Logic of 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]

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.

But your argument does not show that particular conclusion to be false. To show that particular conclusion is false, you would need to outline a counterexample where the proposition “P & P is unknown” is known. Your argument would not show this. Your argument is basically that one can know that a proposition P has a determinate truth value and that its truth value is unknown. Even if successful, at most that can only show that some proposition of the form “P & P is unknown” is true, not that it is known.

There is probably no getting around this: if you want to hold that a proposition of the form “P and ~K(P) ” is known, then you have to reject Axiom B) or T). Application of those axioms to any proposition of this form leads to contradiction.

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.😳

I would dispute that the central issue is defense of agnosticism with respect to O, and I’m certainly not arguing that agnosticism cannot be an appropriate stance with respect to O. It could be depending on specifics, and I’m not disputing that. The central issue is a bit different. The conclusion of DT’s opening argument is that ~O is unknowable. Now, if ~O is false (as it is in your example, since it stipulates that A1 is omniscient and hence O is true), then this conclusion is trivial. Obviously, one cannot know a false proposition, so that’s not interesting. So, the non-trivial, interesting entailment of the conclusion is that even if ~O is true, it is still unknowable.

Now, I disagree with this conclusion and I basically provided two reasons why (I would have a third reason in that I don’t agree with reasonableness of one of the axioms, but I am leaving that aside).

First, DT’s reasoning doesn’t actually go through. I think it would go through if ~O entailed any particular Church proposition(s), but that is not the case because it basically functions as a disjunction over all the Church-type propositions.

Second, the conclusion just does not make sense when we consider the nature of conceivable justificatory conditions for ~O. And this is where I brought up the point about hypothetical examples and evidence. For, if ~O is true and yet unknowable, that more or less reduces to an implication that belief in ~O is unjustifiable. But, there are counterexamples to this, like the number of hairs example. These counterexamples do not need to actually provide conditions under which the evidence justifies belief in ~O; they just need to affirm that it can come down to a matter of evidential weight whether or not belief in ~O is justified (as opposed to being unjustified just in virtue of more basic independent considerations, like the definitions and axioms in play). Twhitehead’s example about truly random future states would also represent a counterexample (since it would entail that at least one Church proposition is true), but that example suffers from an obvious objection: if the future states are truly random, then there seems no good reason to think forward-looking propositions about their outcomes would have determinate truth values. My number of hairs example avoids this objection.

Hope that makes it more clear, maybe.

Originally posted by DeepThought
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 ...[text shortened]... ult will not automatically obey grammatical rules, which is what that type of argument requires.

DT: “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. “

Okay, that’s helpful, as is the belief discussion that follows. I think I’m starting to get it. Thanks.

Originally posted by LemonJello
I would dispute that the central issue is defense of agnosticism with respect to O, and I’m certainly not arguing that agnosticism cannot be an appropriate stance with respect to O. It could be depending on specifics, and I’m not disputing that. The central issue is a bit different. The conclusion of DT’s opening argument is that ~O is unknowable. Now ...[text shortened]... ues. My number of hairs example avoids this objection.

Hope that makes it more clear, maybe.

I would dispute that the central issue is defense of agnosticism with respect to O

Yes, I can see that I read that into a more specific question.

I think it is making it clearer. Thanks.

Originally posted by LemonJello
I would dispute that the central issue is defense of agnosticism with respect to O, and I’m certainly not arguing that agnosticism cannot be an appropriate stance with respect to O. It could be depending on specifics, and I’m not disputing that. The central issue is a bit different. The conclusion of DT’s opening argument is that ~O is unknowable. Now ...[text shortened]... ues. My number of hairs example avoids this objection.

Hope that makes it more clear, maybe.

For, if ~O is true and yet unknowable, that more or less reduces to an implication that belief in ~O is unjustifiable.

So that the general implication is that belief in any P that is (in principle) unknowable is unjustifiable. Clear.

Suppose that the P in question is not “that O”, but “that O is logically possible” [P = Pos(O]. I’m assuming that, then, only Pos(O) would need to be in principle knowable to allow justifiability (even if O itself is not knowable)? In which case, agnosticism is justified or not solely on evidentiary issues pertaining to the possibility.

Originally posted by vistesd
[b] For, if ~O is true and yet unknowable, that more or less reduces to an implication that belief in ~O is unjustifiable.

So that the general implication is that belief in any P that is (in principle) unknowable is unjustifiable. Clear.

Suppose that the P in question is not “that O”, but “that O is logically possible” [P = Pos(O]. I’m assuming t ...[text shortened]... ]agnosticism[/i] is justified or not solely on evidentiary issues pertaining to the possibility.[/b]

So that the general implication is that belief in any P that is (in principle) unknowable is unjustifiable.

Well, probably not strictly, since there are other components to the analysis of knowledge, such as propositional truth and others (and even the analysis of knowledge as justified true belief is a first-order simplification). But, there's no reason here why ~O cannot be true. So, the idea that ~O is unknowable sort of boils down to the idea that belief in ~O is unjustifiable (to first order).

Suppose that the P in question is not “that O”, but “that O is logically possible” [P = Pos(O]. I’m assuming that, then, only Pos(O) would need to be in principle knowable to allow justifiability (even if O itself is not knowable)? In which case, agnosticism is justified or not solely on evidentiary issues pertaining to the possibility.

I'm not sure I understand. By agnosticism here, you are referring to agnosticism with respect to P or with respect to O?

Originally posted by DeepThought
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 ...[text shortened]... ult will not automatically obey grammatical rules, which is what that type of argument requires.

This is a good summary....

Originally posted by LemonJello
I would dispute that the central issue is defense of agnosticism with respect to O, and I’m certainly not arguing that agnosticism cannot be an appropriate stance with respect to O. It could be depending on specifics, and I’m not disputing that. The central issue is a bit different. The conclusion of DT’s opening argument is that ~O is unknowable. Now ...[text shortened]... ues. My number of hairs example avoids this objection.

Hope that makes it more clear, maybe.

The justification argument is interesting. I'm wondering how much higher the justification needs to be before "I believe" can be replaced with "I know". It's helpful to have a proposition to illustrate, I'll go for one which is pretty uncertain - suppose P is "Red Rum [famous race horse from the 70s] will win the race." and some punter believes it enough to put money on. Clearly that they back the horse is not part of the justification, it just demonstrates that they do believe in the horse. Adequate justification may be to do with Red Rum's previous form (a Grand National winner on 3 occasions) and the form of the other horses. However, it can hardly be described as adequate justification for the claim: "I know Red Rum will win this race.", pace arguments concerning the truth of forward looking propositions, since he might fall at the first fence.

Although your hairs example works against any mortal knowing how many hairs are on your head, and may be enough to justify a belief that there is no omniscience, in so far that beliefs require justification, I'm not convinced it's enough to overthrow a not knowable claim in the face of the kind of omnipotent, infallible knowers that our conception of an omniscient entity tends to be. Your argument seems to be that if the proof in the OP were correct then no one would be justified in believing that there is not an omniscient entity. However, I feel that although they could not get that up to a knowledge claim, it wouldn't block their justification in believing that there is no such entity - there being two standards of justification in play. Or maybe I've misunderstood the point you were making.

Edit: Just seen your above two posts, I'm not sure how to modify this in the light of the first. Thanks for your kind words, regarding the second.

Originally posted by DeepThought
The justification argument is interesting. I'm wondering how much higher the justification needs to be before "I believe" can be replaced with "I know". It's helpful to have a proposition to illustrate, I'll go for one which is pretty uncertain - suppose P is "Red Rum [famous race horse from the 70s] will win the race." and some punter believes it enou ...[text shortened]... two standards of justification in play. Or maybe I've misunderstood the point you were making.

Your argument seems to be that if the proof in the OP were correct then no one would be justified in believing that there is not an omniscient entity.

My main counter against your argument is simply that the logic does not actually go through. Based on your terrific summary I just saw, I think you agree with this and we are on the same page there. (I also have some objections against axiom K, but I don't think the logic goes through even with axiom K in play). That's my primary counter-argument.

I would say that number of hairs objection is a secondary counter, probably quite a bit weaker. My basic argument there is that if your argument were correct, then it basically follows -- simply as a matter of nothing more than considerations of definitions and epistemic axioms and whatnot -- that belief in ~O is not justified even if ~O is true. And I think the number of hairs example shows that this just cannot be right. There are examples where we can restrict attention to a specific subset of Church propositions and it has to be a further evidential question (beyond just these definitions and axioms) whether or not at least one member of this subset is true (which, if so, directly entails ~O). Sure, in the number of hairs example, one can always claim that there is some all-knowing god such that all the associated Church propositions in the example are false. And that's always going to be clearly epistemically possible. But, those sorts of considerations are not exempted from evidential scrutiny, either.

Originally posted by LemonJello

So that the general implication is that belief in any P that is (in principle) unknowable is unjustifiable.

Well, probably not strictly, since there are other components to the analysis of knowledge, such as propositional truth and others (and even the analysis of knowledge as justified true belief is a first-order simplification). But, ...[text shortened]... . By agnosticism here, you are referring to agnosticism with respect to P or with respect to O?

I'm not sure I understand. By agnosticism here, you are referring to agnosticism with respect to P or with respect to O?

O. The agnostic does not believe either O or ~O, but is committed to the possibility of O—as opposed to the atheist (assuming that O stands in for “god” of some sort), who is either weakly or strongly committed to ~O. (And I’m thinking a bit of the Batchelor book that I think you mentioned some time back—one can be a weak or a strong agnostic (Batchelor, if I recall correctly, argued for a strong agnosticism, in the Buddhist context).

I am thinking that O, even if true, might not be knowable, and so ~O is also unknowable—but the disjunction (O or ~O) still has truth values. I mean, it doesn’t seem like an incoherent P that (O or ~O), as long as O is meaningfully defined. But, if an unknowable P more or less reduces essentially to unjustifiability, then it seems you have the paradox of a disjunction with valid truth values, that cannot properly be believed in—as a disjunction. (EDIT: Or is it the very unknowability of O that keeps the disjunction "alive" for the agnostic?)

Basically, it seems the agnostic is claiming precisely P = P(O or ~O), and cannot be agnostic toward that proposition.

And I was, likely clumsily, wondering if the seeming paradox can be relieved if P = Pos(O) is in principle knowable, even if the actuality of O is unknowable? (But I now suspect I am just over complicating—and some of that thinking was in response to googlefudge’s claims about claiming knowledge with sufficient probability.)

Originally posted by DeepThought
It doesn't undermine the unknowability of Church propositions because of the argument I made above.

It totally destroys your argument as your conclusion is obviously false.

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.
I don't think you are following me at all.

Originally posted by LemonJello
But your argument does not show that particular conclusion to be false. To show that particular conclusion is false, you would need to outline a counterexample where the proposition “P & P is unknown” is known.

I did.

Your argument would not show this.
Yes it did. If it did not, then explain why not.

Your argument is basically that one can know that a proposition P has a determinate truth value and that its truth value is unknown.
In what way does that differ? If English is significantly different to logic-speak in this case then please translate the sentences in English because I fail to see the difference in my own interpretation.

There is probably no getting around this: if you want to hold that a proposition of the form “P and ~K(P) ” is known, then you have to reject Axiom B) or T).
Once again, I do not need to do any such thing. That is for you and DeepThought to do. All I need to do is show that the conclusion is false. Once that is done, you or DeepThought can sus out the flaw in the argument.

Originally posted by twhitehead
I did.

[b]Your argument would not show this.
Yes it did. If it did not, then explain why not.

Your argument is basically that one can know that a proposition P has a determinate truth value and that its truth value is unknown.
In what way does that differ? If English is significantly different to logic-speak in this case then please tr ...[text shortened]... conclusion is false. Once that is done, you or DeepThought can sus out the flaw in the argument.[/b]

It's not clear to me what it is you are disputing, the ¬K(¬O) result in the OP which was demonstrated not to follow by LJ on the first page, or ¬K(P & ¬K(P))?

Originally posted by DeepThought
It's not clear to me what it is you are disputing, the ¬K(¬O) result in the OP which was demonstrated not to follow by LJ on the first page, or ¬K(P & ¬K(P))?

It is abundantly clear what I am disputing. I have quoted it directly several times .
Here it is again:

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

Now it is entirely possible that I do not understand what that sentence means as it may have non-standard English meaning in it. If so, please explain it in detail.
I am not disputing anything else in your OP, so I do not want to see ¬ or any other such symbols.
I only want to know does my forgetful God satisfy the condition that the above sentence claims is not possible?

Originally posted by twhitehead
I did.

[b]Your argument would not show this.
Yes it did. If it did not, then explain why not.

Your argument is basically that one can know that a proposition P has a determinate truth value and that its truth value is unknown.
In what way does that differ? If English is significantly different to logic-speak in this case then please tr ...[text shortened]... conclusion is false. Once that is done, you or DeepThought can sus out the flaw in the argument.[/b]

You claim to have provided a counterexample to the idea that propositions of the form “P and P is unknown“ represent an unknowable class. So you need to provide an example where a proposition of that form is known. Your argument, again, is that one can know that a proposition has a determinate truth value and that this truth value is unknown. Fine, let’s go through it in steps.

That a proposition P has a determinate truth value translates in logic to (P or ~P). And that this truth value is unknown would translate to (~K(P) & ~K(~P)). So, your argument if successful shows:

K[(P OR ~P) & (~K(P) & ~K(~P))]. (1)

Now, we can ask ourselves: does this provide the counterexample you claim it does?

First, we can check if it simply directly satisfies. So let (P OR ~P) = Q. Does (1) have the form K[Q & ~K(Q)]? Well, it would if it happened to be the case that ~K(P OR ~P) were logically equivalent to (~K(P) & ~K(~P)). That would require ~K(P OR ~P) iff (~K(P) & ~K(~P)). But that clearly does not hold. The <-- direction is particularly problematic: just from the fact that individual disjuncts are not known, it does not follow that their disjunction is not known. So, your example does not directly satisfy as a counterexample of the right type.

Second, we can check if a counterexample of the right type follows from (1) according to valid logic and analytic operations (and let us assume the axiom about knowledge being closed under entailment to give you the best benefit of the doubt). Just consider all of the logical combinations that result from (P OR ~P) & (~K(P) & ~K(~P)). This is logically equivalent to the following:

[P & ~K(P) & ~K(~P)] OR [~P & ~K(P) & ~K(~P)] (2)

But we know some of the atomic components are analytically related. In particular, if P is true, then ~K(~P) follows analytically and if ~K(~P) is false, then it follows analytically that P is false, so P and ~K(~P) rise and fall together; likewise if ~P is true, then ~K(P) follows analytically and if ~K(P) is false, then it follows analytically that ~P is false, so these rise and fall together as well. So (2) can be simplified accordingly, and at the end of the day your (1) is equivalent to:

K[(P & ~K(P)) OR (~P & ~K(~P))]. (3)

Now, the problem is that there is no valid way to get from (3) to the counterexample you want. As DT already correctly pointed out in his assessment of your argument, the knowledge operator does not distribute over a disjunction. So, you cannot make a step here from knowledge of a disjunction to knowledge of an individual disjunct.

So, your argument fails to provide the counterexample that you claim it does. In summary, again, it fails to provide a direct counterexample because from the fact that individual disjuncts are not known, it does not follow that their disjunction is not known. It also fails to entail a counterexample of the right type under further operations because, as DT already pointed out, knowledge does not distribute over a disjunction in the manner your argument would need in order to be successful.

Your argument, if successful, would show that at least one church proposition is true. But it does not show that one is known.