#93 (C&R Chap 10, Part I) - An Introduction to Popper's Theory of Content
Increments
Where Bayesians Fit: Updates vs. Hypothesis Generation
Ben and Vaden argue over whether Bayesianism addresses hypothesis generation or only prescribes how to update degrees of belief given hypotheses.
Back to basics baby. We're doing a couple introductory episodes on Popper's philosophy of science, following Chapter 10 of Conjectures and Refutations. We start with Popper's theory of content: what makes a good scientific theory? Can we judge some theories as better than others before we even run any empirical tests? Should we be looking for theories with high probability?
Ben and Vaden also return to their roots in another way, and get into a nice little fight about how content relates to Bayesianism.
We discuss
- Vaden's skin care routine
- If you find your friend's lost watch and proceed to lose it, are you responsible for the watch?
- Empirical vs logical content
- Whether and how content can be measured and compared
- How content relates to probability
Quotes
My aim in this lecture is to stress the significance of one particular aspect of science—its need to grow, or, if you like, its need to progress. I do not have in mind here the practical or social significance of this need. What I wish to discuss is rather its intellectual significance. I assert that continued growth is essential to the rational and empirical character of scientific knowledge; that if science ceases to grow it must lose that character. It is the way of its growth which makes science rational and empirical; the way, that is, in which scientists discriminate between available theories and choose the better one or (in the absence of a satisfactory theory) the way they give reasons for rejecting all the available theories, thereby suggesting some of the conditions with which a satisfactory theory should comply.
You will have noticed from this formulation that it is not the accumulation of observations which I have in mind when I speak of the growth of scientific knowledge, but the repeated overthrow of scien- tific theories and their replacement by better or more satisfactory ones. This, incidentally, is a procedure which might be found worthy of attention even by those who see the most important aspect of the growth of scientific knowledge in new experiments and in new observations.
- C&R p. 291
Thus it is my first thesis that we can know of a theory, even before it has been tested, that if it passes certain tests it will be better than some other theory.
My first thesis implies that we have a criterion of relative potential satisfactoriness, or of potential progressiveness, which can be applied to a theory even before we know whether or not it will turn out, by the passing of some crucial tests, to be satisfactory in fact.
This criterion of relative potential satisfactoriness (which I formu- lated some time ago,2 and which, incidentally, allows us to grade the- ories according to their degree of relative potential satisfactoriness) is extremely simple and intuitive. It characterizes as preferable the theory which tells us more; that is to say, the theory which contains the greater amount of empirical information or content; which is logically stronger; which has the greater explanatory and predictive power; and which can therefore be more severely tested by comparing predicted facts with observations. In short, we prefer an interesting, daring, and highly informative theory to a trivial one.
- C&R p.294
Let a be the statement ‘It will rain on Friday’; b the statement ‘It willbe fine on Saturday’; and ab the statement ‘It will rain on Friday and itwill be fine on Saturday’: it is then obvious that the informative contentof this last statement, the conjunction ab, will exceed that of its com-ponent a and also that of its component b. And it will also be obviousthat the probability of ab (or, what is the same, the probability that abwill be true) will be smaller than that of either of its components.
Writing Ct(a) for ‘the content of the statement a’, and Ct(ab) for ‘thecontent of the conjunction a and b’, we have
(1) Ct(a) <= Ct(ab) >= Ct(b).This contrasts with the corresponding law of the calculus of probability,
(2) p(a) >= p(ab) <= p(b),
where the inequality signs of (1) are inverted. Together these two laws, (1) and (2), state that with increasing content, probability decreases, and vice versa; or in other words, that content increases with increasing improbability. (This analysis is of course in full agreement with the general idea of the logical content of a statement as the class of all those statements which are logically entailed by it. We may also say that a statement a is logically stronger than a statement b if its content is greater than that of b—that is to say, if it entails more than b does.)
This trivial fact has the following inescapable consequences: if growth of knowledge means that we operate with theories of increasing content, it must also mean that we operate with theories of decreasing probability (in the sense of the calculus of probability). Thus if our aim is the advancement or growth of knowledge, then a high probability (in the sense of the calculus of probability) cannot possibly be our aim as well: these two aims are incompatible.
- C&R p.295
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