what I said here:

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.

But even though we don't say it, I think that this argument shows that such a claim would be true:

1. I believe with justification that John is a bachelor
2. My belief might be wrong
3. John is a bachelor
4. Therefore, my true (from 3) justified belief (from 1) might be wrong (from 2)

The issue, then, is to unpack what "might be wrong" means, which I have tried to do above.

I don't understand how you can deny the conclusion. Maybe you're equivocating and so mean something different by the "might be wrong" in "my belief might be wrong" to the "might be wrong" in "my true justified belief might be wrong"?

Unless you think, as you seemed to before, that this is valid:

I believe with justification that John is a bachelor
My belief might be wrong
Therefore, John is not a bachelor

This entire thread has been devoted to confusing "I have knowledge of something that need not be the case" with "I have knowledge of something that may not be the case." — Srap Tasmaner

(sorry, deleted the previous comment)

I think our conflict is in regards to the prima facie difference between saying:

a. There is a possible world where my belief is false
b. It is possible that my belief is actually false

Given Kp ∧ ◇¬p I trust that you accept (a) is true even if my belief is true?

But I suspect that you claim that (b) is false if my belief is true?

The ambiguity here is crippling. — Srap Tasmaner

I clarified that in the previous comment:

Therefore, I have knowledge that is not necessarily true

In other words, I know that John is a bachelor and it is not necessarily true that John is a bachelor. — Michael

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

There is a contradiction.

A partial solution is to abandon fallibilism:

1. "might be wrong" means either "is not certain" or "is not necessarily true"
2. It is unacceptable to say that we can have knowledge that is not certain
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

Although if, as my gut intuition suggests, certainty is only possible if the truth is necessary, we have a complete solution:

1. "might be wrong" means either "is not certain" or "is not necessarily true"
2. It is unacceptable to say that we can have knowledge that is not certain
3. It is unacceptable 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

But I think that if we wish to remain fallibilists then we must reject (4) and say that it is acceptable to say that we can have knowledge that might be wrong.

It may be tempting to find a third meaning of "might be wrong" that allows us remain fallibilists without rejecting (4), but such a meaning will then entail that this is valid:

I believe with justification that John is a bachelor
My belief might be wrong
Therefore, John is not a bachelor

@Srap Tasmaner

I've tried to collect the entirety of my thoughts below. Perhaps you would clarify exactly which step you take issue with?

One of these is a valid argument, where knowledge is taken to be justified true belief:

Argument 1
I believe with justification that John is a bachelor
My belief might be wrong
Therefore, John is not a bachelor

Argument 2
I believe with justification that John is a bachelor
My belief might be wrong
John is a bachelor
Therefore, I have knowledge that might be wrong

If we accept that Argument 1 is invalid then how are we to interpret "might be wrong"? One interpretation is "is not certain", and so the argument is:

I believe with justification that John is a bachelor
My belief is not certain
John is a bachelor
Therefore, I have knowledge that is not certain

Another interpretation is "is possibly false", and so the argument is:

I believe with justification that John is a bachelor
My belief is possibly false
John is a bachelor
Therefore, I have knowledge that is possibly false

At this point it is important to understand that there is a distinction between "is false" and "is possibly false". "is possibly false" just means "is not necessarily true", and so the argument is:

I believe with justification that John is a bachelor
My belief is not necessarily true
John is a bachelor
Therefore, I have knowledge that is not necessarily true

In other words, I know that John is a bachelor and it is not necessarily true that John is a bachelor.

Nothing so far should be controversial (except to the extent that the fallibilism of the first interpretation is controversial).

However, there is some ambiguity with "is possibly false", as shown by the prima facie difference between these two claims:

a. There is a possible world where my belief is false
b. It is possible that my belief is actually false

If when I claim that my belief might be wrong I am making a claim such as (a) then the interpretation of "might be wrong" as "is not necessarily true" above is sufficient, but if I am making a claim such as (b) then something more must be said.

It may be tempting to simplify matters by saying that (b) is false if my belief is true, but then this argument would be valid:

I believe with justification that John is a bachelor
It is possible that my belief is actually false
Therefore, John is not a bachelor

If this is unacceptable then how are we to interpret (b)? Perhaps as the claim that there is a possible world where the actual world is one of the possible worlds where p? This second layer of possible world semantics is admittedly confusing. Is it coherent? If not then how are we to interpret (b)? Perhaps it is to be interpreted as "my belief is not certain" as explained above?

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"[He could be declared] a Russian spy. Will we try to exchange him to bring Trump to Russia? Will they include Trump on the prisoner exchange list?"

:rofl:

(2) breaks down into cases, right?

(2a) I'm not wrong, and aliens do exist here.
(2b) I am wrong, and aliens do not exist here.

Are both of those cases consistent with premise (3)?

No, they are not. By disjunctive inference, we are forced into the (2a) branch. — Srap Tasmaner

Edit: I've tried to summarise my view below so you might not need to address this comment.

Are you saying that "I might be wrong" means "I'm not wrong"? That doesn't seem to be at all consistent with ordinary use or intention. And if it were the case then this would be a valid argument:

I believe with justification that aliens exist
I might be wrong
Therefore, aliens exist

Or are you saying that "I might be wrong" means "either I'm not wrong or I'm wrong"? That would be consistent with (3), and so my claim that we can have knowledge and might be wrong is true.

But is that all it means? You and others have repeatedly rejected the claim that we can have knowledge and might be wrong, and have said that if we're not wrong then we can't be wrong, and so clearly mean by "I might be wrong" something other than "either I'm not wrong or I'm wrong".

And I wonder if this is consistent with ordinary use and intention. Do these two mean the same thing?

I believe that aliens exist but I might be wrong
I believe that aliens exist and either I'm not wrong or I'm wrong

I should add: if you don't like my translation of (2), and would prefer it to be something like "It is possible there are no aliens here, in this world" then, in the presence of a further premise that there are or are not aliens here, this can only be understood as an epistemic possibility -- that is, as a way of saying I don't happen to know. — Srap Tasmaner

You can understand it as epistemic possibility if you like, but using the phrase "I don't happen to know" is misleading given that:

Either this is a valid argument:

1. I believe with justification that aliens exist
2. I might be wrong
3. Therefore, aliens do not exist

Or it is possible that these three are all true:

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

If we accept that it is possible that all three are true then we accept that it is possible to have knowledge (a justified true belief, given by 1 and 3) that might be wrong (given by 2).

"I might be wrong" here means, it is possible that aliens do not exist. That is, there is a possible world in which aliens do not exist. — Srap Tasmaner

Again, when I say "I might be wrong" I'm not saying "I'm actually right but there's a possible world where I'm wrong". I'm saying "I might actually be wrong".

Either this is a valid argument:

1. I believe that aliens exist
2. I might be wrong
3. Therefore, aliens do not exist

Or it is possible that these three are all true:

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

If you believe that the first is a valid argument then I think yours is the belief in error.

So far as I can see your problems have been addressed successfully. — Banno

I don't think they have. I said this:

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

It is possible that these 3 are true:

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

If these three are true then I have a true belief that might be wrong.

@Srap Tasmaner's response was to say that if 3 is true then 2 is false, which entails via modus tollens that if I might be wrong then I am wrong. This is false. I believe that aliens exist but I might be wrong. It does not follow from this that aliens do not exist.

So answer me: can I be wrong?

Modal logic can be used in epistemic terms as well, where ◻ means "certain" and ◇ means "not certainly not".

And also from close to the start of this discussion:

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. In this case I have fallible omniscience.

And I think certainty is only possible if the truth is necessary, so infallible omniscience requires that all truths are necessary.

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.

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.