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il to include a relevant datum (‘2’ in this case), or that an incorrect one occasionally gets included (‘9’ in this case), sometimes due to something as trivial as a typographical error. These things shouldn’t happen but they do. That’s par for the course so far as the human condition goes. And the rule I have spotted is so powerful that the most reasonable conclusion is that the data set you have supplied is a bit dodgy.’

At this point Mr. Smartypants leaves to advise his supervisor that the candidates are all trouble makers and the job should be re-advertised.

What are the morals of this little tale? The first one is one that was (to the best of my knowledge) first pointed out by the Austrian-born Cambridge philosopher Ludwig Wittgenstein (1889 - 1951). It is illustrated by two of the three answers (the tester’s – and Sue’s - ‘officially correct’ answer, and Helen’s (officially incorrect) answer). Given any finite set of data, there is always more than one rule that is perfectly consistent with the data supplied, but which will generate incompatible projections as to what further data to expect. This is not a matter that can be fixed by supplying a bigger data set. For no matter how large the initial data set, Wittgenstein’s maxim will still apply.

From this, a terminological moral is derivable. It is that while, given a rule and a set of initial data, we can uniquely determine the subsequent data for as long and as far out as we like, the reverse is not true; i.e., given a set of data, however large, then we cannot uniquely determine a rule that is consistent with the data. For example (back to the intelligence test), (i) given the initial set of numbers we were given, and (ii), given (which we were not) an applicable rule, then the rest of the data, for as far out as we like, are uniquely determined. BUT, (i) given the initial set of numbers we were given and (ii), asked to continue the series in a way that accords with a rule which is not merely consistent with the given initial set of numbers (easy, at least in principle, because there is, according to Wittgenstein, an infinite number of such rules) but is uniquely determined by the given initial set of numbers, we cannot do it, quite simply because what is demanded is logically impossible.

We might express this by saying, an initial set of data, plus a rule, can – indeed will, if the rule is unambiguous – uniquely determine a series, but the reverse is not true; an initial set of data always under-determines a rule for its generation.

TASK 1: Do you think this shows there is something shonky about the use of these sorts of questions in intelligence tests for e.g. job applicants? Or do you think the use of such tests is defensible? Why? /Why not?

You will be familiar with this word ‘under-determine’ from our previous discussion of theories. This brings us to the second moral of this tale. Our discussion of rules is (I submit) a perfect model for our previous discussion of theories. It takes us right to the nub of the issue, even to the question of – given there is no uniquely correct theory (rule) determined by the available data (or any possible set of such data) - what considerations make one theory (rule) more or less preferable to another. (Sue’s – and the tester’s rule – was simple; Helen’s is inelegant and more complex and, if Leslie is right, of little useful application.)

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