Inductive Reasoning

Philosophical Reflections XXV

In Philosophical Reflections 24 we started to look in more detail at the theory of knowledge – epistemology – specifically at the nature and validity of concepts. Concept formation is at the base of all our knowledge. Now we look at inductive reasoning, the process by which we form our concepts and build the knowledge based on them.

The Primacy of Induction

Induction is the art of deriving the general from the particular. It is complementary to deduction, which is the art of deriving the particular from the general.

It is often thought that deductive reasoning provides greater “certainty” than inductive reasoning. The rules of deductive logic are ironclad: if an argument is valid and its assumptions are true, then its conclusion is guaranteed to be true. In contrast, the rules of induction seem more “rubbery”: they concern generalisation of the “all” from the “some”, and involve evidence, probabilities and best explanations – which can be wrong, even when the evidence is true.

However, it is critical to note those two requirements of a true deduction: “if an argument is valid and its assumptions are true.” In most cases, those assumptions are statements of general fact. And all “general facts” are based on induction. Even if the immediate assumptions are the result of deduction, at base all judgment of the truth of propositions depends on induction. For example, take the classic syllogism:

All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

How do we know all men are mortal? What is man – and how do we know Socrates is one? All these items of knowledge are a result of the inquiry method described in the earliest Philosophical Reflections: the application of reason and experiment to the evidence of our senses. They all depend upon induction.

In Douglas Adams’s The Hitchhiker’s Guide To The Galaxy, the super-computer Deep Thought started with “I think, therefore I am” and had deduced the existence of income tax and rice pudding before they could turn it off. The belief that you can start from introspection and by deep enough deduction come to truths about the universe is called rationalism. Such attempts are doomed to fail. The universe does not exist inside our heads, but outside: and the only way to learn about the external universe is to look at it. And the only way to learn by looking at things is by inductive reasoning.

Concepts, Induction & Deduction

As noted previously, valid concepts are derived from observed reality. We observe – we note similarities – we think – we identify – we combine and differentiate. Thus concept-formation is basically an inductive process.

Deduction is of course important. It gives concepts their predictive usefulness. Having generated the concept of “snake”, and identifying a particular thing as a snake, we can apply our inductively acquired knowledge of the characteristics of snakes to deductively predict the nature of thissnake. Having previously determined that taipans have a deadly bite, we don’t need to poke our finger in this one’s mouth to realise that we shouldn’t.

It is in the nature of concepts and reality that exceptions occur. And this is where deduction again comes in, in refining our concepts. If the nature of a particular existent contradicts our understanding of the nature of its fellows, then we have learned that our concept is inaccurate, or this existent does not actually belong in it, or we have come across an exception. Which of these it is, is the task of our reason to determine.

Thus the logical certainty of deductive reasoning has a vital role in our thinking. All forms of deductive logic are some form of If A then necessarily B, whose converse is If not B then necessarily not A. The first lets us deduce the nature of particular instances of existing concepts: you don’t have to wait for Socrates to die to know that he is mortal. The second refines or refutes our concepts: if a thousand years passes and Socrates is still alive, we know that either he isn’t a man, or our understanding of the nature of man is incomplete. In Philosophical Reflections 24 I noted that the first rule of knowledge is that all our knowledge must be consistent without contradiction. Deduction is how we achieve this. Concepts are generated inductively and tested and refined deductively.

Blacked Out

A commonly used example of inductive reasoning is:

All observed crows are black.
Therefore all crows are black.

This does exemplify the nature of induction: inducing the universal from the particular. And clearly the conclusion is not certain. Unless we’ve seen every crow – and how do we know that? – maybe there are some rare blue ones.

And yet, whatever one says about the certainty of induction, one thing is clear and must be explained: induction works.

It is easy to show that induction works. If anyone claims it doesn’t, ask them to live without using any knowledge which was gained by induction or any related form of learning! Induction is simply a conscious application of the basic mechanism of learning, which can be summarised as once bitten, often shy; once rewarded, often repeated. This is the fundamental mechanism behind all learned behaviour – and all instinctive behaviour, for that matter. And there is one thing that we can be sure of: nothing that universal and fundamental would have evolved if it didn’t work.

Of course, this simple mechanism of learning is not completely reliable. That is why the rule is not “once bitten, forever shy.” It is just a rule of thumb which is based on the fundamental rule of existence: a thing can only act according to its nature. Therefore if a thing has once acted in a certain way, that way is within its nature, and odds are it is typical of that kind of thing.

Inductive reasoning is the conceptual analogy and improvement of this basic mechanism of perceptual learning, and works for the same reason. That things act and can only act according to their nature is the fundamental basis of deductive reasoning – else no conclusion would follow from any assumption – and inductive reasoning – else we could induce nothing general from any particular – and of reality itself – because A is A and can be nothing else. This leads us to the proof of the validity of inductive reasoning.

Inductive Proof

Let us return to the common, but simplistic, example of inductive reasoning:

All observed crows are black.
Therefore all crows are black.

In fact the blackness of crows is a poor example of induction, because the nature of life (as we saw in the last Philosophical Reflections) includes genetic variability. So we actually expect there to be an occasional non-black mutant crow even if every crow we’ve seen so far is black! Crows are a subset of living things, and we can deduce from our much wider inductions about life in general that all crows will not necessarily be black. As proved by the occasional albino.

In the last Philosophical Reflections we saw why concepts, which group multiple things into a single mental unit, are valid: because those things in fact share essential features. For example, fundamental particles such as electrons are all identical in their properties, and by virtue of the fact that A is A, this extends through all things in the universe (all being made of fundamental constituents interchangable with their fellows). Similar considerations apply to living things, despite their inherent variability, because that variability is variation on a theme.

Since only individual things exist in reality and induction concerns generalities, the method and purpose of induction is to find out the qualities shared by the members of a conceptual group. For if they do not share qualities, no generalisation is possible.

By virtue of the fact that A is A, a thing can only be and act according to its nature. From that springs the true foundation of inductive reasoning:

I observe an instance of B with the quality C.
Therefore, the nature of B allows C.

This foundation is certain. Further observations will reveal whether C is rare, common or universal among B, from which we make the first level of induction: an hypothesis about the nature of B, what I shall call descriptive induction. But such hypotheses are far more certain than a simplistic and uncertain All observed B are C, therefore all B are C. There are subsidiary axioms of great import, and when these are made explicit, so is the true scope of inductive truth. For example in the simplest case, physical objects, the full reasoning is:

If all observed B are C,
And B is a simple physical system (i.e., without volition or genetic variability),
Then since A is A, all B are C under the conditions observed (that is, unless affected by some additional factor not yet encountered).

It is well known that our confidence in the conclusions of inductive reasoning increases with the number and quality of our observations. This is implicit in the above. The quality determines the truth of “all observed B are C”, upon which our conclusion rests. The number determine not the truth of our hypothesis, which is guaranteed within the context stated, but its generality: how robust it is against those unknown additional factors. A single case, for example, might be the result of an unusual set of factors, and practically any other set of factors might change it! While a thousand identical outcomes under a wide range of conditions indicates great generality.

Similar considerations apply to the more complex case of living things in which, as we’ve noted, is complicated by inherent variability. In this case, the full reasoning is:

If all observed B are C,
And B is alive,
Then since A is A, the nature of B includes C under the conditions observed. But since life is life, variation is possible even under the same external conditions, especially if C is volitional.

Once again, our confidence in the generality of B are C depends on the quality and number of our observations. But genetic variability and the effects of volition – including both the rational volition of a human being acting according to his or her chosen values, and the lesser but still complex volition of an animal acting according to its experience and internal state – make prediction less certain than with simple physical systems.

Digging Deeper

In terms of practical knowledge, descriptive induction – a general statement about the qualities of certain existents – is often sufficient for our immediate purposes. The qualities of iron which enable its purification, alloying and strength; how to make glass from sand; how to grow grain, thresh it and bake it into bread – were all determined by descriptive induction, long before the whyof all these things was understood.

However, it is the why which is the most important and powerful part of induction, what I call explanatory induction. Epistemologically, knowing why proves the what: because we know whydiamonds are hard, we know that all diamond crystals are hard. Practically, knowing the why gives us much greater power to reproduce and improve desirable qualities, and to predict new ways of making yet more useful things: and thus better achieve our values and improve our lives, which is the ultimate goal of all knowledge. It is our understanding of atoms and the materials made from them which has enabled us to make computers, cars, and aeroplanes; it is our understanding of biology and mechanics which has enabled us to feed an entire population on the work of a few farmers, to cure disease and put us on the threshold of redesigning our own genes.

And that brings us full circle. In the early Philosophical Reflections 2 I noted that the enquiry method by which we learn consists of observation, reason and experiment. There are few experiments more rigorous than building a technology on the basis of our knowledge. Can you calculate the chance that a jumbo jet would fly if the physical principles on which it was based and the materials science by which it was constructed were incorrect? Likewise, it took many careful experiments to learn that DNA – and sometimes RNA – is the molecule of genetics, and to uncover the genetic code by which it works. That we can now make a gene from simple chemicals, put it in a cell, and thereby make that cell do our bidding – to make a product we want, kill a virus, or cure a genetic disease – is the final seal of approval on that knowledge. That knowledge might not be complete – but it is true.

© 2003 Robin Craig: first published in TableAus.