'Extra-Weak' Underdetermination

I’ll start briefly with a few reminders and fix a little terminology. I then introduce a new(?) sub-species of underdetermination argument, whose premises are logically weaker than existing underdetermination arguments, but still(?) deliver the anti-realist’s conclusion.

Underdetermination arguments in philosophy of science aim to show that an epistemically rational person (= someone who weighs their available evidence correctly) should suspend belief in the (approximate) truth of current scientific theories, even if such theories make very accurate predictions.

A scientific theory T is strongly underdetermined = T has a genuine competitor theory T*, and T and T* make all the same observable predictions. (So the two theories’ disagreement must concern unobservable stuff only.)

A scientific theory T is weakly underdetermined = T has a genuine competitor theory T*, and all the observable data/ evidence gathered thus far is predicted equally well by both T and T*.(†) (So collecting new data could end a weak underdetermination situation.)

Anti-realists then argue from the purported fact that (all/most) of our current scientific theories are undetermined, to the conclusion that an epistemically rational person should suspend belief in (all/most) of our current scientific theories.

Realists can reasonably respond by arguing that even weak underdetermination, in the above sense, is not common: even if one grants that there is an alternative theory T* that is consistent with the data gathered so far, that certainly does not entail that T and T* are perfectly equally supported by the available data is unlikely. There is no reason to expect T and T* would be a perfect ‘tie’ for every other theoretical virtue besides consistency with the data. (Theoretical virtues here include e.g. simplicity, scope, relation to other theories, etc.) The evidential support for a hypothesis is not merely a matter of the consistency of that hypothesis with available data.

At this point, anti-realists could dig in their heels and simply deny the immediately preceding sentence. (The other theoretical virtues are ‘merely pragmatic,’ i.e. not evidential.) But that generates a standoff/stalemate, and furthermore I find that response unsatisfying, since an anti-realist who really believes that should probably be a radical Cartesian skeptic (yet scientific anti-realism was supposed to be peculiar to science).

So here’s another reply the anti-realist could make: grant the realist’s claims that even weak determination is not all that common in the population of current scientific theories, and furthermore that typically our current theory T is in fact better supported than any of the competitors T₁, T₂, ... that are also consistent with the data collected so far. The anti-realist can grant these points, and still reach the standard underdetermination-argument conclusion that we should suspend belief in the truth of T, IF the sum of the credences one should assign to T₁, T₂, ... is at least 0.5.

For example: suppose there are exactly 3 hypotheses consistent with all the data collected thus far, and further suppose
 Pr(T) = 0.4,
 Pr(T₁) = 0.35, and
 Pr(T₂) = 0.25.
In this scenario, T is better supported by the evidence than T₁ is or T₂ is, so T is not weakly underdetermined. However, assuming that one should not believe p is true unless Pr(p)>0.5, one should still not believe T in the above example.

I call such a T extra-weakly underdetermined: the sum of rational degrees of belief one should have in T’s competitors is greater than or equal to the rational degree of belief one should have in T.

We can think about this using the typical toy example used to introduce the idea of underdetermination in our classes, where we draw multiple curves through a finite set of data points:

We can simultaneously maintain that the straight-line hypothesis (Theory A) is more probable than the others, but nonetheless deny that we should believe it, as long as the other hypotheses’ rational credence levels sum to 0.5 or higher. And there are of course infinitely many competitors to Theory A, so it is an infinite sum. The realist, in response to this argument, will thus have to say that that infinite sum will converge to less than 0.5.

The above argument from extra-weak underdetermination is clearly related to the ‘catch-all hypothesis’ (in the terminology above, ~T) point that has been discussed elsewhere in the literature on realism, especially in connection with Bayesian approaches (see here and the references therein). But I think there is something novel about the extra-weak underdetermination argument: as we add new competitor theories to the pool (T₃, T₄… in the example above), the rational credence level we assign to each hypothesis will presumably go down. (I include ‘presumably,’ because it is certainly mathematically possible for the new hypothesis to only bring down the rational credence level for some but not all of the old hypotheses.) So the point here is not just that there is some catch-all hypothesis, which it is difficult(?) to assign a degree of rational belief to (that's the old news), but also that we increase the probability of something like the 'catch-all' hypothesis by adding new hypotheses to it. (I have to say 'something like' it, because T₁, T₂... are specific theories, not just the negation of T.)

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(†): Note that, unlike some presentations of underdetermination, I do not require that T and T* both predict ALL the available data. I take "Every theory is born refuted" seriously. And I actually think this makes underdetermination scenarios more likely, since a competitor theory need not be perfect -- and the imperfections/ shortcomings of T could be different from those of T* (e.g. T might be more complicated, while T* has a narrower range of predictions). Hasok Chang's Is Water H2O? makes this point concretely and (to my mind) compellingly, about the case of Lavoisier's Oxygen-based chemistry vs. Priestley's Phlogiston-based chemistry.