If I believe that I don't know if A or B is the case, does it follow that I believe that it's possible that either A or B are the case?

An epistemic sense has to do with individuals' beliefs, right? — Terrapin Station

Of course! I said that from the very beginning. It is relative to the body of beliefs of a person at some point in time.

If I believe that I don't know if A or B is the case, does it follow that I believe that it's possible that either A or B are the case? — Terrapin Station

Yes, if you are using 'possible' in the epistemic sense; no, if you are using it differently. If I don't know whether or no my girlfriend is home, that doesn't entail that I believe that she is some sort of a Schroedinger Cat living in an unsettled state of both possibly being or not being at home. Mere epistemic possibilities don't logically entail other sorts of possibilities.

Yes, if you are using 'possible' in the epistemic sense; — Pierre-Normand

Well, the epistemic sense is about what I believe. So if I believe that it's not in fact possible for A or B to have obtained--I believe that it's only possible for one of them to have obtained and the other was always impossible (even before one obtained), then how does my saying "I don't know" imply a belief that both are possible? I rather explcitly believe that one was never possible; I simply don't know which one was possible.

Maybe post when you're not so busy, so that it doesn't take so long to get a simple yes or no answer? — Terrapin Station

Really? You can't wait ten minutes? Should I take an appointment with your personal secretary, next time? I am allowed more than one bathroom breaks in a day?

Really? You can't wait ten minutes? Should I take an appointment with your personal secretary, next time? I am allowed more than one bathroom breaks in a day? — Pierre-Normand

Don't you ever have time available where one wouldn't have to wait ten minutes for a response in a conversation?

Don't you ever have time available where one wouldn't have to wait ten minutes for a response in a conversation? — Terrapin Station

You can't possibly be serous. Just listen to yourself. I am patiently replying to your rapid fire quibbles over a simple notion (epistemic possibility) that you could have gotten acquainted with easily on your own with a Google search. We didn't come to a prior agreement that this exchange, which has spanned a few days, ought suddenly to become be a no-break-allowed conversation.

Getting to the won't answer a simple question point. Nice.

Well, the epistemic sense is about what I believe. So if I believe that it's not in fact possible for A or B to have obtained--I believe that it's only possible for one of them to have obtained and the other was always impossible (even before one obtained), then how does my saying "I don't know" imply a belief that both are possible? I rather explcitly believe that one was never possible; I simply don't know which one was possible. — Terrapin Station

How would you describe this in modal logic?

"It is not possible for A or B to have obtained".
¬◇(A ∨ B)

Now, I'm not particularly knowledgable of modal logic, but does De Morgan's theorem apply here, and like this?

¬◇(A ∨ B) ⊢ ¬◇A ∧ ¬◇B

Are you simply asking my personal view, outside of the context of what I was talking about? If so, then yes, certainly. Keep in mind that I'm not a determinist — Terrapin Station

So it's possible that I have a brother.

I don't have a brother.

What does this say about your claim that it's possible that I have a brother? Does it make sense to say that your claim/belief was illusory? I don't think so.

Getting to the won't answer a simple question point. Nice. — Terrapin Station

But I did answer it, didn't I? You're just complaining that it took more than 10 minutes, due to my being busy answering other posts of yours!

That looks okay to me, but I can't guarantee I'm not missing something.

How about getting back to the more substantive comment? I need to leave in less than half an hour, and I'll be in and out prior to that.

So it's possible that I have a brother.

I don't have a brother.

What does this say about your claim that it's possible that I have a brother? Does it make sense to say that your claim/belief was illusory? I don't think so. — Michael

The comments about epistemic possibility being illusory were in a different context--supposing that determinism were true, etc. That's why I qualified that response when I made it.

Well, the epistemic sense is about what I believe. So if I believe that it's not in fact possible for A or B to have obtained--I believe that it's only possible for one of them to have obtained and the other was always impossible (even before one obtained), then how does my saying "I don't know" imply a belief that both are possible? I rather explcitly believe that one was never possible; I simply don't know which one was possible. — Terrapin Station

You are freely mixing up metaphysical and epistemic possibility operators in this paragraph, so the question seems ill posed. I already mentioned that epistemic possibilities don't entail other sorts of possibilities. If you believe that only one among two 'possibilities' can be actual, due to the conjunction of them being nomologically impossible, then this could be formalized thus: EP(◇A xor ◇B), where EP is the epistemic possibility operator, '◇' is a different sort of possibility operator.

That looks okay to me, but I can't guarantee I'm not missing something. — Terrapin Station

You must be missing something. Let's say that A is "I will pick up the cup" and B is "I will not pick up the cup". So ¬◇A ∧ ¬◇B means "it is not possible that I will pick up the cup and it's not possible that I will not pick up the cup". I don't think you meant this.

I already mentioned that epistemic possibilities don't entail other sorts of possibilities — Pierre-Normand

But epistemic possibility necessarily has to do with the individual's beliefs re possibility, no?

You must be missing something. Let's say that A is "I will pick up the cup" and B is "I will not pick up the cup". So ¬◇A ∧ ¬◇B means "it is not possible that I will pick up the cup and it's not possible that I will not pick up the cup". I don't think you meant this. — Michael

Right--for one I misread the "and" as an "or"

Right--for one I misread the "and" as an "or" — Terrapin Station

So unless De Morgan's theorem doesn't apply this way to modal logic, to avoid this consequence you have to abandon your claim "It is not possible for A or B to have obtained".

In other words, it's either not possible for A or it's not possible for B to have obtained.

But epistemic possibility necessarily has to do with the individual's beliefs re possibility, no? — Terrapin Station

What epistemic possibility "has to do with" is rational people's consistent sets of beliefs, and what they are about is specific propositions in relation to those prior sets of beliefs. The content of those propositions can concern ordinary empirical claims, claims of temporal modalities, alethic modalities, mataphysical modalities, or even other epistemic modalities (or anything else that one can intelligibly believe or disbelieve).

For instance, I may claim that, for all I know, for all Donald Trump knows, he will be remembered at the greatest president ever. This could be formalized thus: EPPN(EPDT(P)), where EPPN is the epistemic possibility operator relativized to my own present epistemic perspective, EPDT is the same operator relativized to Trump's epistemic perspective, and P is the proposition that he will be remembered at the greatest president ever.

In other words, it's either not possible for A or it's not possible for B to have obtained. — Terrapin Station

So ¬◇A ∨ ¬◇B. Which, again using De Morgan's theorem, just entails ¬◇(A ∧ B), but then nobody's arguing that.

But presumably what you would accept is ◇A ⊻ ◇B (either A is possible or B is possible). Doesn't that then entail ◇(A ⊻ B)? And so, again, you must abandon your earlier claim that "It is not possible for [either] A or B to have obtained"

¬◇A ∨ ¬◇B. Which, again using De Morgan's theorem, just entails ¬◇(A ∧ B), — Michael

It rather seems to me that applying De Morgan's law to ¬◇A ∨ ¬◇B yields ¬(◇A ∧ ◇B).
If either A or B are impossible, then it's not the case that they're both possible.

It rather seems to me that applying De Morgan's law to ¬◇A ∨ ¬◇B yields ¬(◇A ∧ ◇B). — Pierre-Normand

Might be clearer if I change ¬◇ to IMPOSSIBLE. That gives us:

IMPOSSIBLE(A) ∨ IMPOSSIBLE(B) which entails IMPOSSIBLE(A ∧ B).

Although ¬(◇A ∧ ◇B) also follows.

If either A or B are impossible, then it's not the case that they're both possible.

And also that it's not possible that they're both the case.

But which of these is a proper application of De Morgan's law, I'm not sure. Maybe you're right on that. Or maybe "the negation of a conjunction is the disjunction of the negations; and
the negation of a disjunction is the conjunction of the negations" is broad enough to cover both?

I think "they're both possible" is ambiguous here. Do you mean "A is possible and B is possible" or "either A or B is possible"? I'm saying the second, not the first. — Michael

I was paraphrasing you formula and interpreting '¬◇' as "impossible". I meant "both impossible" also as a paraphrase for "¬◇A ∨ ¬◇B".

Looks like you had this page open for a while before replying. I edited my post a few minutes ago (you weren't showing as online so I didn't think it confusing to re-write it).

Looks like you had this page open for a while before replying. I edited my post a few minutes ago. — Michael

No trouble. I must however retract what I just said. I hadn't actually said "both impossible" originally, but rather "not the case that they're both possible". It is the latter that is the correct paraphrase from the right hand side of my revision of your formula, and it is the proposition that I take to follow from the left hand side thought the correct application of De Morgan's law.

Your own "IMPOSSIBLE(A ∧ B)" also follows indirectly, but it follows from modal logic rather than propositional logic.

From ¬(◇A ∧ ◇B) you can indeed infer ¬◇(A ∧ B) since if

(1) it is not the case that ((there is a possible world where A) and (there is a possible world where B)),

then

(2) there is no possible world where both A and B.

What epistemic possibility "has to do with" is rational people's consistent sets of beliefs — Pierre-Normand

So when you don't want to deal with some particular individual's beliefs, you just claim that that is not a rational person.

Who gets to decide who is rational?

So ¬◇A ∨ ¬◇B. Which, again using De Morgan's theorem, just entails ¬◇(A ∧ B), but then nobody's arguing that. — Michael

It's being argued if epistemic possibility is supposed to amount to both A and B being epistemically possible at some point in time.

But presumably what you would accept is ◇A ∨ ◇B — Michael

A determinist would not accept that. I noted this explicitly already.

So when you don't want to deal with some particular individual's beliefs, you just claim that that is not a rational person. — Terrapin Station

Everyone has inconsistent beliefs but there ought to be a consistent core. Who said I wan't prepared to deal with it on a personal level? We were discussing rules of inference that are valid for epistemic logic, presently. For those rules to apply to the belief systems, or epistemic perspectives, of real persons, there must be a minimal presumption of rationality. When a person believes that P, and also that Q isn't logically consistent with P, she isn't normally prepared to accept that, for all she knows, Q. If she is nevertheless prepared to accept Q as an epistemic possibility, then she must thereby either acknowledge that she doesn't really believe that P for sure, or be prepared to revise her belief that P and Q are inconsistent.

If a person shows no tendency to revise some of her beliefs when they are shown to be mutually inconsistent, then there is no saying what it is that, for all she knows, might be true, and epistemic modal logic breaks down as a means for interpreting her. (Look up Donald Davidson on 'radical interpretation', the 'constitutive ideal or rationality' and also the 'principle of charity'.)

We were discussing rules of inference that are valid for epistemic logic, — Pierre-Normand

We were? That's not what I was discussing. I had said simply said this, which you never addressed:

"So if I believe that it's not in fact possible for A or B to have obtained--I believe that it's only possible for one of them to have obtained and the other was always impossible (even before one obtained), then how does my saying 'I don't know' imply a belief that both are possible? I rather explcitly believe that one was never possible; I simply don't know which one was possible. "

"Then how does my saying 'I don't know' imply a belief that both are possible" wasn't a rhetorical question.

What's the answer to that? That the person isn't rational?