What is the Problem of Induction?
To say that there is a problem with induction is, to my mind, mistaken. To say that there is a problem is to claim that whilst the inductive method works, there is some reason why we should be wary of using it, or that there are some things that don't yet quite work that need ironing out. This is not, however, the case, since the inductive method doesn't work in the way most people seem to believe it does - that is induction does not provide proof of things and there is no way to justify the use of induction for predictions except using the inductive method itself.
The inductive method states that if something has been observed many times in the past then there is a high likelihood that a similar examination will result in a similar observation in the future. But this is clearly not the case as a rule. The only proof that we need is an instance in the past of something that induction would not predict could happen happening, and clearly there are such instances. The famous example that Russell uses is that of the chicken, who, having been fed by the farmer every day of his life, expects to be fed by him on the day when his neck is wrung by the very same man. Clearly the chicken was not being entirely stupid in associating the farmer with his feeding, and it would not be the most obvious assumption to decide that the farmer would eventually kill him.
A simple way of getting round this particular 'problem' would be to suggest that the chicken is simply not intelligent enough to understand that a farmer would not feed him except to fatten him up and if he did understand the rules by which the world operates better he would have realised what fate had in store for him. The problem with this is twofold: firstly such an understanding would obviously require knowledge of every aspect of the universe and how it works, in order to make any proper prediction, and to say that it is possible to understand everything is not necessarily going to be true; secondly there is an implicit presumption in such a claim about the world - and that is that the rules will continue to act exactly as they do now. We can't claim that this in itself is a rule - the continuity of nature - since this gets us nowhere (since we could always enquire whether there was a further rule that ensured that continuity of nature would continue indefinitely). Clearly, however, random events (in the sense that they would not be predicted on the basis of the past, or even on the basis of the most complex rules about the universe) could happen without causing a logical confusion or paradox.
Even if on a larger scale we find it hard to find examples of totally random events occurring, Heisenberg's Uncertainty Principle suggests that for a very very short period of time absolutely anything could happen - the universe could become a doughnut - just as long as it returns to the way it was by the end of the short period of time. Whilst such an anti-instinctual theory is not necessary for a philosopher to conclude that it is possible that anything could happen this principle strongly suggests that anything literally is happening, and all the time.
I find it very hard to understand why so many intelligent minds say in their first breath 'A random event could happen...' and yet are willing to say in their next breath '...so how can we prove one won't?' or 'It's possible that the sun won't rise tomorrow... so how do we show that it will?'. Both these questions contain an obvious contradiction, and yet people are willing to ask them as if they are simply problematic and can be sorted out. If a random event really can happen then obviously there is no way that we could prove that one won't, since you shouldn't be able to prove something which is false. Likewise if there is the remotest possibility that the sun won't rise tomorrow then - obviously - there is a remote possibility that the sun won't rise tomorrow, and no amount of thought should be able to assure us that it is certain that it will.
So do we have to deny that induction works and cease to base our daily routine on the presumption that it does? The simple answer is that we can't. The principle of induction is so deeply entrenched in the way we live our lives that nobody could simply stop using it as a basis for making predictions and taking decisions on a day-to-day basis, and this ties in with the very reason why it is entirely reasonable for us not to reject the method. Basically what induction comes down to in the end is the frequent sub-philosophical justification that 'We can't live without it'. The simple fact is that if the past made no indication about the future and absolutely anything could happen we'd simply have to accept that it wouldn't be possible to make any predictions, or even any judgements about the world based on causal experiences. It may be that there is a different principle underlying the universe that we have not yet discovered, but until we do so there is no point in simply abandoning any attempt to describe the world or make prediction.
Having accepted, therefore, that we cannot prove anything for certain using induction, many philosophers suggest that instead we should merely talk about the probability of something happening, yet this too seems to miss the point about the impossibility of predicting things about the future. This would suggest that every time something happens under certain circumstances it lends additional certainty to the same thing happening under the same circumstances, but this would seem to make little sense if we have concluded that 'anything can happen', and if we conclude that a higher probability makes a certain event more likely how could anything still happen.
A statistician will tell you that the probability that if you roll a fair die you will get a six is one in six, yet, if you enquire further, he'll be able to tell you the probability that if you roll a die six times you'll get a six (66.5%) and yet, if you enquire even further he'll tell you what the probability is that, if you roll six sets of six, you'll roll a six. The probability, however, never reaches 100% (though clearly you don't need a mathematician to tell you this) so the statistician could happily sit rolling a die forever, never once rolling a six, and should not be surprised as he does this. Simply because the lottery has enormous odds against my winning does not prevent me winning every week for a year.
Whilst the sum evidence of all dice-rolling should inductively suggest to us that on average a six should come up every six rolls of the die this clearly does not conclusively tell us what will happen next time we roll a die, or roll it six times, or roll it thirty-six times. A different form of probability, where we were assured that in a certain number of throws a six would be rolled, might allow us to make sub-deductive but deduction like statements which, whilst not concrete about the specifics of when things will happen, would at least allow us to state that they will, and in a general time scheme. But clearly this is not what probability in the normal sense sets out to do, and probability cannot be trusted, even with things of the highest possibility, to produce certain results. Even if probability did have some demonstrable relationship with how often things occur the possibility always exist that the next time someone tries rolling dice they'll find that probability has ceased to have any effect on the world .
Consider, also, how we could assign possibilities for different events. Would we count every instance of the sun rising as a independent event, so that there would be a tiny chance of the sun not rising tomorrow, but an even smaller chance that it would rise tomorrow and the next day? Do we think that in the past the sun has never not risen, or that the sun has never stopped rising? Ultimately how do we decide which are more likely events and which less likely, if we have already conceded that just about anything can happen? If the probability of the sun rising has been worked out to be 99.9999% does that mean that on one day in every million similar days the sun will fail to rise? If the world is running according to rules how can we make such a claim of randomness, and if it isn't, how can we claim that any one thing is more likely than another?
It is clear that the claim that we simply need to convert inductive statements away from positive, certain statements to probabilities will not be easy to carry out, and that very specific rules would be needed to get any idea of our new supposed probabilities before we could even begin to claim we knew what could happen in the future. Unfortunately this is by no means an easy task, and, whilst people have tried to explain it in simple terms we find this is not possible - one of the major elements (as I will discuss later on) of induction is that the more complicated the rules are the better.
When discussing induction Goodman seems to apply words which would fit much better with deductive concepts than induction, and this causes one of the biggest problems with his argument. His first mistake is to talk about which statements can be seen to confirm other statements, and throughout the discussion he wrongly seems to suggest that with induction it is statements that provide confirmation for other statements, rather than evidence. It is clear in induction that statements cannot have an interaction with each other, but purely experimental evidence, since these causal principles are only observable a posteriori (as Hume proved) and anything that can be shown from other statements can be done deductively alone.
He also seems to spend a remarkable amount of time debating why it is that a single piece of evidence shouldn't lead to any belief, and misses one of the fundamentals about induction - that pieces of evidence must be in the plural, and it is meaningless to say 'This ball is red so all balls are red' just as it wouldn't make sense to say 'The universe is infinite so there are many universes'. We cannot extrapolate from single instances - the process of induction presumes that we have evidence of extremes, and tries to interpolate for the bits we haven't yet seen. It is for this reason that having seen many dull colours in nature we presume that all natural colours will be fairly bland and pastel-like, rather than expecting naturally occurring fluorescent yellows and pinks.
This is also the problem with his long debate about whether things could be assigned names which mean different things before and after time t such a 'grue' or 'bleen'. We cannot make predictions that knowingly attempt to extrapolate from what we know to things that we can't, and we could only say that we suspect an object is grue if we have experienced it before and after t. Whilst Russell is correct in picking up on the fact that past futures are different from future future it is only because we presume that these are the same that we are able to make any sort of prediction about the future. If we were to do as Goodman does, and arbitrarily suggest that at a certain time past futures will cease to have a connection to future futures then clearly we can't continue to make predictions with any basis.
Goodman also tries to suggest a probability theory, and almost sounds as if you should just add up the number of times something has happened and divide by the total number of possible times where the presumed causes have happened to find the probability of a connection. This is not the case, and a connection which has never had exceptions must be seen as an entirely separate case from one where even a single exception has been seen. It would be a completely different thing to ask whether the sun would rise tomorrow if there were even a single known instance of the sun failing to rise. If we were to open one of his bags of marbles and find it to be a bag of cubic marbles the suggestion that 'All marbles are round' would have surely been dealt a near fatal blow. Only by arguing that by virtue of being cubic these objects were not marbles might you save such a hypothesis. He even seems to suggest that a negative over-hypothesis (a more general case that has a bearing on a specific hypothesis) has just the same effect as a positive one. Surely a single negative over-hypothesis should cripple any previous belief in a hypothesis which had up to that point had no known exceptions.
The final problem, and I would say one of the biggest with Goodman's model, is that of what we count as a single instance of experiencing something. If for example when trying to discover if a pile of bags of marbles only contained bags of only red marbles, rather than opening and tipping out a pile of bags to discover that every marble inside was red, we kept the necks nearly closed and took out marbles one by one psychologically by the end of the exercise we would be far more convinced that every marble in the pile was red, purely for the fact that we spent longer gathering evidence and more instances of finding single red marbles occurred than of finding bags of red marbles. The attack that we are only concerned with bags of marbles would work if full bags were independent of individual marbles inside but clearly this is not the case - every time we draw out a red marble it adds to the certainty that every marble in the stack is red (particularly as the probability of taking a red marble out of the bag is considerably higher if all the marbles are red than if only some of them are). Yet surely how we examine the marbles can have no effect on the probability that the last unopened bag contains only red marbles.
In fact, this increased reliability based on more and more observations is one of the biggest differences between deductive and inductive arguments. With deduction it is never possible to add an additional premise which can make a valid deduction invalid, so bringing new evidence to an argument should never affect the conclusions reached, and yet almost the opposite is the case with induction, and this in itself is one of the biggest problems with the method - that the more evidence the better and the stronger someone might believe a link formed by induction, but there is always the possibility that new evidence might be the piece that shows the link to be incorrect.
With deduction the more direct the route from premises to conclusion the better, and the more obvious the logic is, and yet with induction the less direct the route the better - whilst there are more things to go wrong in saying that something won't happen because of a large number of rules, it is usually better to state a large number of rules to show a good grounding. For example 'The sun will rise tomorrow because it always has' has less grounding than 'The sun will rise tomorrow because movement of celestial bodies is due to sets of rules, particularly gravity and rules of motion, that have always worked in the past and which seem to conform generally to the principle that things won't change without an external interaction'.
The process of deduction itself usually takes a more direct route - with premises leading to conclusions, as opposed to induction which forms hypotheses about certain premises leading to certain conclusions, basing the hypotheses purely on what would be a meaningful connection, and then appealing for evidence to decide whether the hypothesis works or not. Each of these three differences spring from one of the most fundamental aspects of deduction, and from the aspects that make deduction so powerful - that the process is short, that an argument with fewest steps is best, and that once an argument has been found to be valid no additional data can invalidate it - and in each case induction fails to deliver the simplicity and certainty of deduction.
To sum up, therefore, we don't have any reason to believe that induction should work, but have to accept our instinct that is does, because otherwise we can only know very little about the world. Probability methods of explaining induction seem very flawed since the very idea of probability is almost entirely incompatible with the claim that 'anything can happen'; and there is no problem of induction - it just doesn't work.