Logic of Omniscience

I was looking at the Stanford Philosophy site's article on Epistemic paradoxes [1]. It contains the following argument apparently due to Alonzo Church. Suppose P is some proposition which is not known, then the proposition "P and P is unknown" is itself unknown. The proof proceeds by assuming the converse:

Let K() be a modal operator meaning that it's argument is known by at least one agent.
P & ¬K(P) - states that there is a true proposition P and it is unknown.

We need the following axioms for the modal operator K():
T) K(P) -> P - If P is known P is true because knowledge is justified true belief.
B) K(P & Q) <-> K(P) & K(Q) - knowledge distributes over conjunction.
K) K(P -> Q) -> K(P) -> K(Q)

1. K(P & ¬K(P)) ⊦K(P) & K(¬K(P)) From B
2. K(¬K(P)) ⊦¬K(P) From T
3. K(P & ¬K(P)) ⊦K(P) & ¬K(P) From 1 and 2
4. ⊦¬K(P&¬K(P)) - 3 is a contradiction so the assumption on line 1 was false.

Because the assumption was dismissed the conclusion is unconditionally true. If a proposition is true but unknown then the combined proposition is unknowable. This isn't a great problem for an omniscience as it will know P and therefore ¬K(P) is false and the omniscience isn't required to "know" things that aren't true.

This left me wondering about omnisciences. Can I justify my agnostic stance on the basis of epistemic logic? Let O be the proposition: "There is an omniscient entity.", if there is an omniscient entity then any given proposition P must be known, in symbols O -> (P -> K(P)). To show that it is unknowable that there is not an omniscience I need to argue that:

(P -> K(P)) -> O.

In the case of a single entity I can't, but suppose a collective is good enough. Then since by assumption any given proposition is known by at least one agent there is a smallest set of agents who are collectively omniscient. However they may not be individually infallible so although collectively they know all things, it's no use as not all of them realise which of their beliefs count as knowledge (a confounded omniscience). However it's enough for me to claim they are an omnipotent entity and write:

O <-> (P -> K(P))

This means that

¬O <-> ¬(P -> K(P))

and it's not hard to show that ¬(P -> K(P)) <-> (P & ¬K(P)) which in combination with axiom K gives us: ⊦¬K(¬O)

In other words it is unknowable that there is not an omniscience. If there is an omniscience (even a confounded composite one) then it must know it exists and is omniscient so the statement is just that one cannot know things that aren't true. If the omniscience does not exist then it is impossible to know that it does not exist.

[1] http://plato.stanford.edu/entries/epistemic-paradoxes/#KnowPar
[2] http://plato.stanford.edu/entries/logic-epistemic/

Originally posted by DeepThought
I was looking at the Stanford Philosophy site's article on Epistemic paradoxes [1]. It contains the following argument apparently due to Alonzo Church. Suppose P is some proposition which is not known, then the proposition "P and P is unknown" is itself unknown. The proof proceeds by assuming the converse:

Let K() be a modal operator meaning that i ...[text shortened]... edu/entries/epistemic-paradoxes/#KnowPar
[2] http://plato.stanford.edu/entries/logic-epistemic/

"...the omniscience isn't required to "know" things that aren't true. "

There is no need to know what things are not true if one knows all that is true.
When people are trained to know what a good 20 dollar bill looks like they are just shown
the real thing, and everything that follows that isn't real, isn't.

Originally posted by DeepThought
Can I justify my agnostic stance on the basis of epistemic logic?

That depends on what your 'agnostic stance' actually is.
If your agnostic stance is:
If the omniscience does not exist then it is impossible to know that it does not exist.
then yes, I think you can justify it, and I don't think you need any fancy logic to do so. I don't think anyone would disagree that there can be many things that we cannot know do not exist.
But that is not the same as the stance of many agnostics who believe it is reasonable to believe that such an omniscience might exist. Not the same thing at all.

I might also add that an 'omniscience' is not an entity capable of thought and action but merely a static object with a record of the entire universe. So we could say for example that 'space-time' is an omniscient entity. In fact any other omniscient entity would have to be an exact copy of 'space-time' or and exact copy plus extra stuff. What the extra stuff is, is irrelevant as it is not part of the 'omniscience'. Alternatively an entity that could, at will, travel to any point in space-time and have a look at it, could be thought of as omniscient, with space-time being its 'knowledge'. But it would be incapable of changing space-time
In addition for any such entity we must invent a new 'time' dimension for it to exist in and act in and in which it is not omniscient. So I would argue that a wholly omniscient entity is logically impossible.

Originally posted by twhitehead
That depends on what your 'agnostic stance' actually is.
If your agnostic stance is:
[b]If the omniscience does not exist then it is impossible to know that it does not exist.
then yes, I think you can justify it, and I don't think you need any fancy logic to do so. I don't think anyone would disagree that there can be many things that we cannot kn ...[text shortened]... it is not omniscient. So I would argue that a wholly omniscient entity is logically impossible.[/b]

To be omniscient it has to know things, it has to have beliefs. A static thing cannot do that. Prove that an omniscient entity is logically impossible. If it is then my argument must be incorrect, because if something is impossible then you know it cannot exist. So if one cannot know that an omniscient entity cannot exist then it cannot be a priori impossible.

Originally posted by DeepThought
To be omniscient it has to know things, it has to have beliefs. A static thing cannot do that.

Please define 'belief' in such a way as it requires it to be dynamic.

Prove that an omniscient entity is logically impossible.
If the entity is, as you claim, dynamic, then does it know its own future, or is its omniscience only knowledge of a subset of reality?
If it knows its own future including what it will believe in the future, then I say it is therefore static, but you say that contradicts the definition of belief. The only way out of a static entity is to posit a dynamic entity in another timeline that is not omniscient in that timeline. But even in that scenario, the entity cannot change the universe as doing so would change its knowledge, a logical impossibility.

I also would like you to address the issue that the entity will necessarily have a copy of the universe and essentially be a universe simulator (or more accurately have a complete space-time that has essentially been precalculated). The question then becomes: why do we need the original? Maybe we are nothing more than beliefs in the mind of God. Certainly we would be indistinguishable from them or its knowledge could not be considered omniscient.

Originally posted by DeepThought
I was looking at the Stanford Philosophy site's article on Epistemic paradoxes [1]. It contains the following argument apparently due to Alonzo Church. Suppose P is some proposition which is not known, then the proposition "P and P is unknown" is itself unknown. The proof proceeds by assuming the converse:

Let K() be a modal operator meaning that i ...[text shortened]... edu/entries/epistemic-paradoxes/#KnowPar
[2] http://plato.stanford.edu/entries/logic-epistemic/

My biggest problem with this from a standpoint of practicality [as opposed to simply having fun
for the sake of it] is that we already take it for granted that we cannot know anything absolutely
about the reality we live in [the problem of hard solipsism]. And as such any 'knowledge' about
that reality we hold must always be provisional and probabilistic in nature. This is what science
and the philosophy underlying science teaches us.

Your argument here is talking about knowledge in epistemic and absolute terms. Something is
only known if it is absolutely 100% guaranteed to be true. But this bears no relation to reality
where nothing is known for certain.

The question that has any bearing on practical everyday reality, the thing that actually matters outside
of intellectual debate, is whether a proposition is PROBABLE or not. And how probable [or otherwise]
the proposition is.

So sure, it might be logically possible for a god of whatever description to exist and have created the
universe.

But if the probability of that being the case in reality is worse than the odds of me going out and winning
the national lottery on three consecutive draws with only one ticket each [without cheating] then we can still
'know' beyond all reasonable doubt that that god doesn't exist and should plan and act accordingly.

So if you are looking to 'justify' being agnostic with respect to the existence to any particular god concepts
then I would regard this argument to be lacking and hollow. Because you are essentially saying that any
probability for the truth value of a proposition other than 1 and 0 is in essence equal.

Originally posted by DeepThought
I was looking at the Stanford Philosophy site's article on Epistemic paradoxes [1]. It contains the following argument apparently due to Alonzo Church. Suppose P is some proposition which is not known, then the proposition "P and P is unknown" is itself unknown. The proof proceeds by assuming the converse:

Let K() be a modal operator meaning that i ...[text shortened]... edu/entries/epistemic-paradoxes/#KnowPar
[2] http://plato.stanford.edu/entries/logic-epistemic/

Interesting line of thought, but I do not think it works. For brevity, let us call a proposition of the form "P and ¬K(P) " a Church proposition. I agree that Church propositions are unknowable. Then, regarding the justification of your agnostic stance on omniscience, you would really be on to something if it happened to be the case that ¬O entailed any particular instance of a Church proposition. But it does not. It entails something weaker: only that some Church proposition is true (for some unknown and presumably unspecifiable P). Although justifying any particular instance of a Church proposition is an exercise in epistemic absurdity, justifying the latter weaker condition is not. So, implicit in your argument is something wrong regarding what is needed to justify, say, the position that ¬O.

Perhaps if we consider a concrete example it will make it more clear. Let us consider the question of how many hairs are on LemonJello's head at the moment of this writing. Now, to justify the position that ¬O, all one needs to justify is the conjunction (a) there is an independent fact of the matter concerning the number of hairs and (b) no entity knows this fact. Part (a) seems straightforward and not problematic. Part (b) is debatable on the basis of evidence: one may argue that the evidence reads as such that no entity is in a position of observation to know how many hairs, one may argue the reverse. So, whether ¬O is justified here is debatable on evidential bases, and that is the point: the axioms of epistemic logic do not rule out the justification of ¬O.

I can see the appeal, though. The negation of O entails that some Church proposition is true, and vice versa. And, regarding each and every Church proposition, application of axiom K will entail ¬K(¬O). So, isn’t this enough to justify agnosticism with respect to O? Seems appealing, but the answer is no. Fact is, no particular Church proposition is needed in pursuit of justifying the negation of O (nor could it be used in such pursuit). The distinction here is that you are applying K to particular instances of the Church proposition; but the negation of O does not entail any of these particular instances; it only entails the condition that at least one of them is true.

Originally posted by googlefudge
My biggest problem with this from a standpoint of practicality [as opposed to simply having fun
for the sake of it] is that we already take it for granted that we cannot know anything absolutely
about the reality we live in [the problem of hard solipsism]. And as such any 'knowledge' about
that reality we hold must always be provisional and probabi ...[text shortened]... t any
probability for the truth value of a proposition other than 1 and 0 is in essence equal.

A god is not necessarily omniscient and an omniscient entity not necessarily a god.

I don't understand your last sentence. Nowhere did I make comments about the probability of propositions. The epistemic modal operator K() has some axioms associated with it that require knowledge to be true (K(P) -> P) and what one can deduce about an agent's knowledge (K, T, and B above, others are possible). It does not include axioms to ensure that justification is infallible.

To be honest, I'd assumed people would argue with notion that:

(1) ⊦ ¬O <-> (P & ¬K(P))
in combination with
(2) K(¬O)
entails that
(3) K(P & ¬K(P))

I don't think it's true that it is impossible to know things except probabilistically. The catch isn't the true or false aspect, but the justification. Essentially, anything contingent except one's own existence, runs into your 'brain in a vat' argument (or problem of solipsism as you put it), but that doesn't stop it being true and the justification is not required to be infallible.

Consider the following sentences: "Newton's theory of gravitation is not in agreement with the empirical evidence." is a true statement because we have evidence that disagrees with it. We cannot guarantee that the sentence "Einstein's theory of gravity is true." is knowledge merely because it hasn't yet been disproved, the justification is faulty. But a sentence such as "At this time there is no empirical evidence that Einstein's theory of general relativity is anything other than rigorously true." can count as knowledge as it is properly justified (by the empirical evidence up to now) and makes a sufficiently careful statement that it gets around arguments about quantum effects because it is a statement about the agreement of the evidence with the theory, and the fragment "At this time" makes it robust against future discoveries. What is more it is robust against "brain in vat" scenarios as even if the evidence is really from a virtual reality it still agrees with the theory.

So I think it is possible to know things which are contingent other than cogito ergo sum with absolute certainty.

Originally posted by LemonJello
Interesting line of thought, but I do not think it works. For brevity, let us call a proposition of the form "P and ¬K(P) " a Church proposition. I agree that Church propositions are unknowable. Then, regarding the justification of your agnostic stance on omniscience, you would really be on to something if it happened to be the case that ¬O entailed an ...[text shortened]... of these particular instances; it only entails the condition that at least one of them is true.

It took me so long to make my previous post you'd written this in the meantime. I was wondering about that in a fairly confused manner in the time since I'd written the OP. I need to think about this and get back later.

Originally posted by DeepThought
I don't understand your last sentence.

Kelly's favourite argument is that everyone holds beliefs and thus there really is no difference between what he believes and what those that believe differently from him believe.
Similarly many agnostics will say that the existence or non-existence of God cannot be known and they live their lives as if the chance of God existing is 50/50.
Googlefudge is saying that even when something cannot be known as in rigorously logically proven, you can still have very very good reasons for thinking it to be the case.
If a real number were picked at random I would be justified in saying 'the number picked is not 42'. I could not rule 42 out logically and cannot strictly be said to know that it is not 42. But to say I am agnostic about it would not really be equivalent to the usual meaning of 'agnostic'. I would bet my life that it is not 42.

Originally posted by DeepThought
It took me so long to make my previous post you'd written this in the meantime. I was wondering about that in a fairly confused manner in the time since I'd written the OP. I need to think about this and get back later.

It is a bit confusing.

Let’s try looking at it this way. For not-O to hold, we just need it to be true that some Church proposition is true. Therefore, not-O is like a disjunction, where the disjuncts are Church propositions. Now, the interesting observation here is that all of those individual disjuncts are themselves unknowable. But, does it follow that not-O is unknowable too? Seems to me the answer is clearly no, since it could be that at least one of the disjuncts is true and that one has evidence sufficient to justify the position that at least one of the disjuncts is true, even if all the disjuncts happen to be individually unknowable. The number of hairs example I raised earlier was supposed to show something like this. In that example, the particular subset of Church propositions that are of interest are “P and not-K(P)” where the P = There are exactly n hairs on LemonJello’s head at the moment (n=1,2,3,….). One could have good reasons to think one of those Church propositions is true, even if Church propositions are categorically unknowable. So, the interesting upshot is that unknowability/unjustifiability is not closed under disjunction: you can have knowledge/justification of a disjunction even where all the individual disjuncts are categorically unknowable.

As an aside, it reminds me of a parallel. Consider a lottery of N tickets (N >> 1) where one and only one ticket wins. Then, of each and every ticket, one can be justified in holding that this particular ticket will not win. But, the same does not extend to the conjunction of all these propositions (which entails that no ticket wins). So, the interesting upshot is that justification is not closed under conjunction.

Originally posted by DeepThought
A god is not necessarily omniscient and an omniscient entity not necessarily a god.

I don't understand your last sentence. Nowhere did I make comments about the probability of propositions. The epistemic modal operator K() has some axioms associated with it that require knowledge to be true (K(P) -> P) and what one can deduce about an agent's knowle ...[text shortened]... e to know things which are contingent other than cogito ergo sum with absolute certainty.

A god is not necessarily omniscient and an omniscient entity not necessarily a god.

Yes.

I don't understand your last sentence. Nowhere did I make comments about the probability of propositions.

Indeed. However, you presented your post as a possible 'justification' for being agnostic.

As the argument [if it holds, a subject I currently take no view on, but note that others are disputing] is
basically along the lines that 'being agnostic is justified' IF 'a proposition cannot be logically and necessarily
proven to be true or false': I respond by saying that this is in essence claiming that all probabilities other than
1 and 0 are equivalent.

I don't for a moment think that you actually believe that, so consider this a Reductio ad absurdum refutation of
the idea that the argument as presented is a justification for agnosticism.

To be honest, I'd assumed people would argue with notion that:

(1) ⊦ ¬O <-> (P & ¬K(P))
in combination with
(2) K(¬O)
entails that
(3) K(P & ¬K(P))

Well, my problems with the argument run deeper than it's formulation. I haven't yet, and probably wont, gone to the
trouble to fully parse the argument as I haven't yet seen a reason I should care. Because even if it is true, my deeper
objections still hold.
And I have no doubt that others more conversant in logical notation will tackle the issue more than adequately.

Consider the following sentences: "Newton's theory of gravitation is not in agreement with the empirical evidence." is a true statement because we have evidence that disagrees with it.

Do we?

How do you know that our 'knowledge' of the 'empirical evidence' is not itself a delusion?

The problem you face is that any point where you make a claim about actual reality [as opposed to knowledge
about knowledge for example] you hit the solipsism problem.

So I think it is possible to know things which are contingent other than cogito ergo sum with absolute certainty.

I believe it could be possible to 'know things' with absolute certainty other than cogito ergo sum... Just not
things about the reality we inhabit.

P is a proposition about reality.
I could [potentially] know "that I believe P" with absolute certainty [and have that belief be justified under whatever criteria
necessary for a knowledge claim, so I thus "know that I know P"].
But while I thus "know that I know P" absolutely. That doesn't mean that "I know P" absolutely, because to know P absolutely
I would have to have overcome the problem of hard solipsism, which I don't believe is logically possible.

Originally posted by LemonJello
It is a bit confusing.

Let’s try looking at it this way. For not-O to hold, we just need it to be true that some Church proposition is true. Therefore, not-O is like a disjunction, where the disjuncts are Church propositions. Now, the interesting observation here is that all of those individual disjuncts are themselves unknowable. But, does it foll ...[text shortened]... ticket wins). So, the interesting upshot is that justification is not closed under conjunction.

As something of a late edit:

Let's just run with the lottery example again. The proposition that some ticket will win acts as basically the disjunction of the individual propositions "Ticket 1 will win", "Ticket 2 will win", etc, up to "Ticket N will win". Now, even if it were impossible to justify any of the individual disjuncts, one is clearly justified in believing the disjunction. So, this is another example that shows that even if unknowability applies to all disjuncts, it may well not apply to the disjunction. When you consider, again, that not-O is basically a disjunction on the Church propositions, this should make you doubt that a blanket stance of agnosticism toward O is justified just on the basis of unknowability of the Church propositions.

Originally posted by googlefudge

A god is not necessarily omniscient and an omniscient entity not necessarily a god.

Yes.

I don't understand your last sentence. Nowhere did I make comments about the probability of propositions.

Indeed. However, you presented your post as a possible 'justification' for being agnostic.

As the argument [if it hold ...[text shortened]... ave to have overcome the problem of hard solipsism, which I don't believe is logically possible.

LJ's point, if I've understood what he is saying, is that what we can deduce from ¬O is that there exists a Church Proposition. But what I've deduced is there exists and what I need is a specific instance. This isn't automatically ruled out since we can know of a proposition without knowing it. I know the proposition "God exists." I don't know if it is true or not so I can "know of" without knowing. One could use a computer to generate all grammatically correct statements less than some length and then look for unknown proposition candidates. The candidates would be things that we do not know the truth of. Such as "There is a tea pot in orbit around the Sun.". The problem is that one cannot rule out the possibility that some agent (other than an omniscient one which we've ruled out by assumption) knows the truth of this proposition. So I can't get between K(¬O) and K(P & ¬K(P)). So the reductio argument fails. Although I'm thinking about that.

The point of Critical Rationalism (Popper's falsification theory) is that evidence against is far stronger than evidence for. We might delude ourselves that incorrect evidence supports a theory, but I don't see a way that we could be mistaken, on the basis of numerous measurements of the precession of Mercury, that Newton's theory could possibly be rigorously true. A "brain in a vat" argument doesn't work here because our theory is about the way the world appears to be rather than the base reality in an implausible scenario.

Your last bit doesn't work. You have "I believe P", then you have "I know I believe P" which somehow turns into "I know I know P". The problem is that you mean two different things by "know". The first has infallible justification (pace arguments about whether we know our own minds) the second has fallible justification. So if you can't be sure that you know P you can't be sure that you know you know P. The fallibility of your justification for claiming knowledge of P passes up the chain. So if you insist that you know P you have to accept that you only know you know P with the same level of fallibility as you know P.

Originally posted by LemonJello
It is a bit confusing.

Let’s try looking at it this way. For not-O to hold, we just need it to be true that some Church proposition is true. Therefore, not-O is like a disjunction, where the disjuncts are Church propositions. Now, the interesting observation here is that all of those individual disjuncts are themselves unknowable. But, does it foll ...[text shortened]... ticket wins). So, the interesting upshot is that justification is not closed under conjunction.

I know the coin toss will come up heads or tails, but not I know that it will come up heads or I know it will come up tails.

I'm not sure about the conjunction part. I'm relying on K(A&B) <-> K(A)&K(B) (the necessity part assumes there is only one agent doing the knowing) and it is a fairly well accepted axiom of epistemic logic. Your example seems to violate that. Suppose the raffle has happened but we don't know which ticket (let's make it a raffle as there does not have to be a winner in a lottery, there does in a raffle). Then it doesn't break under conjunction as our justified belief that ticket 1 did not win is true, our justified belief that ticket 2 did not win is true etc., until we get to the winning ticket when we may well have a justified belief that the ticket did not come up but it did so we didn't know it was not the winning ticket.

I understand the argument (for non-distribution of justification). I just don't see how knowledge can distribute over conjunction of propositions if justification cannot.