* Fred R. Berger (1977). Studying Deductive Logic
* A.C. Grayling (2001). An Introduction to Philosophical Logic
Warren Goldfarb (2003). Deductive Logic
* Maria Konnikova (2013). Mastermind: How to Think Like Sherlock Holmes
1.3 Induction
I (Julian Baggini) have a confession to make. Once, while on holiday in Rome, I visited the famous street market, Porta Portese. I came across a man who was taking bets on which of the three cups he had shuffled around was covering a die. I will spare you the details and any attempts to justify my actions on the grounds of mitigating circumstances. Suffice it to say, I took a bet and lost. Having been budgeted so carefully, the cash for that night’s pizza went up in smoke.
My foolishness in this instance is all too evident. But is it right to say my decision to gamble was ‘illogical’? Answering this question requires wrangling with a dimension of logic philosophers call ‘induction’. Unlike deductive inferences, induction involves an inference where the conclusion follows from the premises not with necessity or definitely but only with probability (though even this formulation is problematic, as we’ll see).
Defining induction
Perhaps most familiar to people is a kind of induction that involves reasoning from a limited number of observations to wider generalisations of some probability. Reasoning this way is commonly called inductive generalisation. It’s a kind of inference that usually involves reasoning from past regularities to future regularities. One classic example is the sunrise. The sun has risen regularly each day, so far as human experience can recall, so people reason that it will probably rise tomorrow. This sort of inference is often taken to typify induction. In the case of my Roman holiday, I might have reasoned that the past experiences of people with average cognitive abilities like mine show that the probabilities of winning against the man with the cups is rather small.
But beware: induction is not essentially defined as reasoning from the specific to the general. An inductive inference need not be past–future directed. And it can involve reasoning from the general to the specific, the specific to the specific, or the general to the general.
I could, for example, reason from the more general, past‐oriented claim that no trained athlete on record has been able to run 100 metres in under 9 seconds, to the more specific past‐oriented conclusion that my friend had probably not achieved this feat when he was at university, as he claims. Reasoning through analogies (see 2.4) as well as typical examples and rules of thumb are also species of induction, even though none of them involves moving from the specific to the general. The important property of inductive inferences is that they determine conclusions only with probability, not how they relate specific and general claims.
The problem of induction
Although there are lots of kinds of induction besides inductive generalisations, that species of induction is, when it comes to actual practices of reasoning, often where the action is. Reasoning in experimental science, for example, commonly depends on inductive generalisations in so far as scientists formulate and confirm universal natural laws (e.g. Boyle’s ideal gas law) only with a degree of probability based upon a relatively small number of observations. Francis Bacon (1561–1626) argued persuasively for just this conception of induction.
The tricky thing to keep in mind about inductive generalisations, however, is that they involve reasoning from a ‘some’ in a way that in deduction would require an ‘all’ (where ‘some’ means at least one but perhaps not all of some set of relevant individuals). Using a ‘some’ in this way makes inductive generalisation fundamentally different from deductive argument (for which such a move would be illegitimate). It also opens up a rather enormous can of conceptual worms. Philosophers know this conundrum as the problem of induction. Here’s what we mean. Take the following example:
1 Almost all elephants like chocolate.
2 This is an elephant.
3 Therefore, this elephant likes chocolate.
This is not a well‐formed deductive argument, since the premises could possibly be true and the conclusion still be false. Properly understood, however, it may be a strong inductive argument – if the conclusion is taken to be probable, rather than certain.
On the other hand, consider this rather similar argument:
1 All elephants like chocolate.
2 This is an elephant.
3 Therefore, this elephant likes chocolate.
Though similar in certain ways, this one is, in fact, a well‐formed deductive argument, not an inductive argument at all. One way to think of the problem of induction, therefore, is as the problem of how an argument can be good reasoning as induction but be poor reasoning as a deduction. Before addressing this problem directly, we must take care not to be misled by the similarities between the two forms.
A misleading similarity
Because of the general similarity one sees between these two arguments, inductive arguments can sometimes be confused with deductive arguments. That is, although they may actually look like deductive arguments, some arguments are actually inductive. For example, an argument that the sun will rise tomorrow might be presented in a way that can easily be taken for a deductive argument:
1 The sun rises every day.
2 Tomorrow is a day.
3 Therefore, the sun will rise tomorrow.
Because of its similarity with deductive forms, one may be tempted to read the first premise as an ‘all’ sentence:
The sun rises on all days (every 24‐hour period) that there ever have been and ever will be.
The limitations of human experience, however (the fact that we can’t experience every single day), justify us in forming only the less strong ‘some’ sentence:
The sun has risen on every day (every 24‐hour period) that humans have recorded their experience of such things.
This weaker formulation, of course, enters only the limited claim that the sun has risen on a small portion of the total number of days that have ever been and ever will be; it makes no claim at all about the rest.
But here’s the catch. From this weaker ‘some’ sentence, one cannot construct a well‐formed deductive argument of the kind that allows the conclusion to follow with the kind of certainty characteristic of deduction. In reasoning about matters of fact, one would like to reach conclusions with the certainty of deduction. Unfortunately, induction will not allow it. There’s also another more complex problem lurking here that’s perplexed philosophers: induction seems viciously circular. It seems in fact to assume the very thing it’s trying to prove. Consider the following.
Assuming the uniformity of nature?
Put at its simplest, the problem of induction can be boiled down to the problem of justifying our belief in the uniformity of nature or even reality across space and time. If nature is uniform and regular in its behaviour, then what’s been observed past and present (i.e. premises of an induction) is a sure guide to the so far unobserved past, present, and future (i.e. the conclusion of an induction).
The only basis, however, for believing that nature is uniform is the observed past and present. We can’t then, it seems, go beyond observed events without assuming the very thing we need to prove – that is, that unobserved parts of the world operate in the same way as the parts we observe. In