Hacking On Koopman

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\title Defending Koopman’s logic of support.

Koopman’s logic of support is one of the most historically important formalisations of the foundations of credence, with only de Finetti’s system being a serious competitor to it at the time of its publication in 1940 [[CITE]]. Arguably it is still a plausible system all these years later. I do not intend to show in this paper that it is correct; rather, I will contribute the project of resurrecting it by tying up a loose end in an argument of Hacking’s which purports to show that Koopman’s system is defective.

Hacking says:

is one possible defect in Koopman’s logic which had better be recorded. His logic must be a trifle too strong, for it includes the principle of anti-symmetry:

$$\hboxIf h|e \le i|d, \hbox then \neg i|d \le h|e.$$

is cast on the principle if we put $e = d = $ a box contains 40 black balls and 60 green ones; a ball will shortly be drawn from the box'. Let $i = $the next ball to be drawn from the box will be black’, and let $j = $ `the next ball to be drawn from the box will be green’. I take it that e supports both i and j; without e or some other piece of evidence, there would be no support for either of those propositions, but given e, there is some support. Now i and j are contraries, so j implies ¬i. By the thesis of implication quoted earlier, it follows that e supports ¬i at least as well as j.

Hacking goes on to show that his example values of e, d, i & j, together with anti-symmetry, cause problems. But that is not the fault of anti-symmetry at all. e supports j and j deductively entails ¬i, but e doesn’t support ¬i. So Hacking’s example contains an internal inconsistency. Hence we have a problem with the values of the variables, without even mentioning anti-symmetry. It is no surprise that the problem persists when we add the principle of anti-symmetry. That anti-symmetry fails to fix the problem does not mean that it is anti-symmetry which has introduced the problem. I conclude that Hacking’s example fails to show us anything about whether we should accept anti-symmetry.

A natural question is whether the example can be changed to avoid internal inconsistency, in such a way that the fixed version supports Hacking’s view that the principle of anti-symmetry needs to be amended.

Firstly, the most obvious, minimal changes to the example which avoid the internal inconsistency do not support Hacking’s view. I take it that such a minimal change is …

How does this work out in Bayesian language? It depends on the priors for i and j.