After reading a certain amount of philosophy, it is easy to become confused as to exactly what
the problem of induction is. For our purposes, however, the problem of induction is very simple
and straightforward. Why is it justified for a system to assume that a pattern it has observed in its environment or itself will continue to be a pattern in its environment or itself in the future?
Leibniz (1704) was the first major philosopher to comprehend the seriousness of the problem
of induction. He did not presume to have solved it, but rather claimed to have a research
programme which would eventually lead to its solution. His "Universal Characteristic" was to be
a systematization of all human knowledge in mathematical form, in such a way that the various
possible answers to any given question could be assigned precise probabilities based on the
background knowledge available.
But although Leibniz made important advances, Hume (1739) was the first to give the
problem of induction its modern form. He gave a sequence of brilliant arguments to the effect
that human knowledge is, in fact, induction. For instance, he spent a great deal of effort
demonstrating that the "fact" that one thing causes another, say that fire causes smoke, is
"known" to us only in the sense that we have seen it do so consistently in the past, and therefore assume it will continue to do so in the future. At the time this was an largely novel insight into the nature of human knowledge. The crux of his analysis of induction was his argument that it is fundamentally unjustifiable; however, without the conception of knowledge as inductive, his clever insight in this regard would have been meaningless.
After all — to summarize and simplify Hume’s argument — how do we knowinduction works?
Either we know it by "divine revelation" or "deduction" or some other principle which has
nothing to do with the specific properties of the real world, or we know it because of some
reasoning as to specific properties of the real world. If the latter, then how do we know these
specific properties will continue into the future? This assumption is itself an act of inductive
reasoning! So the two alternatives are: justify induction by some a priori principle unconnected
with the real world, or justify induction by induction. And the latter is inadmissible, for it is circular reasoning. So induction is just another form of dogmatism.
I take it for granted that Hume was right — induction is unjustifiable. But nonetheless, we
execute inductions every day. As I see it, the practical problem of induction is the problem of
coming up with a simple, general, useful model of the universe according to which induction is
indeed possible. This is quite distinct from the philosophical problem of induction. In solving the practical problem, we are permitted to justify induction in terms of some principle divorced from observable reality. The objective is to find the best way of doing so.
A SIMPLE EXAMPLE
Consider the sequence 1010101010…. Given no background knowledge except that there will
indeed be a next term, and that it will be either 0 or 1, simple intuitive inductive reasoning
indicates that the next term should be a 1. One reasons: "Every time a 0 has occurred, it has been followed by a 1; hence with probability 1 this 0 is followed by a 1."
Similarly, given the sequence 010101001010101010… and the same minimal background
information, one could reason: "Eight out of the nine times a 0 has occurred, it has been followed by a 1; hence with probability 8/9 this zero is followed by a 1."
But this sort of reasoning is, of course, plagued by serious problems. It makes the implicit
assumption that the probability distribution of 0’s and 1’s will be the same in the future as it
was in the past. So it makes an inductive assumption, an assumption as to the "regularity" of the world. There is no "a priori" reason that such assumptions should be justified — but we
intuitively make them, and they seem to work fairly well.
INDUCTION AND DEDUCTION
Many people would be willing to accept logical deduction on faith, and justify induction in
terms of deduction. This would be one way of solving the practical problem of induction;
unfortunately, however, it doesn’t seem to work. Even if one does take it on faith that the
universe is constituted so that the familiar rules of deductive logic are valid, there is no apparent way of solving thepractical problem of induction. The rules of deductive logic give no reason to assume that the regularities of the past will continue into the future.
In one sense this is a technical point, regarding the specific forms of deductive logic now
known. It might be argued that we simply don’t know the true deductive logic, which would
justify induction. But that is not a very convincing argument; it is certainly not something on
which to base a theory of mind.
And modern physics has added a new wrinkle to this controversy. In 1936, Von Neumann and
Birkhoff proposed that a special non-classical "quantum logic" is required for the analysis of
microscopic phenomena. Over the past few decades this suggestion has evolved into a
flourishing research programme. Mittelstaedt (1978) and other quantum logicians contend that
the choice of a deductive system must be made inductively, on the basis of what seems to work
best for describing reality — that what we call logic is not absolute but rather a product of our particular physical experience. If one accepts this postulate, then clearly there is no point in trying to justify induction deductively, for this would be just as circular as justifying induction inductively.