We use inductive reasoning when we reason from a few examples to a generalisation. For example:
· We have observed 1000 swans to be white
· Therefore, all swans are probably white
· The sun has risen every day throughout recorded human history
· Therefore, it will probably rise tomorrow morning.
However, there is a fatal flaw in this type of reasoning, as the philosopher David Hume pointed out in the 18th century. We can never know for sure that a conclusion reached by inductive reasoning is true. In the first of my examples, we would have to observe every swan in the world in order to be completely sure of the validity of the conclusion. In fact, black swans do exist, so the conclusion is actually false. In the second and more interesting case, we can only draw this conclusion by insisting that the laws of nature will remain the same tomorrow as they have in the past (I am here ignoring the possibility that the sun will be destroyed by some event consistent with current natural laws). But, how can we know this? The natural answer is that they have always been the same in our experience, so it makes sense that they should be the same tomorrow as well. However, this in itself is an inductive argument – reasoning from observations in the past to those in the future. We can’t justify induction by recourse to induction can we, as this is just circular reasoning?
So, we seem to have a predicament. According to Hume’s problem with induction, we have no rational reason to believe that the sun will rise tomorrow just because it has always done so in the past. This is not making the weak claim that the sun will probably rise tomorrow, but there is a small probability that the laws of nature might suddenly change so that it will not. Rather, Hume is saying that we cannot conclude that it will rise tomorrow based upon previous observations of it rising - as there is no reason at all to suppose that the laws of nature will not suddenly change. This is a particularly acute problem for science in general, as it relies heavily on inductive reasoning to generalise from a few observations to theories and laws of nature.
How might we attempt to resolve this problem? A.J. Ayer held (in his book Language, Truth, and Logic) that the problem of induction was actually a fictitious one, as there is no solution, and all genuine problems are at least theoretically capable of being solved. According to him, if we take it to be a tautology, then we cannot deduce from this matters of empirical fact. On the other hand, if one approaches it empirically, then one assumes what one is setting out to prove.
It seems to me that we have no entirely satisfactory solution to this problem, but we may attempt solutions along the following lines. Firstly, we might say that we have no choice but to take induction as an epistemic practice that is in need of no further justification. A slightly more satisfying solution comes from utilising the ideas of reliability that I discussed in the previous question. We might say that we are justified in using induction because induction is a reliable way of forming true beliefs, in the same way that my perception is a reliable way of my forming true beliefs. Of course, this then becomes a circular argument, as its reliability up until now doesn’t guarantee its reliability in the future, unless we assume inductive reasoning is valid.
Karl Popper attempted a solution based upon his idea of falsification. He proposed that science does not in fact evolve by means of inductive reasoning but, rather, by means of the falsification of theories. Popper held that, to be considered properly scientific, a theory must be capable of being falsified. Under this scenario, we can avoid inductive reasoning by relying instead upon modus tollens (e.g. if all swans are white then we will not find any black swans; we have found black swans; therefore, not all swans are white etc). Critics would say that science does not in fact evolve this way, relying instead upon inductive reasoning.
Another possible solution is by recourse to Occam’s razor. We might reason that the hypothesis that the laws of nature will change in some way such that they invalidate our previous conclusions based upon inductive reasoning is a less parsimonious hypothesis than the one that they will just remain the same, as it involves the introduction of additional ad-hoc elements. This approach also seems to be applicable to Nelson Goodman’s idea of grue, in which something is grue if it is observed to be green before time t, and blue thereafter. This seems less parsimonious than the idea that it would just remain green – not because we are defining grue in terms of green and blue, which seems intuitively less simple, but because we are specifying some arbitrary change in one of its properties at time t, as opposed to no change at all.
In the end, perhaps the least unsatisfactory solution is a pragmatic one. Even though we perhaps lack a complete justification for inductive reasoning, it is still (at least indirectly) rational for us to continue to use it, as we want to form true beliefs about the world and, of any method of inference, induction is the one that will maximise the number of true beliefs about the world that we will obtain and minimise the number of false ones.
An interesting question I saw raised is whether this has any implication for arguments to disprove the existence of God. If they rely upon inductive reasoning, are they equally vulnerable to the problem of induction?
Well, perhaps, but fortunately many of the arguments against God’s existence rely upon deductive reasoning, which is entirely different. These are arguments of the form: all dogs are mammals, Rover is a dog, therefore Rover is a mammal etc.
In deductive reasoning, there is a distinction between a valid deductive argument, and a sound one. A valid deductive argument is one that is correctly formed so that the conclusion follows inevitably from the premises. There is no necessity for the premises to actually be true, merely for the conclusion to follow logically from them. For example:
Premise 1: all dogs are three-legged animals
Premise 2: all three-legged animals are from Mars
Conclusion: therefore, all dogs are from Mars
This is a valid deductive argument, despite the fact that both premises (and the conclusion) are clearly false.
By contrast, a sound deductive argument is one in which we have the additional constraint that the premises are true. For example:
Premise 1: all dogs are mammals
Premise 2: no mammals are animals with scales
Conclusion: therefore, no dogs are animals with scales
How about deductive arguments to disprove God’s existence? Can I formulate one that is sound? Theists in particular might say that the premises in any such argument are false, as we don't have a clear-cut situation as in my examples above. However, in the end it comes down to plausibility and degree of reasonableness. Can I construct valid deductive arguments that disprove God's existence, and which are plausible? Well, for example, I would say that this argument is valid and plausible, but not necessarily sound:
Premise 1: if the Christian God exists, with the usual properties of omnipotence, omnibenevolence, and a particular interest in human beings, then he would not allow any more than the absolute minimum amount of evil possible (natural and man-made) to exist in the world (this follows from God's properties)
Premise 2: the amount of evil that exists in the world far exceeds this absolute minimum (from the empirical evidence)
Conclusion: therefore, the Christian God doesn't exist
Here is another one:
Premise One: If a merciful and compassionate God wants us all to be saved (as many Christians believe), then he would provide clear and unambiguous information about his message to all humans, as this is necessary for salvation (by definition).
Premise Two: This clear and unambiguous information is not provided to all humans (from the empirical evidence).
Conclusion: Therefore, such a God does not exist.
These are valid deductive arguments, and I would further contend that they are plausible ones too. Of course, whether they could be classified as sound depends very much upon one's worldview. This is a consequence of the fact that we know of no knockdown argument to disprove God's existence (but the same applies equally with the arguments for God's existence). It all turns upon degree of plausibility and reasonableness, and I find the valid arguments that seek to disprove God's existence to be far more plausible and reasonable than those that seek to prove it. The weight of plausible arguments against God's existence also forms a cumulative case.
Now, one may take issue with either or both of the premises in my arguments above, but they are nonetheless valid deductive arguments. That is, if we accept the premises, then the conclusions must be true. Moreover, any argument that attempts to disprove the existence of God by deducing what predictions are entailed by the God hypothesis, and then finding these predictions unfulfilled, has proceeded by falsification. It is therefore a deductive argument that doesn't suffer from the induction problem.