The implication of this is that if p is not necessarily true then I can know that p is true even if it is possible that p is not true. — Michael

I'm a bit lost, which is not unusual when we get into more formal logic. I can see two possible meanings for "it ain't necessarily so," 1) it is not logically required that it must be true and 2) there is some doubt about whether it is true. I would have thought that 1 is the proper meaning. If so, then there is not contradiction between "it is true" and "it is not necessarily true."

If so, then there is not contradiction between "it is true" and "it is not necessarily true." — T Clark

I'm not saying that it's a contradiction? I'm just explaining that p → □p is not a valid inference.

I'm not saying that it's a contradiction? I'm just explaining that p → □p is not a valid inference. — Michael

Agreed. Here is where I get confused:

The implication of this is that if p is not necessarily true then I can know that p is true even if it is possible that p is not true. — Michael

I assume we are operating under Justified True Belief rules. Given that, I don't think your statement is true. A true statement would be "p is true, even though it could have been otherwise."

I assume we are operating under Justified True Belief rules. Given that, I don't think your statement is true. A true statement would be "p is true, even though it could have been otherwise." — T Clark

How is the symbolic different to what I used?

Kp ∧ ◇¬p

How is the symbolic different to what I used?

Kp ∧ ◇¬p — Michael

Sorry. I'm not very good with logical symbols. I guess I misunderstood. We can leave it at that.

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

Couldn't we be "wrong" about this conclusion?

Doesn't knowledge presuppose the fallibility of knowing?

Wouldn't "omniscience" consist in (1) knowing the entire set of non-truths and (2) knowing that truths belong to a subset of the set of non-truths (corollary: knowing what we do not know (contra – not knowing what we do not know))?

Couldn't we be "wrong" about this conclusion? — 180 Proof

That's what I'm suggesting. The conclusion is counterintuitive, so something is probably wrong somewhere but I can't see where.

1. ∀p: Kp (premise)
2. ∃p: ¬□p (premise)
3. ∃p: Bp ∧ ◇¬p (from 1 and 2)

It could be that this is a reductio ad absurdum against 1: it is logically impossible to know everything. Or it could be that all truths are necessary. Or it could be that "Bp ∧ ◇¬p" is not the definition of "I could be wrong".

Or maybe it really is the case that it’s possible to be wrong even if you know everything.

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 as the argument above shows.

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.

Or maybe it really is the case that it’s possible to be wrong even if you know everything. — Michael

Yeah, in other words, 'knowing everything' that is true, not-true & unknowable.

And I think certainty is only possible if the truth is necessary, so non-fallible omniscience requires that all truths are necessary. — Michael

I don't think knowledge entails "certainty" (Peirce-Dewey, Popper-Taleb); only logic & mathematics (i.e. tautologous syntactic transformation systems) produce "necessary truths" (Spinoza, Hume, Witty).

Given that knowledge that p entails belief that p it then follows that I could be wrong:

6. Bp ∧ ◇¬p (from 5) — Michael

Is that the right parsing? If you know that p then p is true, after all, and you could not be wrong about p being true, even if p, in some other possible world, might have not been true...

That is, that p might have been false does not imply that you are wrong that p is, as things turned out, true.

The cat is indeed on the mat, you know the cat is on the mat, it is true that the cat is on the mat, you believe that the cat is on the mat, but the cat might have been elsewhere.

Is that the right parsing? If you know that p then p is true, after all, and you could not be wrong about p being true, even if p, in some other possible world, might have not been true... — Banno

Exactly my thoughts. This seems to be an epistemic version of a modal scope fallacy where the possibility that not-p entails some possibility of not-p as a conjunction with knowing-that-p. But this is impossible: while p is possibly false, there are simply no worlds where p is both known and false (these worlds are contradictory, i.e. impossible).

So in virtue of knowing that p, we know that p is actually true, which is perfectly consistent with the fact that the truth of p could've been otherwise.

:up:

Conflating "omniscience" with "god" (which god?) is a non sequitur. — 180 Proof

:up: :100:

Is that the right parsing? If you know that p then p is true, after all, and you could not be wrong about p being true, even if p, in some other possible world, might have not been true...

That is, that p might have been false does not imply that you are wrong that p is, as things turned out, true.

The cat is indeed on the mat, you know the cat is on the mat, it is true that the cat is on the mat, you believe that the cat is on the mat, but the cat might have been elsewhere. — Banno

Well, there's a difference between these two:

1. I believe p but I am wrong
2. I believe p but I could be wrong

How do we formulate these in symbolic logic? This is my attempt:

1. Bp ∧ ¬p
2. Bp ∧ ◇¬p

It makes sense to me. "I believe p but I could be wrong" means "I believe p and it's possible that not p".

If not these then what?

Exactly my thoughts. This seems to be an epistemic version of a modal scope fallacy where the possibility that not-p entails some possibility of not-p as a conjunction with knowing-that-p. But this is impossible: while p is possibly false, there are simply no worlds where p is both known and false (these worlds are contradictory, i.e. impossible). — Kuro

Again, from the OP:

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

This is a valid argument. To deny the conclusion you must deny one of the premises.

Note that I'm not saying:

◇(Kp ∧ ¬p)

From here:

1. It is possible that I know everything and am wrong about something
2. I know everything and it is possible that I am wrong about something

The former is false but the latter seems possible as the argument above shows.

If omni science includes omni physics? Would it also include omni metaphysics? Omni evil? Omni good? Omni error. Does omniscience suggest knowledge of all possible errors?

Does omniscience suggest knowledge of all possible errors? — universeness

@Rocco Rosano quoted a phrase in his first post which says: ‘‘An omniscient being needs no logic and no mathematics,’’
It is so interesting, indeed. Trying to answer your question I guess omniscience doesn't suggest knowledge of all possible errors because it is not a logical entity. When we debate about errors inside logic we have to start with the premise that the thing is logical itself, so we conclude it makes "errors" because it doesn't fit what we consider as "logic" or "success", etc...
But omniscience cannot fit those parameters because it isn't logical since the beginning.

I agree, but I think the inevitable landing zone for all omni posits is called paradox.
I mean paradox as a purely logical proposal (or of propositional logic). Therefore we end up in the zone of neither true or false. To me this is like the measurement problem in quantum mechanics. Only when you make the measurement and the 'wave collapses,' will you get an outcome which is 'true' or 'false' this is what we call reality as far as I can project what that word refers to.
Logic comes from Plato's logos, yes? and that reference has serious religious connotations as do the omni references but to me, they just don't hold up to any definition I can hold as part of 'reality', not now, not in the past and not in the future. You can poke too many holes in the omni posits, especially when they refer to deities. If one is omniscient then how can its creation be flawed? Unless it was deliberate and therefore Christopher Hitchen's claim would be true. You were made sick and commanded to heal yourself on the threat of eternal punishment if you don't. What kind of omniscient monster would create rules like that?
Omnipotent must follow from omniscience if knowledge is power. But you land on paradox straight away with omnipotent. Can an omnipotent create something more powerful that it?
Can an omniscient create new knowledge it never knew before?
Can an omnipresence expand?
Can an omnibenevolent do evil?
They all end in paradox!
So all that can then be claimed is as you and Rocco suggests all omnis cannot be approached via logic or propositional logic to be more precise but then the omnis must be metaphysical or metalogical, 'after logic' or 'beyond logic' so does that mean we must employ a label like omnimetaphysical or omnimetascience or omnimetascient or omnisupernatural? How about omniwoowoo!

Agreed with all your argument and post :up:

If one is omniscient then how can its creation be flawed? — universeness

A selfish one. Or at least someone or something who makes us remember that it is over of all criteria and goes beyond to all the limitations of possibilities. Writing this answer I am acknowledging that this has no sense. I guess this is what a omnipotent looks like in the infinite universe of metaphysics or quantum mechanics.
"We", thus, their "creation" are flawed because the omnipotent entity would not create something clever and mightier than itself. But we end up in a paradox again as you perfectly explain previously.
I see a paradox here because his perfection is senseless is he pretends to elaborate just flawed creations.

How about omniwoowoo! — universeness

:eyes: :sparkle: This one goes beyond to any category of reason!

Agreed with all your argument and post — javi2541997

Well, in that case, I hope I am correct. :up:

A selfish one. Or at least someone or something who makes us remember that it is over of all criteria and goes beyond to all the limitations of possibilities. Writing this answer I am acknowledging that this has no sense. I guess this is what a omnipotent looks like in the infinite universe of metaphysics or quantum mechanics — javi2541997

Its been a long time theistic claim that humans cannot approach god or understand god using human intellect or inquiry. Only belief is required and obedience. The epitome of a nefarious sociopolitical tenet, if ever there was one. I remain a fan of the Greek 'cosmos,' the universe is knowable and I refute and reject the omnis (perhaps, apart from omniwoowoo or maybe just omniwoo) and their supernatural connotations, its nothing but omniwoo!

Some of you are reading too much into the word "omniscience". This isn't a discussion about God or anything like it. This is just a discussion about the below argument and how to interpret the conclusion.

Kp ≔ x knows p
Bp ≔ x believes p

1. Kp ⊨ Bp (premise)
2. ∀p: Kp (premise)
3. ∃p: ¬□p (premise)
4. ∃p: Bp ∧ ◇¬p (from 1, 2, and 3)

I am not saying that there is anything that satisfies the second premise, I'm just looking at what would follow were there to be something that knows everything.

I would appreciate it if you could keep on topic and not discuss unrelated issues.

Ok, but could we have more lay terms please.
So does 1. translate into:
1. For all values of p, Kp (does his mean K is a function or process performed on values of p? Are you using the colon to indicate a ratio?)
2. 3p (3 multiplied by p) :(ratio?) ¬(not) □ (what does this symbol indicate?) p
3. I got bored looking up maths symbols at this stage. I know ^ means to the power of but what does B represent and what are you using the small rhombus shape for?
I taught maths to higher grade level at secondary schools before going full computing and I studied maths pure and applied up to 2nd year uni but I will be rusty to say the least. Perhaps you need someone like @jgill to answer. You would have to explain the maths symbology you use a little more for me to make any useful contribution.

I don't want to turn this discussion into a lesson on symbolic logic so I'll just refer you to these:

List of logic symbols
Modal logic

As I mentioned in the previous post (as an edit, so you may need to refresh to see it), Kp means "x knows p" and Bp means "x believes p" ("x" being some hypothetical person and "p" being some state of affairs).

Ok I see I am better to keep my Computing hat on as you are using ^ as the logical AND operator etc.
I will do my own translation and perhaps comment later if I come up with any points I have not already touched on in what I have already typed.

EDIT: Ok, thanks for the edits you made above it makes things a little clearer.

I would appreciate it if you could keep on topic and not discuss unrelated issues — Michael

Defending my own typings, I thought I was on topic. Your title is 'The paradox of omniscience,' which I quite clearly responded to. The term omniscience strongly relates to theism and the symbology in your OP is cryptic to say the least. Are there many people on TPF with quite advanced mathematical knowledge? Do you not want to open your OP up to as many members as possible? I assume you do, based on your attempt to provide further explanation in your edits.

the symbology in your OP is cryptic to say the least. — universeness

It really isn't. It's very basic modal logic.

It really isn't. It's very basic modal logic — Michael

Once you explain it or a person, understands it based on their own research then sure, it becomes basic, but not until then. A good teacher does not make inaccurate assumptions regarding the previous knowledge of those they are attempting to teach or even create a discussion with. Best to explain as clearly and fully as you can.

'all physical theories, their mathematical expressions apart ought to lend themselves to so simple a description 'that even a child could understand them.'
Albert Einstein.

I would apply this to your descriptions of 'basic modal logic,' if you want to encourage as wide a variety of responses as possible.

A good teacher... — universeness

I'm not here to teach. In fact I specifically posted this to get answers from people more knowledgeable than me because, as I said, the conclusion seems counterintuitive.

Best to explain as clearly and fully as you can. — universeness

That's the point of symbolic logic. Ordinary language is often vague and ambiguous and open to misunderstanding. Symbolic logic allows us to clarify our terms and better make sense of inferences.

I'm not here to teach. — Michael

I typed

those they are attempting to teach oreven create a discussion with. — universeness

Symbolic logical allows us to clarify our terms and better make sense of inferences. — Michael

I agree, once all who can or want to contribute, understand the symbology used in the logic you are presenting. My complaint towards you is a minor one.