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THE PROBLEM OF INDUCTION It seems that empirical observations are crucial to science. But a key question arises: What role can observation play in coming up with scientific theories? Can theories by confirmed by observation? Let us see why these issues are pressing. 1 Deductive reasoning The nice thing about deductive logic is that it licenses conclusions that follow necessarily from premises. 1. Socrates is human. 2. All humans are mortal. Therefore: 3. Socrates is mortal. But deductive logic cannot tell us whether the premises are true, and so cannot tell us whether the conclusion is true. We might summarize this as follows: deductive logic provides certainty, but at the price of offering no new empirical information. 2 Kinds of reasoning Hume suggests that there are two kinds of reasoning: Relations of ideas: Deductive subjects such as logic and math. To hold the premises but deny the conclusion is contradictory. E.g., Socrates is a human, all humans are mortal but Socrates is not mortal. Therefore, one can deduce conclusions from premises by thought alone. Matters of fact: Empirical sciences. Conclusions are not entailed by premises—can consistently hold the latter but deny the former. E.g., the sun has always risen but will not rise tomorrow. Requires investigation into the world. 3 Induction We can never observe all instances of a phenomenon across all of space and time. So, in investigating the world, we move from samples to wholes. This is absolutely central to empirical investigation of any kind. This is known as inductive reasoning (what Hume called reasoning about matters of fact). Hume: Induction can never lead to justified beliefs. 4 Temporal induction Of particular importance to us is drawing conclusions about the future on the basis of past or present experience. For example: I’ve always liked Mars bars, so I’m sure I’ll like this one. Air Canada has had very few accidents, so it’s safe to fly with them. I’ve given blood for years without problem, so I’m sure it’s safe to do so. Smokers have been shown to be at high risk for lung cancer, so if you don’t smoke you’re less likely to get lung cancer. We reason in similar ways all the time. Indeed, it is hard to see how we could get on in life without induction. So why doubt that induction is justified? 5 Hume’s doubt Hume: all such reasoning works only if things go on as they have before. I.e., only if past experience is a reliable guide to future experience. Without this assumption, the reasoning is unjustified. But how can you be justified in believing the future will be like the past? This can’t be proven deductively because it is not a contradiction to assume that the future will change. E.g., we can consistently hold that the sun has always risen but won’t rise tomorrow. So we need some other way to justify inductive reasoning. 6 First answer In past experience, the future has resembled the past. When I have reasoned inductively (as if the future would resemble the past), I wasn’t disappointed. So I won’t be disappointed in the future. Hume: What could this prove? Just because the future has resembled the past in the past, it doesn’t follow that it will resemble the past from now on. If you assume this, you are arguing in a circle! I want to know what the justification is for assuming the future will be like the past. If you simply assume the future is like the past, you haven’t given me a reason to believe you. This answer isn’t even an argument. It is just a repetition of the conclusion we need to prove. 7 Second Answer The past/present causes the future. If we know the (past/present) cause we know the (future) effect. So, induction is justified. Hume: You only have the idea of causation because, in the past, you have seen things go together all the time. Hence, the very concept of causation is based on past experience. To assume that causation will continue as it has is to assume the future will be like the past. This is just circular reasoning again. 8 Summary of the problem Hume: the attempts to answer the problem of induction cannot succeed. 1. Deduction won’t work. 2. Induction just begs the question: This argument is: the future will resemble the past because in the past the future has resembled the past. But this is just an instance of induction, so we need to justify it. This leads to an infinite regress. If we stop the regress anywhere, then we are simply stating: “Induction works because induction works”. 9 Sceptical “solution” Hume: we can’t stop ourselves from reasoning inductively. It is human nature. Experience forms in us the habit of assuming the future will resemble the past. However, there is no reasoning that can justify this habit. We just do it, and that’s all there is to say. 10 What about laws of nature? Third answer: but contemporary science is very successful. Its success is based on discovering exceptionless laws of nature. So, we are justified in concluding that nature follows exceptionless laws. Reply: This is still circular reasoning. Laws have been exceptionless up until now—how do we know that will continue? Secondly, even if there are exceptionless laws, how could we know we’ve discovered them unless we’ve experienced phenomena throughout all time? What’s at issue is not the nature of laws, but the nature of our knowledge of laws. 11 Probability and induction Answer 4: the more we notice a uniformity, the higher the probability it will continue. The more often we see two things together, the more probable that they will appear together next time. We can’t prove that they always appear together, but as we gather enough evidence the probability will approach 1. Problem: The number of observed cases is always finite. It is possible that the number of unobserved cases is effectively (or actually) infinite. Hence, the probability may never reach anything close to 1. At any rate, there is no way of proving that it will. 12 A disastrous conclusion If Hume is right, we are in trouble. All empirical science seems to depend at least in part on induction to draw conclusions about the world. If induction is irrational, so is all of science. Everyday reasoning depends on induction. If induction is irrational, it is no more rational: To take an elevator than jump off a roof. To eat bread than ingest arsenic To take an aspirin for a headache than commit suicide Etc. So something has gone wrong here, but what? 13 Popper’s “solution” Popper’s argument: Science is the best means of gaining knowledge. Science does not proceed by induction. Science proceeds by “conjectures and refutations”. This only requires deductive logic. Therefore, the problem of induction needn’t be solved. To begin, recall Popper’s view of science: Scientific theories must take risks—they must make predictions that can be falsified by experiment. But how do we test a scientific hypothesis? 14 Testing theories For Popper, testing a theory, T, means using deductive logic to determine what the theory entails. I.e.: If T, then Q. Then, one tests to see whether Q is true. We know that if not-Q, then not-T. This is because the following is a valid argument form: If T, then Q. Not-Q Therefore, not-T If Yogi is a bear, then Yogi is a mammal. Yogi is not a mammal. Therefore, Yogi is not a bear. 15 Example: Testing Einstein’s Theory The star is in fact located here. The star looks like it coming from here. 16 Popper’s reconstruction of the test Here: T = General Relativity. Q = light will be deflected by angle Had not-Q been observed, we’d have concluded not-T. 17 Passing the test What can we conclude if Q is observed (as it was)? Well: If T, then Q. Q Therefore T is an invalid argument. Compare: If John is in Guelph, then John is in Ontario. John is in Ontario. Therefore, John is in Guelph. So, observing Q does not confirm T. 18 Corroboration Popper: If we observe Q, then T is corroborated (we’ve seen this before): Still in the running. Not falsified. Worth investigating further. Key point: none of this requires induction! Empirical investigation proceeds by deductive conjecturing (and refutations). 19 Summary The problem of induction casts a great deal of knowledge in doubt. Hume: we must remain skeptical—much knowledge is impossible. Popper: we can give up induction: proceed by deductive conjecture and refutation. Question: is Popper’s stance a solution or an evasion of the problem? 20 Some further issues Is Popper’s view oversimplified? Suppose we had observed that the angle that light deflects ≠ ? Would we have really given up on General Relativity? Probably not (recall Kuhn, Lakatos). Even Popper admits this but then the question becomes: What part of the theory do we give up? Can we ever perform a test to determine which hypothesis should be surrendered? We’ll examine this issue soon. 21