Beliefs: True, False, or Probable?

@Mr B,

Mr B wrote:

In The Problems of Philosophy, Russell discusses the principle of induction: roughly, if A and B have been found associated a great number of times, and never dissociated, the it is probable A will be associated with B in a fresh instance.

I believe, for example, the sun will probably rise tomorrow, since it has risen every other day. But I can't say in what the truth of this belief subsists. Tomorrow will come, and the sun will either rise or not rise - in won't probably rise.

Russell also says that experience can neither prove nor disprove one's belief in the principle of induction. But I don't understand his reasons, and his argument loses me at that point. This is completely because, as I stated, I don't see what the truth of the principle has to do with experience at all. What could one possibly experience to validate the belief that a A will probably be associated with B? As before, either A will or will not be associated with B in experience.

Russell is not concerned with whether it is true that the Sun will rise tomorrow. He, no doubt, believes it is true just as firmly as you or I believe it is true. For him, the issue is whether we know it is true, and whether we would be justified in thinking we know it is true. The problem is about knowledge and justification, not about truth. And the question here is whether induction (by enumeration, the only kind of inductive justification Russell considers) is enough justification for it to be true that we know that the Sun will rise tomorrow. And Russell's answer, like Hume's before him, is no, it isn't. This raises the question why Russell thinks the justification by enumerative induction is not enough, and also raises the question of what it is he means by "knowledge". For, if he means by knowledge, certainty without the possibility of error, then clearly enumerative justification is not sufficient for knowledge, since we can have all the inductive justification we can have and it still would be possible that the Sun not rise tomorrow. But, then, that is in the nature of inductive justification. Inductive justification cannot produce certainty. So to fault inductive justification for not producing certainty (if that is what Russell demand knowledge be, namely certain) is like faulting a baseball player for not scoring a touchdown. Baseball is not about scoring touchdowns, and induction is not about scoring certainty.

@kennethamy,

I must still be confused. Or perhaps he has a bad way of making his point. Russell seems to say that principle of induction cannot be proven by experience. (Roughly, if A and B have been found associated a great number of times, and never dissociated, then it is probable A will be associated with B in a fresh instance.) My reaction is that of course it can't be proven, for what could prove the belief that the sun will probably rise tomorrow?

But if he just wants to make the case that there can be no certain knowledge of things with which we aren't acquainted, or that it is misplaced to "fault inductive justification for not producing certainty," why does he constantly emphasize the word "probably?"

@Mr B,

Mr B wrote:

I must still be confused. Or perhaps he has a bad way of making his point. Russell seems to say that principle of induction cannot be proven by experience. (Roughly, if A and B have been found associated a great number of times, and never dissociated, then it is probable A will be associated with B in a fresh instance.) My reaction is that of course it can't be proven, for what could prove the belief that the sun will probably rise tomorrow?

But if he just wants to make the case that there can be no certain knowledge of things with which we aren't acquainted, or that it is misplaced to "fault inductive justification for not producing certainty," why does he constantly emphasize the word "probably?"

The principle of induction is this: that the unobserved (and unobservable) will resemble the observed. It is on this principle that we believe that because the Sun has been observed to rise many times in the past, that it will rise tomorrow (what is not observed). And Russell argues (as did Hume before him) that this principle cannot be proved inductively because that would be using induction to prove the principle of induction, and cannot be prove deductively, because then we would need premises from with the principle could be deduced, and how would we justify such premises?

The statement that the Sun will probably rise tomorrow can be proven in the same way that we can prove that the last marble in an urn will turn out to be red, because all the other marbles we have already drawn from the urn have been red. The more red marbles we draw from the urn, the more probable it is that the next marble we draw will also be red. The we expect the unobserved to resemble the observed. The question of the justification of induction is whether that expectation is justified, and to what extent it is.

Your question about why Russell talks about probability if he is hankering after certainty is an interesting one. I think Russell believes that inductive justification does not even justify probability claims, because he thinks that the principle of induction cannot be justified.

@kennethamy,

Thanks, kennethamy. I understand the case you are making. But I would never get it out of Russell's writing. Or maybe his claim is subtler than I understand it to be.

@Mr B,

"induction" means that reality is essentially mythical;

In ordinary terms:

Que Sera, Sera,
Whatever will be, will be
The future's not ours, to see
Que Sera, Sera
What will be, will be.

Smile

@perplexity,

perplexity wrote:

"induction" means that reality is essentially mythical;

In ordinary terms:

Que Sera, Sera,
Whatever will be, will be
The future's not ours, to see
Que Sera, Sera
What will be, will be.

That is fatalism, and fatalism not only has nothing to do with induction, but it cannot be true if inductive inference is correct. For, fatalism says that human actions cannot effect anything, and induction says that human actions are like any other events, and can effect future events. In fact, fatalism can be shown to be false inductively. For instance, the fatalist says that it makes no difference whether a soldier takes precautions to avoid being shot, because que sera, sera, if he is meant to be shot, he will be shot whatever he does. But induction tells us that soldiers who take precautions are less likely to be shot than soldiers who do not take precautions. So, not only are you mistaken that fatalism is the same as induction, but if fatalism is true, inductive inference is false, and conversely.

@kennethamy,

:eek:

Some bad news for narrow minded bullet dodgers:

The future extends from now to infinity.

You are going to be a long time dead.

@perplexity,

perplexity wrote:

:eek:

Some bad news for narrow minded bullet dodgers:

The future extends from now to infinity.

You are going to be a long time dead.

The wisdom! The wisdom! Who needs logic and facts when we have such wisdom?

@Mr B,

Mr B wrote:

In The Problems of Philosophy, Russell discusses the principle of induction: roughly, if A and B have been found associated a great number of times, and never dissociated, the it is probable A will be associated with B in a fresh instance.

I believe, for example, the sun will probably rise tomorrow, since it has risen every other day. But I can't say in what the truth of this belief subsists. Tomorrow will come, and the sun will either rise or not rise - in won't probably rise.

Russell also says that experience can neither prove nor disprove one's belief in the principle of induction. But I don't understand his reasons, and his argument loses me at that point. This is completely because, as I stated, I don't see what the truth of the principle has to do with experience at all. What could one possibly experience to validate the belief that a A will probably be associated with B? As before, either A will or will not be associated with B in experience.

Mr B,

:)At the outset, is it not the problem of confuseing correlation with causation?

@Mr B,

Mr B wrote:

In The Problems of Philosophy, Russell discusses the principle of induction: roughly, if A and B have been found associated a great number of times, and never dissociated, the it is probable A will be associated with B in a fresh instance.

I believe, for example, the sun will probably rise tomorrow, since it has risen every other day. But I can't say in what the truth of this belief subsists. Tomorrow will come, and the sun will either rise or not rise - in won't probably rise.

Russell also says that experience can neither prove nor disprove one's belief in the principle of induction. But I don't understand his reasons, and his argument loses me at that point. This is completely because, as I stated, I don't see what the truth of the principle has to do with experience at all. What could one possibly experience to validate the belief that a A will probably be associated with B? As before, either A will or will not be associated with B in experience.

The principle of induction is that the inference from the observed to the unobserved is a valid inference. That is is more probable than not that what is unobserved will continue to be like the observed. We can prove a proposition only in two way: deductively or inductively. Clearly, to try to prove induction by induction fail because of circularity. If we tried to give a deductive proof, then we would have to have premises from which the conclusion would follow deductively. But how would such premises be, themselves, proved? Deductively or inductively? And the same problem arises. The only other possibility is for the premises not to require proof. And that would necessitate that the premises be self-evident. But no one has ever come up with such a self-evident premise. In his final philosophical work, Human Knowledge tried to come up with such a self-evident premise. But it is generally agreed that he failed to do so.

@kennethamy,

kennethamy wrote:

The statement that the Sun will probably rise tomorrow can be proven in the same way that we can prove that the last marble in an urn will turn out to be red, because all the other marbles we have already drawn from the urn have been red. The more red marbles we draw from the urn, the more probable it is that the next marble we draw will also be red. The we expect the unobserved to resemble the observed. The question of the justification of induction is whether that expectation is justified, and to what extent it is.

I'm not sure what Russel intended but ...

This example with the urn and marbles has another insight to offer. From the evidence we have so far (we have retrieved only red marbles) it is a "natural" conclusion that subsequent marbles will also be red. However this is where the evil of plausibility (that is the human emotional experience of truthishness) gets us in trouble.

For an urn filled halfway with white marbles, then filled the rest of the way with red marbles, our intuition gives an answer directly opposed to reality. In this more specific scenario the more red marbles we've seen so far, the less likely it is that we will get a red one on the next pull.

Now this is obviously an abstract scenario, but the same principle applies to all inductive reasoning of the "sun rising tomorrow verity." The point here is that cataloging the past does not necessarily tell you anything useful about the future at all (not even in a statistical sense.) However we tend very strongly to believe that it does, often to our own detriment.

Deductive reasoning is not of much use in cases like this though. As we do not know how the urn was prepared, the only answer deduction can lead us to is that we do not know at all what color the next marble will be. The same is true for the rising of the sun, if all we know about it is that it rised every day of our lives. However, via the long history of deduction based scientific investigation, we can apply newtons laws to a model of the solar system and say with great certainty that the sun will rise tomorrow.

Another problem with induction is that a "satisfying" conclusion that has been arrived at via induction can, and very often does, halt further inquiry which may have lead to some enlightening idea down the road. History in fact provides an excellent example with regard to the sun rising issue (Galileo and Pals.)

I hope this was helpful;
-Quatl

(As an aside: I mentioned our plausibility emotion, because people generally accept inductive arguments when they think the answer is plausible to them. In other words they "like the idea." It's an excellent way to manipulate people if you're pushing an agenda. All sorts of social institutions use it to great effect daily.)

[edit: semi-developed thoughts from later in the evening beyond this point]

Statistics is the mathematical equivalent of induction and is prone to the same basic problems. Perhaps that's why Russel refers to probability? (I assert they are equivalent in a deep way, that is: that induction is our minds' implementation of statistical mathematics; or more accurately a sub/superset.)

Induction, and statistical methods both are powerful tools for leading us towards interesting realms in which to apply deduction. This for me is their "proper" place in the order of thought, but I seem to be in the minority on that assessment. A great many people seem to value their intuition above any form of deduction ("confirmation bias" etc. if someone is interested, you'd find it in the psych section of your local internet.)

These methods are also better than nothing. If you don't have the information, or tools to get the information, you'd need to apply deduction rigorously, then intuitive methods may get you someplace other than nowhere. These sorts of problems are very common, in fact these categories of problem are the most common of those we encounter day to day.

Perhaps humans are "optimized" for induction for these reasons. Which would give me a satisfying answer to my recurring pseudo-existential crisis of why people act remarkably dumb most of the time, even those folks who seem otherwise intelligent (I would consider myself an example pretty often.)

@Quatl,

Quatl wrote:

For an urn filled halfway with white marbles, then filled the rest of the way with red marbles, our intuition gives an answer directly opposed to reality. In this more specific scenario the more red marbles we've seen so far, the less likely it is that we will get a red one on the next pull.

How did you reach this conclusion?

@kennethamy,

kennethamy wrote:

How did you reach this conclusion?

In the modified scenario there are white marbles in the jar as well as red. The more and more red marbles the proportion of marbles left in the urn that are white is increasing. The likely hood that the next pull will be a white marble increases accordingly.

In the scenario I gave the situation is exaggerated because all the white marbles are in the lower half of the urn, however the same statement is true if the marbles are mixed together.

Numbers sometimes help: start with 50 red marbles and 50 white in the urn all mixed up. say we reach in 10 times and pull out a red one. at this point there are 40 red and 50 white left. There are more possible "choices" that would end up white than there are which would end up red. As long as we keep getting "lucky" and pulling red marbles the ratio will continue to shift.

Recall that the person drawing the marbles DOES NOT KNOW the contents of the jar. If I had seen 30 red marbles come out of the jar I personally would think it was pretty likely that the next one would be red as well. So much so that I'd be momentarily surprised to see a white one on the 31st pull.

Ask yourself though, take the original urn what is the probability of drawing a blue marble from the urn at any point?

The answer is that there is no such thing as probability in this case (rather we cannot calculate it) as we have insufficient information. But if you asked most people to guess what color they'd draw next after the last ,say, 10 pulls had been red they would most likely think of the red pulls as strong evidence against the draw of a blue marble. (but what if the urn has 20 red marbles and one blue one inside?!)

If we know the contents we can get a better guess using probability, that is to say the math, not our intuition of how probability works which is nearly always wrong! We could make a statement about some particular jar that we have a such and such percent chance of getting a green marble on a given pull. We could even refine our estimate by considering how prior pulls have affected the probability. Even when doing this though we cannot predict with certainty the exact result of the next pull!

The problem with our minds is that we believe that we can. We strongly believe it! So strongly in fact that most people who have learned the mathematics of probability continue to guess completely wrong in scenarios like this.

A related example is coin flips, it is possible to flip a coin 5 times and get heads each time. Most people would say that the next flip is less likely to he heads than tails, since there is a 50% chance of each and we haven't seen a tails yet. This proposition is also false! Statistics don't actually work that way, but as I've said we believe that they do.

I don't mean to derail the discussion too much, but the lack of understanding of statistics, probability, and intuitive "proofs" has a very large effect on our everyday lives. So I think it's important to get the point across if I can.

@Quatl,

Quatl wrote:

I am sure you have an answer, but how do we know that there are not now 98 red marbles in the jar, and just 1 white marble?

@kennethamy,

We don't know in the scenario I offered in the first post, and that's the point. I'll try a different format.

The observer in either your scenario or the one I offer has the same information.

Specifically that:
#1:the urn contains marbles
#2:so far, when one was retrieved it was red. (that is we have taken a some number of red marbles from the urn.)

You proposed that this evidence supports the following proposition:
"The next marble will probably be red"

What I was trying to say, is that this proposition is not actually supported by the observed evidence.

Primarily; since we don't know the true contents of the Urn at the beginning, we cannot construct a probability that bears a relationship to the truth. Even though we (that is most people) would tend to come to the same conclusion that you did.

The conclusion simply does not follow from the evidence.

The fact that it would "appear reasonable" to us to assume that it does, is an illustration of the problems we encounter trying to utilize intuition in a way that is rational (I mean rational in the sense that our conclusion bears a relationship to the reality, which in this example it does not.)

Again our expectation is:
"We are more likely to get a red marble next time we draw"
our reason for believing this is:
"Every marble has been red so far"

The problem is illustrated by the fact that we conceive of a class of scenarios where the above would be untrue. One example being that marbles of some color other than red are present in the urn. Now if we had some other evidence we could increase our certainty perhaps. For example If the urn is in a village where there is a strong cultural habit of storing only one color of marble in one urn.

Still this is only ever able to get you to a certain point. That is the kind of intuition that Russel is referring to cannot ever give you a prediction of how a specific event will occur "in fact". The closest you can come is to a high probability. The problem is worse than he seems to be saying however, because out intuition in these cases can lead us to blatantly untrue beliefs if we ignore what we don't know. Importantly, in practice, we usually do ignore the missing information!

Which means that if we are seeking to approach the truth, we must be cautious of our intuition (and statistical and probabilistic methods as well.) They will, by their nature, often lead us in the wrong direction.

I hope this is clearer;
--Quatl

PS, I don't want to give the impression that intuition is useless, as I don't believe that. If the question was to criticize deductive reasoning, I could give you various issues that we encounter with those methods as well. There are many tools in our heads, and the trick is using them all effectively.

@Mr B,

Mr B wrote:

Thanks, kennethamy. I understand the case you are making. But I would never get it out of Russell's writing. Or maybe his claim is subtler than I understand it to be.

I think that case is pretty clearly stated in The Problems of Philosophy. Remember the chicken example? Russell's is much the same argument as David Hume's who really started the problem of induction. The point is that all of our reasoning in science is inductive, and so, based on the inductive principle. But, on Hume's and Russell's account, this principle is not provable. So it seem as if although we have no alternative other than inductive reasoning (we infer from the observed to the unobserved-that is how most of us know there is a country called "Japan" for instance) that kind of inference is not justifiable. It is not, of course, that we should not use it. We have (as I said) no alternative. But, according to Hume (and Russell) what we call inductive reasoning, while not irrational, is non-rational. We do it. We have no alternative to doing it. But we cannot justify doing it,

@Quatl,

Quatl wrote:

We don't know in the scenario I offered in the first post, and that's the point. I'll try a different format.

The observer in either your scenario or the one I offer has the same information.

Specifically that:
#1:the urn contains marbles
#2:so far, when one was retrieved it was red. (that is we have taken a some number of red marbles from the urn.)

You proposed that this evidence supports the following proposition:
"The next marble will probably be red"

What I was trying to say, is that this proposition is not actually supported by the observed evidence.

Primarily; since we don't know the true contents of the Urn at the beginning, we cannot construct a probability that bears a relationship to the truth. Even though we (that is most people) would tend to come to the same conclusion that you did.

The conclusion simply does not follow from the evidence.

The fact that it would "appear reasonable" to us to assume that it does, is an illustration of the problems we encounter trying to utilize intuition in a way that is rational (I mean rational in the sense that our conclusion bears a relationship to the reality, which in this example it does not.)

Again our expectation is:
"We are more likely to get a red marble next time we draw"
our reason for believing this is:
"Every marble has been red so far"

The problem is illustrated by the fact that we conceive of a class of scenarios where the above would be untrue. One example being that marbles of some color other than red are present in the urn. Now if we had some other evidence we could increase our certainty perhaps. For example If the urn is in a village where there is a strong cultural habit of storing only one color of marble in one urn.

Still this is only ever able to get you to a certain point. That is the kind of intuition that Russel is referring to cannot ever give you a prediction of how a specific event will occur "in fact". The closest you can come is to a high probability. The problem is worse than he seems to be saying however, because out intuition in these cases can lead us to blatantly untrue beliefs if we ignore what we don't know. Importantly, in practice, we usually do ignore the missing information!

Which means that if we are seeking to approach the truth, we must be cautious of our intuition (and statistical and probabilistic methods as well.) They will, by their nature, often lead us in the wrong direction.

I hope this is clearer;
--Quatl

PS, I don't want to give the impression that intuition is useless, as I don't believe that. If the question was to criticize deductive reasoning, I could give you various issues that we encounter with those methods as well. There are many tools in our heads, and the trick is using them all effectively.

Since we have no knowledge of the contents of the jar, why should we think that taking a red marble from the jar does not increase the chances that the next marble will be red? If it doesn't, then there is just as great a chance that we will next draw a blackbird.

@kennethamy,

kennethamy wrote:

Since we have no knowledge of the contents of the jar, why should we think that taking a red marble from the jar does not increase the chances that the next marble will be red? If it doesn't, then there is just as great a chance that we will next draw a blackbird.

We should not think it because it is not true.

We can only truthfully say that we don't know in this scenario what color the next marble will be and that's the best we can do in this case.

We could also make a decision that the jar is entirely filled with red marbles and go get a sandwich instead. I don't argue that this isn't a good guess, but it's still just a guess. Let me put it this way, if we had such a jar before us and I bet you a year's salary that the next marble would not be red would you take the bet? If you would, you may want to avoid casinos, as they will feast on your wallet.

The point is that the previous marbles that we have drawn do not influince the color of the next one drawn (even if it did turn out to be red.) If we think it does we are acting on lies and in the long run it very well may bite us (nutronium marbles are very dangerous Smile

)

Having read your post above this one I am having difficulty seeing where you disagree with me in principle. As that post pretty much says exactly what I've been saying Smile

@Quatl,

Quatl wrote:

PS, I don't want to give the impression that intuition is useless, as I don't believe that. If the question was to criticize deductive reasoning, I could give you various issues that we encounter with those methods as well. There are many tools in our heads, and the trick is using them all effectively.

This isn't much of a comment on the conversation... but I just wanted to say that I like the way you phrased that, especially the last sentence. I've been realizing what an important concept it is. Smile

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