Logic of Omniscience

Originally posted by DeepThought
If there is a God and God is omniscient and the future is not set then yes, I suppose I am saying that.

That actually puts a burden on us because our choices would be ours, which is as I do
believe it to be.

Originally posted by DeepThought
I did not mention a Creator God in the OP,

I know that. But the OP does try to show that nobody, creator God or otherwise, could know that a particular proposition exists but is unknown. Since you invoked a creator God as presumably knowing certain rules about the universe, I can similarly invoke him as knowing certain rules about the universe, including possibly creating a proposition that is unknown.

If the result of a coin toss were truly random then the proposition "The coin will come up heads." does not have a definite truth value and no agent can know it.
Then we are agreed that if things are truly random an omniscient entity cannot know the future. Another side effect of truly random things is that there cannot exist anything outside the universe.

A forgetful Creator God would not be omniscient as then there would be propositions that are true but not known to it.
But a forgetful Creator God could presumably know of a proposition, not know its truth value, and also know that nobody else knows the propositions value thus violating the OP argument.

If propositions about the future are not either true or false then even an omniscience would not know them, because there is nothing to know. I do not believe this undermines the notion of omniscience.
Neither do I. It does however undermine the notion of an omniscience that knows the future, which to be honest is the only type of omniscience worth happening. I can get the other kind using cctv.

Why would an omniscient entity necessarily have knowledge about its own future if its future is undetermined, in whatever space it lives in?
Again, it may not. I am fairly sure I specifically said that 'if an omniscient entity could know its own future then it would be static'. I didn't say it was a necessity.

Incidentally what is normal to time is space. You seem to have disjoint spaces, coordinates in disjoint spaces are not normal to each other, or related in any way.
Correct, I was in error. My claim remains the same. A dimension or space that is disjoint from another necessarily has a static relationship between the two. Each space is a 'fact' to the other.

Originally posted by twhitehead
I know that. But the OP does try to show that nobody, creator God or otherwise, could know that a particular proposition exists but is unknown. Since you invoked a creator God as presumably knowing certain rules about the universe, I can similarly invoke him as knowing certain rules about the universe, including possibly creating a proposition that is unk ...[text shortened]... ther necessarily has a static relationship between the two. Each space is a 'fact' to the other.

In the OP I attempted to show that if an omniscience did not exist then it would not be knowable that it did not exist. If on the other hand it did then it would be knowable, if only to the omniscient entity.

Your forgetful creator God argument works in the sense that then there would be two Church type propositions one of which is true and rule out an omniscience. But to be any use you would have to show that there is an agent who knows that some proposition is either true or false and that no one else knows the truth value. In other words you'd have to show that a creator God is necessarily forgetful.

I see no reason why metaphysical randomness should rule out the existence of things not in the universe.

Originally posted by DeepThought
In other words you'd have to show that a creator God is necessarily forgetful.

I really can't see where you got that from. I am trying to show that the following may not be true:

Suppose P is some proposition which is not known, then the proposition "P and P is unknown" is itself unknown.

I am claiming that a forgetful creator God could know "P and P is unknown".
If a forgetful creator God could violate the claim then it is neither necessary for a creator God to exist or for a creator God to be forgetful for your claim to have been proven false. The claim is claimed to be universally logically true regardless of the existence or non-existence of forgetful creator Gods.
If a creator God could know that nobody else knows P, and itself know P and subsequently forget P without forgetting that it formerly knew P, then there is necessarily a flaw in your proof of the claim.

I see no reason why metaphysical randomness should rule out the existence of things not in the universe.
As stated before, if our universe is disjoint from some other existence, then the relationship between us is necessarily static (by definition). Therefore, our future exists in the larger existence and any claim that it does not exist is false. Therefore, true randomness of the form that the future doesn't exist, also doesn't exist.

Originally posted by twhitehead
I really can't see where you got that from. I am trying to show that the following may not be true:

Suppose P is some proposition which is not known, then the proposition "P and P is unknown" is itself unknown.

I am claiming that a forgetful creator God could know "P and P is unknown".
If a forgetful creator God could violate the claim ...[text shortened]... false. Therefore, true randomness of the form that the future doesn't exist, also doesn't exist.

So you are arguing with Alonzo Church's proof that for every unknown proposition there is a proposition that is unknowable. We have:

Axiom 1) K(P) -> P
Axiom 2) K(P & Q) -> K(P) & K(Q)

Axiom 1 follows because of the definition of knowledge. Axiom 2 states that knowledge distributes over conjunction.

Assumption: K(P & ¬K(P))

From axiom 2 this gives us: K(P) & K(¬K(P))

From axiom 1 we have: K(P) & ¬K(P)

Since this is a contradiction the assumption was wrong and we unconditionally have that: ¬K(P & ¬K(P))

There is no possibility that the proposition P & ¬K(P) if true can be known. My argument in the OP failed in the attempt to connect this to an omniscience. To invalidate it you would need to argue against axiom 2, which I don't think is viable. Although time does change the argument I don't think it does in a way that helps you. Since your hypothetical creator God knows ¬K(P) and ¬K(¬P) all you can construct is:

K((P & ¬K(P)) v (¬P & ¬K(¬P)))

and you run into the same problem I did in my argument in the OP. Namely that you cannot get from this to:

K(P & ¬K(P)) v K(¬P & ¬K(¬P))

which you need to show that a Church proposition is knowable.

Regarding your second point, if the future is set then you cannot construct an argument ruling out an omniscience based on metaphysical randomness.

Originally posted by DeepThought
So you are arguing with Alonzo Church's proof that for every unknown proposition there is a proposition that is unknowable. We have:

Axiom 1) K(P) -> P
Axiom 2) K(P & Q) -> K(P) & K(Q)

Axiom 1 follows because of the definition of knowledge. Axiom 2 states that knowledge distributes over conjunction.

Assumption: K(P & ¬K(P))

From axiom 2 th ...[text shortened]... hen you cannot construct an argument ruling out an omniscience based on metaphysical randomness.

Okay, DT—after the contradiction, the symbolic logic gets beyond me. Can you explain in lay terms why you can’t connect this to an omniscience (which, as I recall, had to do with the distribution of justification versus the distribution of knowledge—which I also did not really comprehend)?

EDIT: I am re-reading LJ's argument & your replies . . .

EDIT 2: Thus far, I am wondering why the possibility that O remains unjustified. And all you need for Pos(O) is a justified belief in the possibility, which I do not see as leading to a contradiction (but i might be missing much).

Originally posted by DeepThought
So you are arguing with Alonzo Church's proof that for every unknown proposition there is a proposition that is unknowable.

No. You have changed the wording. I am arguing against the OP claim that:

Suppose P is some proposition which is not known, then the proposition "P and P is unknown" is itself unknown.

To invalidate it you would need to argue against axiom 2,
No, actually, I do not. Showing that the conclusion is false is more than sufficient. If the conclusion is false, the argument is flawed. I have no need to find the flaw, or even understand the argument.

Regarding your second point, if the future is set then you cannot construct an argument ruling out an omniscience based on metaphysical randomness.
And?
My second point was demonstrating that if the future is not set, then nothing can exist outside the universe. Or are you referring to a point from another post?

Also, if the future is not set then reality is being created continuously universe wide, and although it might be hard to rule out an omniscience via pure logic, it seems kind of obvious that there cannot be one as it would have to be both part of the universe and simultaneously recording every event that takes place and storing it thus requiring a universe-wide monitoring system plus universe capacity of storage.

I think relativity might also throw a spanner in the works as there is no set point at which reality is being created universally, which would violate omniscience.

Originally posted by twhitehead
No. You have changed the wording. I am arguing against the OP claim that:

Suppose P is some proposition which is not known, then the proposition "P and P is unknown" is itself unknown.

[b]To invalidate it you would need to argue against axiom 2,
No, actually, I do not. Showing that the conclusion is false is more than sufficient. I ...[text shortened]... is no set point at which reality is being created universally, which would violate omniscience.[/b]

¬K(P&¬K(P)) = the proposition P and P is unknown is unknown.

My second point is solid. For your argument to work you need knowledge to distribute over disjunction and it doesn't.

LJ: “Moreover, if we visit hypothetical examples, it is clear that not-O is justifiable and knowable in principle, given the right sorts of evidential access and conditions.”

I guess what I’m missing—since the point is to defend agnosticism with regard to O*—is why the following would lead to any contradiction:

Suppose there are two agents (or two classes of agents), viz: A1 is omniscient. A2 is not omniscient.

It seems logically possible that [O & ~K(O)] for all A2.

Isn’t that sufficient to keep agnosticism alive in the logical realm? That is, O is justifiable though not-knowable in principle?

However, [Pos(O)] would need to be knowable in principle?

Sorry for all the late edits: I'm grappling as I go, and my mind feels like creamed corn today . . . .


_____________________________________________

* Although I realize that your OP was more restrictive, in terms of using Church Props.😳

Originally posted by DeepThought
My second point is solid.

Please be clearer. You keep referring to numbered points that are not numbered.
What is your second point?

For your argument to work you need knowledge to distribute over disjunction and it doesn't.
None of my arguments require that. If you disagree then please be specific about which argument and why you think it requires that.

Originally posted by vistesd
Okay, DT—after the contradiction, the symbolic logic gets beyond me. Can you explain in lay terms why you can’t connect this to an omniscience (which, as I recall, had to do with the distribution of justification versus the distribution of knowledge—which I also did not really comprehend)?

EDIT: I am re-reading LJ's argument & your replies . . .

EDIT ...[text shortened]... the possibility, which I do not see as leading to a contradiction (but i might be missing much).

I've defined omniscience sufficiently widely that I can say that there is an omniscient entity if and only if all propositions that are true are known. The converse of this is that if and only if there is not an omniscience then there exists some proposition which is both true and unknown.

We have:

(1) ¬O <-> ∃P (P & ¬K(P))

The statement there exists is the same as a string of disjunctions. Propositions are sentences in some language and therefore form a countable set, so we can enumerate them, P1, P2, P3, ... (*)

This means we can write:

(2) ¬O <-> (P1 & ¬K(P1)) or (P2 & ¬K(P2)) or (P3 & ¬K(P3)) or ...

Now, hoping to derive a contradiction I looked at K(¬O), it is known that there is not an omniscience:

(3) K(¬O) <-> K(∃P P & ¬K(P))

which is the same as:

(4) K(¬O) <-> K[(P1 & ¬K(P1)) or (P2 & ¬K(P2)) or (P3 & ¬K(P3)) or ...]

For clarity I'll replace P1 & ¬K(P1) with C1 and so forth, so we have:

(5) K(¬O) <-> K(C1 or C2 or C3 or ...)

Now, there are two problems, one is that in getting this far as I've assumed an axiom called K which assumes that knowledge is preserved by entailment. It is not clear that that is the case. However that is not the serious flaw, the killer is that my argument relies on this:

(6) K(C1 or C2 or C3 or ...) -> K(C1) or K(C2) or K(C3) or ...

but knowledge won't distribute in that way. It is not true that if I know C1 or C2, I either know C1 or know C2, because I may not know which one is true.

My point in reply to twhitehead (which is the post you replied to) is as follows:

Suppose a proposition is known to be either true or false but it's truth value is not known. If the proposition is true it is not known to be true, and if false then it's converse is true and not known to be. So we have that:

(P & ¬K(P)) or (¬P & ¬K(¬P))

Let's make C = P & ¬K(P), and C* = ¬P & ¬K(¬P). I'm using * to mean dual as there is a sort of duality between the two propositions. So we have:

C or C*

and given that we can claim to know this we have:

K(C or C*)

but we cannot get from this to:

K(C) or K(C*)

which twhitehead needs for his objection to work (although he just denied he was claiming that so I don't know what he's arguing).

Regarding edit 2, I think you are right as believing a proposition does not entail said proposition is true. So where K(P) -> P of necessity we have ¬(B(P) -> P), in other words a belief isn't stopped from being a belief by the object of the belief not existing or not having the property that it is believed to have. Compare a belief and a knowledge claim that "all swans are white.", the agent who believes that is mistaken due to the existence of Cygnus atratus, however it would not stop their belief from being a belief. Their "knowledge" that "All swans are white." would never have been knowledge because the proposition is required to be true.

The logic for the proof of the unknowability of some propositions requires that K(P) -> P, but for belief we do not have B(P) (meaning some agent believes P) implying P. So we cannot get between:

B(P) & B(¬B(P))

and

B(P) & ¬B(P).

Note that B(P) means that any agent believes the proposition, so that B(¬B(P)) is not incoherent, agent 1 can believe P and agent 2 not believe P and not believe that anyone else could believe it. In the case of a single agent they might be mad.

(*) In case anyone thinks that you can use a diagonalisation argument to dispute this, it won't work as one cannot construct a grammatical sentence by making a list of grammatical sentences and choosing the first letter so that it is different from the first letter of the first sentence, the second letter different from the second letter of the second sentence and so forth. The result will not automatically obey grammatical rules, which is what that type of argument requires.

Originally posted by twhitehead
Please be clearer. You keep referring to numbered points that are not numbered.
What is your second point?

[b]For your argument to work you need knowledge to distribute over disjunction and it doesn't.
None of my arguments require that. If you disagree then please be specific about which argument and why you think it requires that.[/b]

I'm talking about your forgetful god argument.

You seem to be basing your argument on the following scenario: An agent knows P, but forgets it. However what they do know is that no other agent knows P and that P is knowable in the sense that it has a definite truth value. You then claim that such an agent invalidates my opening argument. Is this a fair statement of your argument?

Originally posted by DeepThought
I'm talking about your forgetful god argument.

You seem to be basing your argument on the following scenario: An agent knows P, but forgets it. However what they do know is that no other agent knows P and that P is knowable in the sense that it has a definite truth value. You then claim that such an agent invalidates my opening argument. Is this a fair statement of your argument?

Yes.

Originally posted by vistesd
LJ: “Moreover, if we visit hypothetical examples, it is clear that not-O is justifiable and knowable in principle, given the right sorts of evidential access and conditions.”

I guess what I’m missing—since the point is to defend agnosticism with regard to O*—is why the following would lead to any contradiction:

Suppose there are two agents (or two cla ...[text shortened]... ___

* Although I realize that your OP was more restrictive, in terms of using Church Props.😳

That sounds reasonable to me, but I need to think about it. We can make our modal operator specific to a particular agent, they usually use a subscript, but that's possible, but painful in these forums, so I'd suggest something like: K(a, P) where a is the agent and P is the proposition. That would mean that agent a knows proposition P.

LJ's argument seems to me to be that, in principle, one could construct a disjunction over Church propositions and argue that one must be true. twhitehead's argument seems to be an attempt to do that.

As I understand it his argument relies on God (or whoever) being selectively omniscient. That is to say at any one time does not necessarily know the truth of any given proposition, but is able to find out or forget.

Going the other way doesn't seem to produce any problem. Suppose there is a proposition currently unknown to any agent, I'll use Russel's tea pot as an example. The proposition would be "There is a tea pot in orbit about the sun.". Before 1st April 2016 it is unknown that there is such a thing in orbit, but astronomers spot it on the first of April. Then since it was unknown before the 1st of April the associated Church Proposition was unknowable. However after that date the Church proposition is no longer true. But since false propositions are unknowable in a trivial way it's knowability hasn't changed.

Again using Russel's teapot imagine that God knew it was there but deliberately forgot. Then before the time of forgetting the Church proposition isn't true (and so is unknowable for that reason) and afterwards is true but still unknowable because we don't know about the teapot and God has deliberately forgotten.

So I don't see any immediate problem with the concept, it doesn't seem to do any damage to the knowability of the Church propositions, just their truth value.

Originally posted by twhitehead
Yes.

Yes, that would undermine omniscience but you have to show that such an agent exists. It doesn't undermine the unknowability of Church propositions because of the argument I made above. It also doesn't undermine the possibility of selective omniscience as a sufficiently powerful entity could find out the truth again on a later date.