Logic of Omniscience

Originally posted by DeepThought
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 o ...[text shortened]... & ¬K(P) From 1 and 2
4. ⊦¬K(P&¬K(P)) - 3 is a contradiction so the assumption on line 1 was false.

Ok, now my head hurts.

Thanks for that.

Originally posted by DeepThought
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 ...[text shortened]... don't see how knowledge can distribute over conjunction of propositions if justification cannot.

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.

The lottery example does not violate this axiom. The lottery example stipulates that there is one and only one winner. In the example, the proposition “It is not the case that Ticket n will win” is justifiably believed for all n. But, since one of these propositions is false, not all of these propositions are known. So, that the conjunction is not known does not violate the axiom, since not all of the conjuncts are known. The example is just designed to show that justification is not closed under conjunction; it does not threaten the same against knowledge.

As an aside, I find it rather dubious that knowledge is closed under entailment, but I’m not sure I am prepared or motivated to argue that here.

I just don't see how knowledge can distribute over conjunction of propositions if justification cannot.

There are two directions to the distribution you are referring to. The lottery argument attacks the <-- direction: that from justification of each conjunct, it does not necessarily follow that the conjunction is justified. Again, this can be shown without threatening a similar claim about knowledge because a conjunct can be justified and yet false. The lottery example does not purport to show that justification does not distribute in the other direction -->.

At any rate, I’ll have to give your opening argument more thought. It is quite interesting.

Originally posted by LemonJello

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.

The lottery example does not violate this axiom. The lottery example stipulates that there is o ...[text shortened]...

At any rate, I’ll have to give your opening argument more thought. It is quite interesting.

Regarding the entailment point, I thought of that as the weak aspect of the argument. If an agent knows A and knows A -> B then one can imagine possible worlds where they fail to believe B. But suppose (A -> B) & K(A) then if A -> B is not known to them they have no reason to start believing B. As far as I could tell from what the author of the Stanford article was saying it's okay provided that (A -> B) is a tautology, in other words a necessary truth, which seems tempting to accept. If I know that not all swans are white then I'm certainly justified in my claim to know that some swans are not white.

The problem seems to be at the belief formation stage, since the justification for our agents claim to know B is that it is a logical consequence of A, which is known to them, and truth is preserved in classical logic. So it seems to depend on the nature of belief. If it is true that what our brains form beliefs about are strings of symbols (or sounds) then it looks problematic because the agent isn't necessarily going to start believing proposition B. But if the internal storage of beliefs is something else that our brains then map to propositions in order to be able to discuss them it looks reasonable as a belief can then map to more than one proposition, so then my belief that not all swans are white would be the same belief that some swans are not white.

Originally posted by DeepThought
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 ...[text shortened]... u have to accept that you only know you know P with the same level of fallibility as you know P.

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".

As I struggle to wade through all this (enjoyably, I add!), I am reminded here of Wittgenstein in On Certainty. The statement “I know I believe P” adds nothing to the statement “I believe P”, except, perhaps, the implication that there is/was doubt (about whether or not I actually believe). Wittgenstein's point, as I understand it, is the cognitive dissonance that would be implied by an assertion that one believes, but has doubt that s/he believes; if one says "I believe", we do not normally question one's certainty about that believing itself. If one adds the apparent redundancy of "I know" to that belief statement, that strange locution is more likely to raise doubt in our minds about that person's confidence in their own statement of belief.
___________________________________________

EDIT:

I’m not sure exactly what googlefudge is arguing (and I'm tired and it's late), and maybe this is way off the mark, but—

To say “I know with a 95% probability that P is true” is just to say that “There is a 95% probability that P is true”—and I know that fact. That is not the same as asserting: “P is true.” (One of my early statistic professors would have given such an assertion an “F”.)

Originally posted by vistesd
[b]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".

As I struggle to wade through all this (enjoyably, I add!), I am reminded here of Wittgenstein in On Certainty. The statement “I know I believe P” adds nothing to the statement “I believe P”, except, perh ...[text shortened]... “P is true.” (One of my early statistic professors would have given such an assertion an “F”.)[/b]

You actually enjoyed that?

Originally posted by vistesd
[b]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".

As I struggle to wade through all this (enjoyably, I add!), I am reminded here of Wittgenstein in On Certainty. The statement “I know I believe P” adds nothing to the statement “I believe P”, except, perh ...[text shortened]... “P is true.” (One of my early statistic professors would have given such an assertion an “F”.)[/b]

Close, but not quite. 🙂

My point was that you could potentially know things about your thoughts [belief,
knowledge claims etc] absolutely. But still not know anything about reality with
absolute certainty. As the latter requires a solution to the hard solipsism problem.
[I am not even certain that logical impossibilities are impossible]

So you could [potentially] know with certainty that Newtonian gravity is not supported
by what you observe about reality.
You cannot however, know with certainty that those observations are real.
Therefore you cannot know with certainty that Newtonian gravity is not supported by
reality.

The distinction being that you could [potentially] have perfect knowledge about supposed
facts that reside inside your brain and be able to use those facts to derive other facts and
have absolute certainty that those resultant facts are logically consistent with the original facts.
But you can never have perfect knowledge that the original facts are actually true. You cannot
know that with absolute certainty.

[Side note: Can god get around the hard solipsism problem other than 'by definition'. If god
cannot get around the hard solipsism problem, then god cannot be omniscient. ?]

I'm 'pressing' this point because it seems to me that a foundation of the argument in the OP is
that IF an omniscient being is not logically impossible or logically required [supposedly making the
probability of such a thing existing 0 or 1] then agnosticism about such a beings existence is
justified.
I disagree, as all our 'knowledge' that DT accepts we have about reality is probabilistic and inductive
at root, and thus not absolutely certain. Then it must be true that you can be gnostic [claim to know]
things for which the probability of truth is >0 and <1 .

The reductio ad absurdum was to point out the implication of the original post that one cannot know
anything that isn't logically certain with 100% probability one way or the other.

Which leads to my answer to the question in the OP " Can I justify my agnostic stance on the basis
of epistemic logic?" to be no.

Originally posted by googlefudge
Close, but not quite. 🙂

My point was that you could potentially know things about your thoughts [belief,
knowledge claims etc] absolutely. But still not know anything about reality with
absolute certainty. As the latter requires a solution to the hard solipsism problem.
[I am not even certain that logical impossibilities are impossible]

So yo ...[text shortened]... stion in the OP " Can I justify my agnostic stance on the basis
of epistemic logic?" to be no.

I can accept that all knowledge about reality (as opposed to of our beliefs about reality) is probabilistic. But then is there some non-subjective, non-arbitrary threshold for P(x) < 1 where one can claim “knowledge” (other than p(x) itself)? 0.90? 0.95? 0.99? Certainly I can be “more sure” with the increased probability (confidence level).

The common definition of knowledge in epistemology is “justified true belief”—that is, a belief that is justified (not just a guess), and happens to be true.

It seems to me that you are redefining knowledge, but unless there is some objective “threshold probability” that would be generally recognized and accepted (even if such a threshold itself might vary, say, across disciplines), then to say that “I know” is just to say that I have a subjective “sureness”. That might be true (the Pyrrhonian Skeptics thought so, and argued pretty cogently).

But, either way, you seem to arguing for a redefinition of “knowledge” to mean something like either:

(1) “a justified belief that is true with a probability of [some P < 1]”, or

(2) “a justified belief that is believed to be true with [some P < 1]”.

As I say, I think, on analysis, the first really reduces to “a justified belief that has a [some P <1] of being true”.

[It occurred to me after my first post, that the “justified belief” [JB] could itself be a probabilistic statement, itself held to be true with [some P< 1].]

In any event, the so-called “Black Swan” problem will always remain (as [some P(~x) > 0]). I don’t see “agnosticism” as just being a statement that P(x) = .50. So, to belabor the point, either (a) there is some objective, non-arbitrary “probability threshold” beyond which agnosticism is an invalid position; or (b) every statement (about reality, as you say) can be said to be “agnostic to [some P > 0]" (which is just the flip-side of saying “gnostic to [some P <1 ]” ). One can argue about “reasonableness”—but that is subject to all the foregoing, with respect to reasonableness instead of knowledge.

At some point, I am reminded of the following quite by W (in On Certainty):

“I am sitting with a philosopher in the garden; he says again and again 'I know that that’s a tree', pointing to a tree that is near us. Someone else arrives and hears this, and I tell him: 'This fellow isn’t insane. We are only doing philosophy.”

(Note: I’m an untrained philosophical “hack”, who just enjoys this stuff.)

_________________________________________________

I am still thinking about your approach here in terms of W’s about the world being the world “of facts, not things”—where by “fact” is meant a kind of “that” statement. For example, if I say, “There is a chair” (pointing), I am essentially saying “that a chair is [/i]there[/i]”. That is, we unable to observe “bare things” absent the relationships (in this example, spatial)—what I recall W as saying “how they hang together” (and I am going to have to revisit this).

Anyway, that’s where this conversation is stimulating my own inquiry.

Originally posted by vistesd
I can accept that all knowledge about reality (as opposed to of our beliefs about reality) is probabilistic. But then is there some non-subjective, non-arbitrary threshold for P(x) < 1 where one can claim “knowledge” (other than p(x) itself)? 0.90? 0.95? 0.99? Certainly I can be “more sure” with the increased probability (confidence level).

The comm ...[text shortened]... ave to revisit this).

Anyway, that’s where this conversation is stimulating my own inquiry.

I was taking the probability aspect to be part of the justification. The statement is either true or not true and one either believes it or one does not. So a statement such as "Paracetamol (aka acetaminophen) reduces pain in patients with headaches with 95% confidence.", means that the researchers believe that paracetamol reduces headache pain and their justification for the statement is statistical.

I'm still responding with some dissonance to googlefudge's approach of using extreme skepticism to argue against religious agnosticism. My current thinking is that if the proposition is wrapped up in such a way that it takes account of the various uncertainties in the world then it can count as knowledge even in the face of a Cartesian demon. Roughly the difference between the statements "I have a cup of coffee" and "I have a cup of what appears to be coffee.".

As a complete aside I found a quite nice joke in one of the Stanford articles: Rene Descartes was in a bar and had almost finished his drink. The barman asked if he would like another. "I think not", said he, and was never seen again!

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/

There is a piece of scripture that touches upon this.

Romans 1:19-21New International Version (NIV)

19 since what may be known about God is plain to them, because God has made it plain to them. 20 For since the creation of the world God’s invisible qualities—his eternal power and divine nature—have been clearly seen, being understood from what has been made, so that people are without excuse.

21 For although they knew God, they neither glorified him as God nor gave thanks to him, but their thinking became futile and their foolish hearts were darkened.

Originally posted by DeepThought
I was taking the probability aspect to be part of the justification. The statement is either true or not true and one either believes it or one does not. So a statement such as "Paracetamol (aka acetaminophen) reduces pain in patients with headaches with 95% confidence.", means that the researchers believe that paracetamol reduces headache pain and the ...[text shortened]... . The barman asked if he would like another. "I think not", said he, and was never seen again!

I was taking the probability aspect to be part of the justification. The statement is either true or not true and one either believes it or one does not. So a statement such as "Paracetamol (aka acetaminophen) reduces pain in patients with headaches with 95% confidence.", means that the researchers believe that paracetamol reduces headache pain and their justification for the statement is statistical.

Thanks. And my question to Google is, “At what ‘probability threshold’ can one claim to know, on the basis of that justification?” And I brought up the so-called “Black Swan” problem for the same reason (I think) that you mentioned falsification.

But the flip side of believing some statement (S) to some probability (confidence) < 1, is always (it seems to me) P(~S) > 0.* So the question becomes, at what point is one no longer agnostic under conditions of such non-certainty? [A common Bayesian technique seems to be, based on my limited reading, to define “agnosticism” fairly strictly as P = .5—to start the iterative process. But, for epistemological considerations, I think that would be ludicrously strict. Only reason why I mentioned it.]

I am more interested in Pyrrhonian skepticism than Cartesian skepticism. But the question of when a claim to know is justified is also the question of when doubt is justified. Here is an interesting essay on Wittgenstein’s On Certainty, in which W seems to claim that there are certain “hinge propositions” whose very indubitability renders both statements of knowledge and skepticism incoherent. (I’m still reading it, and it’s been a long time since I read OC). He mentions the cogito.

http://www.philosophy.ed.ac.uk/people/full-academic/documents/WittOnScepticism.pdf

“So Wittgenstein is offering a picture of the structure of reasons such that that which we are most certain of cannot be properly claimed as known or, for that matter, properly doubted.”

You know that I used to disagree with you about agnosticism. I am now more inclined to think that I was wrong (but I'm not certain! 😉 ).

________________________________________

* My second ( think) statistics professor (in a statistics for economics course) really did grade an "F" for anyone who said that their statistical analysis "proved" the alternative hypothesis, even at a .99 confidence level. Anyone who committed such a "sin", after being warned in the very first class, had to rewrite their paper to get a passing grade.

Originally posted by DeepThought
Regarding the entailment point, I thought of that as the weak aspect of the argument. If an agent knows A and knows A -> B then one can imagine possible worlds where they fail to believe B. But suppose (A -> B) & K(A) then if A -> B is not known to them they have no reason to start believing B. As far as I could tell from what the author of the Stanfo ...[text shortened]... n my belief that not all swans are white would be the same belief that some swans are not white.

Yes, claims regarding closure of knowledge under various types of entailment seem to me to rely implicitly on other non-trivial claims regarding belief topology, which makes me think it is quite dubious they can function satisfactorily when taken axiomatically. But I will set aside any of those objections I might have against the axioms themselves in play.

I have reviewed again the opening argument. My objection would be the same. If the negation of O happened to entail any particular Church proposition, then the argument would have force. However, the negation of O entails only the proposition that at least one Church proposition is true (again, similar to how the truth of a disjunction only tells you that at least one of the disjuncts is true). This distinction is critical to the argument. For, although it follows straightforwardly according to the axioms in play that Church propositions are unknowable (since they represent conjunctions that lead to contradiction when distributed over), the same reasoning does not transfer to this entailment of O (which does not represent a conjunction). Thus, the application of axiom K to the actual entailments of not-O (which are not Church propositions) does not lead to any clear problem for the knowability of not-O. (Moreover, any particular Church proposition does entail the negation of O; but when you apply axiom K to this, it just leads to vacuous truth, with nothing substantive regarding the knowability of not-O. )

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. Since not-O basically takes the form of a disjunction over the Church propositions, a typical attack line for justifying not-O could be to restrict attention to a particular subset of relevantly related Church propositions and then argue for the proposition that at least one member of this subset is true.

How about this for simple?

Logical paradoxes do not negate omniscience

YouTube

Originally posted by vistesd
[b]I was taking the probability aspect to be part of the justification. The statement is either true or not true and one either believes it or one does not. So a statement such as "Paracetamol (aka acetaminophen) reduces pain in patients with headaches with 95% confidence.", means that the researchers believe that paracetamol reduces headache pain and their ...[text shortened]... , after being warned in the very first class, had to rewrite their paper to get a passing grade.

Thanks for the link, I'll give it a read.

Imagine a universe in which the future is not real until it happens. The universe is also finite and it is possible to know everything about said universe up to the present time and an entity exists that does know everything up to the present. The future is truly random and the entity knows this to be true.
The entity will not know what will happen the next day.
Is the entity omniscient? If not, then I think it violates the OP argument in that we know there are unknowns that cannot be known and that no omniscient entity exists.

Originally posted by twhitehead
Imagine a universe in which the future is not real until it happens. The universe is also finite and it is possible to know everything about said universe up to the present time and an entity exists that does know everything up to the present. The future is truly random and the entity knows this to be true.
The entity will not know what will happen the n ...[text shortened]... nt in that we know there are unknowns that cannot be known and that no omniscient entity exists.

But then propositions such as "There will be a sea battle tomorrow." are not true and so are not knowledge.