2006-03-22

Quine's Paradox

We all (being reasonable persons and not fanatics) are trapped by Quine's Paradox: namely, to believe a statement p is to believe that p is true, so I believe that each of my beliefs is true. Yet I also believe that some of my beliefs (I know not which) will turn out to be false if and when tested. (I believe I left my glasses in my bedroom today, but beliefs of this sort have turned out to be false often enough...)

So:

I believe that each of my beliefs is true;
I believe that some of my beliefs are false.

Saith Quine: "I for one had hoped for better from reasonable persons."

17 comments:

david said...

Allow degrees of certainty between 0.0 and 1.0 and the apparent paradox vanishes in a puff of fuzzy logic.

John Cowan said...

Alas, not so. I was entirely certain that my glasses were in my bedroom, but I was also entirely wrong -- they were on my desk the whole time. Or at least, if the certainty was not 1.0, it was as certain a belief as any belief I hold.

No, the trouble comes in with beliefs about beliefs.

david said...

Your certainty that your glasses were in the bedroom was, say, 0.98, but then, presumably, your certainty that that particular belief was wrong would have been around 0.02.

What am I missing? To me, this seems like a phoney paradox built on a rounding error. If you limit yourself to boolean logic and round any non-zero value up to 1.0 (i.e. if it's not false, it must be true), then you can end up with a 1.0 certainty that your glasses are in the bedroom and a 1.0 certainty that your belief might be wrong. As long as you allow probabilities between 0.0 and 1.0, however, I cannot see a paradox.

Joseph Scott said...

When does a belief become knowledge? Using the lost glasses example, I believe that they are on the bed. If I then go and look on the bed and find them there then I now know that they were on the bed, it is no longer belief because I know it. If it turns out they weren't on the bed and were in fact on the desk then my belief that they were on the bed turned out to be wrong and I now know they were on the desk. At no point did I belief that they were on the desk, I skipped belief and went directly to knowledge.

Looking at the definitions of words is helpful here. One definition of believe is 'to accept as true'. As in accepting that your belief that the glasses were on the bed was true. One definition of know is 'to perceive directly'. Once you found the glasses on the desk belief wasn't involved, you found out first hand.

James Ahlschwede said...

Fixed:
I believe that each of my beliefs is true;
I believe that some of my beliefs may be false.

There may come a time when all my beliefs are perfectly aligned with reality; it just seems somewhat unlikely. :)

Dean Edwards said...

Beliefs are a sure sign of insanity.

Mark said...

I'll believe it when I see the Godel numbering. 8-)

John Cowan said...

James: I at least believe that some of my beliefs are false, not merely that they may be. I believe of each belief that it may (turn out to) be false.

Dean: how many times a day do you check that you still have your housekeys with you before you rely on your belief that they are still in your pocket? Most beliefs are trivial and true.

Arnt Richard Johansen said...

I believe that the problem here lies, at least in part, with intensionality. You believe that some of your beliefs are false, but you don't know which. That is different from believeing that some of your beliefs, namely a, b, and c, are false. Now that would have been self-contradictory! So, there are different senses of "some", and you have to figure out which are which.

Dan Nugent said...

At least it works for First Order Logic!

Dean Edwards said...

John, I either know or don't know if I have my house keys. If I'm not sure then I check my pockets.

John Cowan said...

Surety isn't knowledge. Knowledge involves truth (as well as other conditions): we can't know something that's false.

More precisely: if we say "I know p" and then it turns out that p is false, we retract our earlier claim: "I thought I knew p, but it turns out that I didn't."

Ardeshir Mehta said...

The statement:

"to believe a statement p is to believe that p is true, so I believe that each of my beliefs is true. Yet I also believe that some of my beliefs (I know not which) will turn out to be false if and when tested. (I believe I left my glasses in my bedroom today, but beliefs of this sort have turned out to be false often enough...)"

... is ill-phrased. In the second part the word "belief" should have been replaced by "suspicion" and "believe" by "suspect", thus:

"to believe a statement p is to believe that p is true, so I believe that each of my beliefs is true. Yet I also believe that some of my SUSPICIONS (I know not which) will turn out to be false if and when tested. (I SUSPECT I left my glasses in my bedroom today, but SUSPICIONS of this sort have turned out to be false often enough...)"

This is more accurate, and the paradox is also thereby removed!

John Cowan said...

Alas, no. The issue runs considerably deeper than mere suspicions, to very strong beliefs indeed, complete with factitious evidence.

For example, I believed (and thought I had verified) that I put my medicine back in my backpack on Saturday night. Close inspection on Sunday showed that it wasn't there: that was a belief that turned out to be false, based on a verification that turned out to be defective.

Would I have said "I know my medicine is in my backpack" on Saturday? Yes. Would I say so now? Clearly not.

Ardeshir Mehta said...

John Cowan said...

'Alas, no. The issue runs considerably deeper than mere suspicions, to very strong beliefs indeed, complete with factitious evidence.

'For example, I believed (and thought I had verified) that I put my medicine back in my backpack on Saturday night. Close inspection on Sunday showed that it wasn't there: that was a belief that turned out to be false, based on a verification that turned out to be defective.

'Would I have said "I know my medicine is in my backpack" on Saturday? Yes. Would I say so now? Clearly not.'

But then on Saturday you did not *really* know that your medicine was in your backpack, did you? You only *thought* you knew. Which is equivalent to saying that you *strongly supsected* - or even *VERY strongly suspected* - that you knew!

People at one time "knew" that the Earth was flat. That was not *genuine* knowledge, though, was it. It was a mere suspicion.

And one more thing. "To believe a thing is to *believe* it to be true", yes; but a *belief* that a thing is true does not necessarily *make* it true! In other words, even the word "belief" is used in this case in the sense of a VERY strong suspicion.

John Cowan said...

You're quite right, Ardeshir: you only think you disagree with me. It's true that I didn't know but only thought (and said) that I knew. It's true that people didn't (couldn't) know the Earth was flat, though they would have assured you that they knew it. Furthermore, it's true that not all beliefs are knowledge.

But we only know what is knowledge and what is false belief retroactively. For each belief, I have a meta-belief that the belief is true (and a meta-meta-belief, and so on); this meta-belief has some strength, perhaps not a strong as the original belief.

I also have a meta-archi-belief, however, that each of my beliefs are true (for some strength), including the metas and meta-metas. In other words, I have no beliefs for which my meta-belief is that the belief is false.

Lastly, experience teaches me to hold another meta-archi-belief which says that some of my beliefs (but not the metas or meta-metas) are false. These two meta-archi-beliefs are contradictory, but I am perfectly justified in holding both of them.

Anonymous said...

My belief that P counts as knowledge that P iff:

1. P
2. I believe that P
3. If 1, then 2
4. If not 1, then not 2