Based on the Problems of Philosophy by Bertrand Russel.
Will the sun rise tomorrow?
My rational mind tells me that whether or not I will wake up to another sunny morning is completely unknown to me, as this would essentially mean predicting the future. A meteor may strike the sun - there are infinite possibilities as to what might happen tomorrow. We cannot extrapolate from the fact that it has risen in the past that it will continue to do so, because we are aware of the possibility that the sun will die one day- it may even be tomorrow. We have no exact knowledge of the sun; all that we know is through predictions of science based on previous data. Keeping this in mind, we cannot assume that the probability of it rising tomorrow is greater than the probability of anything else, say gravity overcoming it and turning it into a black hole, therefore it is with no certainty that I can say that the sun will rise tomorrow. Without a complete and conclusive knowledge of all the possibilities that might arise tomorrow, I have no reasons for either doubting or believing it. Be default, I assume that it will rise because it is a situation I am familiar with. I cannot imagine any other way, therefore, I live with the assumption that it will rise. Is the principle of uniformity based on inductive reasoning? The principle of uniformity states everything that happens is an instance of a general law to which there are no exceptions. (Problems of Philosophy, page 63, paragraph 3) Induction is essentially: when two events are associated, and have never been dissociated from each other, the more often they are associated, the greater the probability that in a fresh case, they will be associated again. As the number of times they are associated increases, the probability approaches certainty (Problems of Philosophy, page 66,paragraph 3) But how are they related?
Uniformity states that everything is an instance of a law, with NO exceptions. However, the determination of such a law is impossible because that would involve the examination of every possible instance of the law, which is impossible as there are an infinite number of such instances. The only way we can come to any kind of conclusion about a law is by observing a finite number of the instances of the law. Instinctively, we assume that the greater the number of times the instance has repeated itself without contradiction, the higher the probability that the law of which we assume the instance is, is true, and this is the essence of induction. This way, induction is an essential tool required for the assumption of uniformity. However, logical validity of the principle of induction is only illusory. There is no reason for the probability to increase with every instance. To analyze probability, we must have a complete sample space. Say we are examining the probability of picking a black ball out of a bag of mixed balls. If we know the number of balls in the bag and their natures, we can quantitatively assign a probability to it However, when we talk about the probability of other phenomena like gravity, etc., we do not know the number of instances, and we can never know as number of instances is infinite. Therefore, proposing a law purely on the basis of induction is not valid as it leads to contradiction, as invariably, things change. For example, by induction, with each day I live, the probability of me living the next day grows indefinitely. But I do know that I will ultimately die, which clashes with the accumulated high probability that I will live the next day. Mathematically, the probability at the point at which I die should have been decreasing up to that point for it to be logically valid. If I live for ten extra days, the principle of induction is essentially telling me that I am less likely to die ten days later than tomorrow, which is baseless. Some might argue that the principle of induction has always worked in the past, and from empirical evidence, it does appear that the frequency of the occurrence of an event is related to its probability. However, this reasoning is circular as it again uses induction to prove it. Another argument is that if we disregard induction, then we will have no law and no basis upon which to live our lives. I agree that induction is indispensable as it is instinctive, but that doesn't mean it is conclusive. We live our lives with the assumption that things are going to be as they always were merely because we don't have a choice - we cannot imagine any other way, and when things do turn out to be different, like the sun not rising, we adapt and that gets added to our knowledge. But by no means can we, at any point, assume certainty because anything can happen at any time without the slightest warning, in complete discordance with the mentally generated 'probabilities' that are associated with them, which makes all our lives wholly, and fantastically uncertain.