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? — Michael

I think the logical entailment is wrong. As per my reply...

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. — Isaac

It doesn't seem to matter what we put into that syllogism (right term?), it seems to come out wrong.

After all...

If I'm not {right about this} then I can't be {right about this}
Therefore, if I can be {right about this} then I am {right about this}.

It doesn't seem to matter what we put into that syllogism (right term?), it seems to come out wrong. — Isaac

Because premises of that form are almost always false. You cannot go from "not p" to "not possibly p".

The only time the premise is true is when p is necessarily true.

Consider the conclusion.

◇p → p

If it is possible that I am wrong then I am wrong

Do you believe that this is false?

Because the premise is almost always false. You cannot go from "not p" to "not possibly p". — Michael

If I'm not a cat, I can't possibly be a cat. I could have been, but I cannot actually be at the same time as I'm not.

If I'm not a cat, I can't possibly be a cat. I could have been, but I cannot actually be at the same time as I'm not. — Isaac

You conflate "it is possible that I am wrong" and "I am wrong". These two mean different things:

1. ◇p (it is possible that I am wrong)
2. p (I am wrong)

And so the two similar, but different, claims are:

3. ¬p ∧ ◇p (I am not wrong and it is possible that I am wrong)
4. ¬p ∧ p (I am not wrong and I am wrong)

When you say "but I cannot actually be at the same time" you are referring to 4 being false, not 3. 3 is true if ¬p is not necessarily true.

Are you saying it's possible to be x at the same time as not being x?

Are you saying it's possible to be x at the same time as not being x? — Isaac

These are two different claims:

1. ◇(¬p ∧ p)
2. ¬p ∧ ◇p

Translated:

3. It it is possible that I am both not wrong and wrong
4. I am not wrong and it is possible that I am wrong

1 and 3 are false, 2 and 4 are true.

Returning back to this to explain it:

5. ◇p → p
6. If it is possible that I am wrong then I am wrong

You seem to disagree with 5 and 6. If you disagree with 5 and 6 then you accept that there is some situation where ◇p is true and p is false. In other words, you accept that there is some situation where 2 and 4 are true.

1 and 3 are false, 2 and 4 are true. — Michael

I disagree. 2 and 4 are false at any given time. It is not true that something can not be the case at the same time as it is possible that it's the case. The very fact that it's not the case completely prevents it from also being the case. Since we've just established that something is categorically preventing x from being the case (the fact that it's not the case), we can't at the same time say it's possible x is the case.

2 and 4 are false at any given time. — Isaac

Then you agree with this:

1. ◇p → p
2. If it is possible that I am wrong then I am wrong

Then you agree with this:

1. ◇p → p
2. If it is possible that I am wrong then I am wrong — Michael

I don't see how. Your argument for which that was the conclusion was wrong. One clearly cannot use such an argument in such cases. The example of the cat shows that.

I don't see how. — Isaac

Because you say that ◇p ∧ ¬p can never be true. If you say that ◇p ∧ ¬p can never be true then you say that ◇p ∧ p is always true. If ◇p ∧ p is always true then ◇p → p.

Because you say that ◇p ∧ ¬p can never be true. — Michael

No. I said that ◇p ∧ ¬p cannot be true at the same time. Your implication ◇p ∧ p is for all times.

You don't seem to understand what "possibly" means.

Because of the law of excluded middle, this is true:

1. ◇p ∧ (¬p ⊻ p)
It's possible that I'm wrong and I'm either right or wrong.

This then means that for any given p, one of these is true:

2. ◇p ∧ p
It's possible that I'm wrong and I'm wrong

3. ◇p ∧ ¬p
It's possible that I'm wrong and I'm right

If you say that 3 is false for all p then you are saying that 2 is true for all p. If 2 is true for all p then:

4. ◇p → p

The problem here is with using modal logic terms together with knowledge. The 'possibility' of something is a measure of our uncertainty about it, so once we know x is the case, the possibility P(x)=1 which is the same as just x.

The problem here is with using modal logic terms together with knowledge — Isaac

I'm not even talking about knowledge at this point.

I'm just talking about claims like "I believe that you are American but it's possible that I'm wrong". My claim is true if my belief is right and my claim is true if my belief is wrong.

I don't have any knowledge of your nationality whatsoever.

Remember also that "I am possibly wrong" can also imply "I am possibly right".

The 'possibility' of something is a measure of our uncertainty about it, so once we know x is the case, the possibility P(x)=1 which is the same as just x. — Isaac

So you reject fallibilism and claim that knowledge requires certainty?

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

I think it is "could have been wrong" not "could be wrong" the latter is a contradiction.

I think it is "could have been wrong" not "could be wrong" the latter is a contradiction. — Janus

It's not a contradiction.

p ⊬ □p
¬□p
∴ p → ◇¬p

p being true does not entail that p is necessarily true
p is not necessarily true
Therefore, if p is true then p is possibly false

This is a valid argument.

I agree that any contingent p is not necessarily true. But what does "necessarily true" mean? Does it mean could not be false or could not have been false?

But what does "necessarily true" mean? — Janus

Necessity and possibility are defined by each other:

□p (necessarily p) is equivalent to ¬◇¬p ("not possible that not-p")
◇p (possibly p) is equivalent to ¬□¬p ("not necessarily not-p")

Then just look to the ordinary understanding of the words "possibly" and "necessarily" to inject some actual substance into the meaning.

I don't think you've answered the question.

I don't think you've answered the question. — Janus

I did. That it is necessary that p is true is that it is not possible that p is false.

That it is necessary that p is true is that it is not possible that p is false. — Michael

If p is necessarily true it is not possible that p could have been false. If p is contingently true is it is not possible that p is false, but it is possible that p could have been false.

If p is necessarily true it is not possible that p could have been false. If p is true is it is not possible that p is false, but it is possible that p could have been false. — Janus

I think you should read up on modal logic. Here is how possibility and necessity are actually defined:

□p ≔ ¬◇¬p
◇p ≔ ¬□¬p

You're introducing temporal logic. Temporal logic isn't modal logic.

You're introducing temportal logic. Temporal logic isn't modal logic. — Michael

I thought the very basis of modal logic is that contingent truths, which cannot be false (obviously) could have been false (which means could be false in other possible worlds).

I thought the very basis of modal logic is that contingent truths, which cannot be false (obviously) could have been false (which means could be false in other possible worlds). — Janus

No, it's just saying something like "aliens exist" is not necessarily true and not necessarily false, therefore it's possibly true and possibly false.

No, it's just saying something like "aliens exist" is not necessarily true, therefore it's possibly true and possibly false. — Michael

But that's not different from what I'm saying. So, "aliens may or may not exist" is an epistemological, not an ontological, statement; ontologically speaking aliens either exist or they don't, and if "aliens exist" is true, then "aliens exist" cannot be false (unless conditions changed such as they became extinct), not in this world at least.

If "aliens exist" is true, then "aliens exist" cannot be false — Janus

If "aliens exists" is true then "aliens exist" is not false.

"is not" does not mean "is not possibly". ¬p does not mean ¬◇p.

Again, see the valid modal logic:

p ⊬ □p
¬□p
∴ p → ◇¬p

Both of these are true:

1. "aliens exist" is possible true
2. "aliens exist" is possibly false

One of these is true:
3. "aliens exist" is true
4. "aliens exist" is false

Therefore, either these three are true:

1. "aliens exist" is possible true
2. "aliens exist" is possibly false
3. "aliens exist" is true

Or these three are true:

1. "aliens exist" is possible true
2. "aliens exist" is possibly false
4. "aliens exist" is false