Wednesday, March 26, 2008

Fitch's argument. Where I have gone wrong?

Any metaphysical modal wizards out there that can help me? Here's a line of reasoning that I think is true:
1. In order to wonder whether p is the case, one must be able, at least partially, to understand the content of p.
2. To understand the content of p is to able, at least in some cases, to know that p when it is clearly evident that p.
3. If one didn't know that p when p was clearly evident, then one would not understand that p.

4. Therefore if S understands the content of p, it must be possible for it to be clearly evident that p.
5. If S wonders whether or not "p" where p is a proposition expressible by a declarative sentence, then it must be possible to for S to know that p.
6. If S wonders whether or not p, then (at least in some cases) it is possible that p is true and it is possible that p is false.
7. Therefore it is possible to know a proposition that is false.

Conclusion: Knowledge is not necessarily factive.

Applications

It is possible to know that (p and nobody knows that p).
There are many instances of contingently necessary propositions, for example
Jack doesn't know that (there is extra terrestial plant life and Jack doesn't know that there is extra terrestial plant life).

I'm not interested in hearing people disagreeing with the conclusion, I expect that most people do. But it is possible that most people are wrong. What I want to know is which bit of the argument is wrong. I expect there is a scope fallacy going on, or perhaps an illicit blurring of epistemic and metaphysical necessity.

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6 Comments:

Blogger Brandon said...

Not a metaphysical modal wizard, but I'm willing to toss in an idea.

A question for clarification. I take it your idea is that (4) comes from (2), (5) from (1), (2) and (4). I'm a little unclear about (6), though. I suppose it comes from (1) and (4)?

(5) tells us that if S wonders if p is the case, it is possible that S knows p (namely, under the condition, given (1) and (4), that p is clearly evident, which, I assume, means: p and it is clear that p). This suggests that (7) should be understood in the following way: It is possible that S knows ~p (namely, under the condition that p and it is clear that p). If p is true (which is possible), it would follow that ~p is false, but it would also follow that it is not the case that p is clearly evident; in which case the condition for S knowing p would fail. So (7), unless I'm mistaken, does not imply that knowledge can be non-factive, because it is consistent with these being the only possible states:

if S knows p, p is true
if S knows ~p, ~p is true.

What you'd need to show that knowledge is not necessarily factive is the possibility of one of these:

S knows p and p is false
S knows ~p and ~p is false.

But the argument doesn't require that either of these be possible.

Does that sound right?

1:12 AM  
Blogger bloggin the Question said...

Thanks Brandon, exactly the kind of thing I had in mind. Think of the propositional attitude of wondering whether or not p. Given that S is in this state with regard to p it is both possible for S to know p and it is possible that p is false. Lets talk about this in terms of worlds. WS is the actual world where S wonders whether p, and p is contingent. In W1 S knows that p and in W2 ~p. Neither W1 nor W2 are the actual world, so it seems that in the actual world both are possible. So while, as you say, there is no world where (S knows that p and ~p) it seems that both conjuncts are accessible from WS, though not from each other.
To flesh out with an example. I am about to toss a coin. I wonder whether it will land heads. It is possible that it will land heads. If it lands heads I will know that it lands heads (since it will be evident, or clearly true). So that it is possible that I will know that it will land heads. It is also possible that it will land tails.
suppose it lands tails. This counterfactual is true - if it had been heads I would have known it.
Why am I bothered? Because Fitch's argument is bogus, but is used to defeat verificationism and intuitionist logic, both of which I find intuitively plausible.
Fitch's argument:
Verificationism: All true propositions are knowable
Plausible premise: Some things will never be known.
Therefore there is a proposition which is true (p and no one knows that p)
But which is necessarily unknowable.
The argument relies on the fact that there is no possible world where (p and no one knows that p) is both true and known. But does that mean that (p and no one knows that p) is unknowable? I think not. There are clear procedures by which it could be known, otherwise it doesn't really make sense. Just because p is false, it doesn't mean it is unknowable.
We now know that there was a time when H2O was water, and no one knew that it was. When reading a dectective story we may know that (in the story) Jim's body is lying in the woods and know one knows this fact.
One more example: I could know what it means for a person to be dead. If X was evidently dead then I would know that X was dead. So it is possible for me to know that X is dead just so long as X is a person. I am a person, so it is possible for me to know that I am dead. But of course in a way, it would be impossible for me to know that I am dead, because I would be dead.

12:45 PM  
Blogger Lee Walters said...

In your reply to Brandon you write "it is both possible for S to know p and it is possible that p is false". True but this does not deny the factivity of knowledge. This gives poss (SkP) & Poss (~P).

But to deny the factivity of knowledge you need Poss (SkP & ~P) and this you do not have.

In your coin example I think your reasoning goes wrong. You write

A. It is possible that it will land heads.

B. If it lands heads I will know that it lands heads (since it will be evident, or clearly true).

C. So that it is possible that I will know that it will land heads.

D. It is also possible that it will land tails. suppose it lands tails. This counterfactual is true - if it had been heads I would have known it.

But you are not entitled to C. Rather you can only have "So it is possible that I will know that the coin landed heads." You have illicitly smuggled in knowledge of a future tense claim.

Re your original post. What makes you think 4 is true? (and perhaps 2 depending on how you meant it). Take Goldbach's conjecture. I perfectly well understand this but maybe this will never be clearly evident to me or anyone else. Of course that is not to deny that for those cases where P can be clearly evident, understanding P requires being able to kP when P is clearly evident.

9:25 AM  
Blogger bloggin the Question said...

Thanks Lee, I'll just deal with your last point as I think this is where the meat is. I understand this "every even number is the sum of two primes". I don't know whether it is true or not. (although for every practical purpose I know that it is true). As you say, it may never be clearly evident to anyone. I guess you are thinking of the possibility that it is true, but there is no proof. If it were false (which I sincerely doubt) then there would be a number that was divisible by two but not the sum of two primes. I clearly understand this for divisibility by two since I can just check the last digits. Primes are a little more difficult to understand, but given enough time, I could check any number for primeness. As the numbers got larger, my certainty would go down, since the possibility of error would go up.
If it were true and there was a proof, then in so far as I could understand the proof, I could understand Goldbach's conjecture. I think it would be fair to say that, if I did not understand the proof, then I do not fully understand Goldbach's conjecture.
Now for the real intuitionist versus Classical case: If Goldbach's conjecture was true and there is no proof. The intuitionist would just deny that this is a possibility, saying that this is a case where the law of excluded middle does not apply, so the second conjunct contradicts the first. The challenge to my understanding would not be in "prime" or "even" but in "every" as quantified over numbers. There is no such thing as "every even number". For any set of numbers, however large, I have a proceedure for deciding whether Goldbach's conjecture is true for every number in that set. But there is no set that contains all the even numbers. In this sense, I don't know whether "all the even numbers" refers.
Since Pythagorus, people have been mystical about numbers, and why not? But if the set of all even numbers cannot be defined by its members, then it must be defined by a function. So Goldbach's conjecture can be given as the conditional:
If the number of xs is even, then the xs can be partitioned into mutually exclusive and jointly exhaustive ys and zs such that the number of ys is prime and the number of zs is prime.
This conditional is certain as far as I am concerned. It might not be "true" to a mathematician who required a simple proof. But it is not clear that conditionals have truth values in the same way as simple propositions. It is clearly a good assumption to make, and for any actually countable quantity, then you would be a fool to offer any odds other than 0 : 1. There is no proceedure to settle the bet for the whole set of even numbers, but then there is no such thing.
When you say that Goldbach's conjecture may never be clearly evident to anyone, then I admit that this is a possibility, especially if you require a proof for it to be clearly evident. But all this means is that no one will ever know that Goldbach's conjecture is true. The counterfactual: "If there was a proof of GBC then GBC would be evidently true to a mathematician who discovered the proof" would still hold. My modal problem is that I don't want to let go of this even though many modal logicians would want to claim that the antecedent of a counterfactual can't be necessarily false, and mathematical proofs exist in all worlds or no worlds. I just think this is unrealistic and puts to heavy a constraint on modal reasoning. Thinking of the case of a counterexample to GBC, it seems absurd that it is impossible for me to recognise a counterexample simply because it is impossible that there exists a counterexample, unless there actually is a counterexample. It just sounds like nonsense to me. I would be able to recognise a counterexample if one existed, I have the necessary skills. Possession of these skills does not settle the existential claim.

12:52 PM  
Anonymous Lee W said...

Thanks.

Obviously there is the stuff about intuitionism vs classical logic but lets leave that aside. You make the following claim the part of which seems false

"If it were true and there was a proof, then in so far as I could understand the proof, I could understand Goldbach's conjecture. I think it would be fair to say that, if I did not understand the proof, then I do not fully understand Goldbach's conjecture."

I perfectly well understand Goldbach's conjecture or any other simple claim of that sort - Fermat's last theorem - but there is no chance in hell I can come to understand the proof. Similarly I can understand the claim "my car is blue" without knowing everything about cars or blue(ness).

In any case as I and I think Brandon said (I've only scanned) you are committing a modal fallacy - you need both claims within a single operator to show they are compossible. It is not enough to show each is possible.

See you soonish

10:03 AM  
Blogger bloggin the Question said...

"you need both claims within a single operator to show they are compossible"
Surely I can accept this and accept also that some things are not compossible, but still claim that it is possible to know things such that it is not compossible that I am justified in believing and are true. For example, "the last human is now dead". I know how this sentence would be verified, so it is verifiable by me. But in a sense it is not knowable by me since as you point out the truth of the proposition known is not compossible with the truth of my knowing it. As many people have pointed out, if this non compossibility means that the conjunction is unknowable, then it has the absurd consequence that only propositions that are or will be known are knowable. This is clearly not the correct analysis of the factivity of knowledge or of knowability. One more example: I take this counterfactual to be true:
Had I been at the battle of Hastings I could have known whether or not it Harold was shot in the eye.
This entails:
1. If I (counterfactually) had been at the Battle of Hastings and Harold had been shot in the eye then it would be possible that I knew that Harold had been shot in the eye.
2. If I (counterfacutally) had been at the Battle of Hastings and Harold had not been shot in the eye, then it would be possible that I knew that Harold had not been shot in the eye.
Add this to the premise that no one knows that Harold was shot in the eye, and no one knows that he wasn't, and the absurd factivity of Fitch's argument, we end up with a contradiction.

11:37 AM  

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