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Probability, pt. 2

In induction, skepticism on 15/06/2011 at 4:11 pm

Take the statement A, “It will snow on Friday” and the statement B, “It will not snow on Saturday”. The content of the conjunction AB (“It will snow on Friday and it will not snow on Saturday”) will be greater than or equal to any of its components. The more a statements says about the state of affairs, the greater its content. However, note that the probabilities assigned to either A or B require that the probability of the conjunction AB will be smaller than either of the conjuncts A or B.

If we define 0 as ‘false’ (such as a contradiction: for instance “The ball is both all blue and all red at once.”) and 1 as ‘true’ (such as a tautology: for instance, “A prince is a prince.”) and the possible values between 0 and 1 are all the possible assigned probabilities, then AB will always be more improbable or as probable than either the conjuncts A or B. For instance, if we assign the number .5 to A and .5 to B, then AB equals .25. The more a statement says, the less probable it is.

That is to say, with progressive content of our theories (or when a succession of theories increase in their predictive power) they become more and more improbable.

Expressed as follows, where C(A) is ‘the content of the statement A‘ and C(AB) is ‘the content of the statement AB‘:

C(A) ≤ C(AB) ≥ C(B)

In contrast to the probability calculus,

p(A) ≥ p(AB) ≤ p(B)

Now here’s the kicker: strictly universal statements (“for all x, if x, then y”) have an immense content. They say a great deal about the state of the world. In fact, their content is without bounds, for the universal statement, if it were true, will hold in all locations and at all times. Or to look at it in a different light, strictly universal statements are expressed as an infinitely long conjunction of the form C(ABC…) and are thus vastly far more improbable than any finite conjunction.

This increase of improbability holds true for theories that predict more exact events than their predecessors. For instance, take the predictive statement c, “The ball will fall within one meter of the coordinates x,y,z” and the more exact predictive statement d “The ball will fall within one millimeter of the coordinates, x,y,z.” C(d) is less probable than C(c), for d may be false while c remains true. The reason is obvious: if c is true then d must be true, but if d is true then c may not be true and may in fact be false.

The inescapable conclusion is that as we increase the exactness of our theories (or as the theories prohibit more states of affairs) their probability decreases as well.

Now, I may not know many things, but I do know one thing: scientists want interesting theories. They are not just interested in true theories, but theories that are interesting to us, theories that solve our problems. I would think it is a truism that scientists want theories that increase in precision (up to where it is necessary) and theories that increase in breadth. Reciting all the true existential or tautological statements available to scientists is uninteresting. But as we’ve seen, theories that say a lot about what we don’t know, that is, interesting theories that make bold predictions, are also immensely improbable.

The historical record in science shows a progression towards theories that increase in content by unifying disparate fields and increasing the exactness of predictions far beyond their predecessors. We adopt increasingly more content-laden and exact scientific theories. Therefore, the stated goal of the modern inductivist of acquiring highly probable theories and the historical record and stated aim of science are in conflict. We adopt increasingly more improbable scientific theories.

Science is normatively (and, furthermore, ought to be) concerned only with the objective relations between theories, problems, proposed solutions, and so on, not with personal or collective beliefs, convictions and other mental states. Boring theories have very little logical content and as high a predictive value as possible; interesting theories have immense logical content and as low a predictive value as possible. The inductivist gains nothing by replacing the word ‘true’ with the word ‘probable’ and replacing the word ‘false’ by the word ‘improbable.’

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