Closer to the truth (4): Realistic realism

In the previous entry I introduced my definition of a concept of empirical verisimilitude of a proposition H given the evidence E (or how close to the truth H seems under the light of data E), Vs(H,E), as the combination of the similarity or coherence between H and E, and that between E and the whole truth T, which was equivalent to p(H,E)/p(HvE) (i.e., the Bayesian ‘degree of confirmation’ of H given E, multiplied by the inverse of the prior probability of the disjunction of H and E, which we can also interpret as the ‘informativeness’ of HvE). As I announced then, I will now proceed to present some of the nice mathematical consequences that can be derived from that formula and that can be interpreted as methodological norms: assuming that the function Vs represents the epistemic utility a proposition H (theory, model, conjecture, law, etc.) provides to a scientist when E are the accepted empirical data, we would like to have at least some criteria telling in what cases the value of Vs is higher or lower for different propositions or for different sets of data. (By the way, though my proposal is neutral as regards the philosophical interpretation of the probability function p, my own preferred understanding of it is as a Bayesian subjective degree of credence, which can be different for different individual scientists, so that Vs(H,E) can be also different for different people even when the logical relations between H and E are objectively given).

One important previous consideration is that Vs always has a non-negative value, but, contrarily to a probability function, it can take any positive finite value (whereas probability functions have a maximum value of 1). Nevertheless, the value 1 still has a special significance, for it gives the (constant) verisimilitude of a tautological proposition Taut (one that is true no matter how the world happens to be, and hence it does not need any empirical data to be ‘confirmed’; we can interpret all mathematical theorems as propositions of this type, ignoring by now the philosophical question of whether all mathematical truths are ‘tautologies’, a position known as logicism and defended by Bertrand Russell, for example). For, obviously, in this case both p(Taut,E) and p(Taut v E) also equal 1. This allows to derive our two first theorems:

(1) If H is a non-tautological proposition confirmed by E (i.e., if E entails H), then Vs(H,E) > 1 [= Vs(Taut,E)],

because, in this case, p(H,E) = 1 > p(HvE).

(2) If H correctly explains E (i.e., if H entails E), then Vs(H,E) > 1 if and only if p(H,E) > p(E),

because, in this case, p(H,E) = P(H)/p(E), and p(HvE) = p(E) (since HvE is equal to E).

These two results invite us to consider the value of 1 (=Vs(Taut)) as a kind of minimal level of verisimilitude one theory or proposition has to surpass in order to be considered ‘minimally valuable’: if your hypotheses is at most as verisimilar (remember ‘seemingly close to the full truth’, not just to ‘being true’) as a tautology, then it cannot be seriously taken as a purported description (no matter how simplified) of ‘the whole truth about the matter’. Theorem (1) asserts that all non-tautological scientific propositions which happen to be empirically confirmed automatically surpass that limit, and are hence ‘valuable’ (to at least some positive extent) as knowledge of the empirical world. Theorem (2) is more profound, referring to theories that are not confirmed by the empirical evidence, but that are conjectures that explain that evidence (but that can be falsified by new evidence, and hence, it is still possible that they are wrong). It imposes a threshold on how implausible a theory H has not to be in order to be considered as a ‘serious’ alternative, for what (2) says is that, in order to be better than a tautology, H must have a degree of confirmation on the light of E, or H’s posterior probability, (p(H,E)) which is at least as high as the prior probability E had to be true in principle (p(E), i.e., before knowing whether E was true or not). Stated otherwise, E, the sum of all our empirical knowledge about the world (or the portion of the world which is relevant for the scientific endeavor in question), is what it is, but in principle it might have been different (for purists: ‘in principle’, given our ‘background beliefs’, or whatever basic ideas about the world we had before engaging in the scientific problem we are trying to solve in this case, and that are not debunked by our discoveries; of course, under my subjectivist Bayesian interpretation, these ideas can be different for different scientists, for example for members of different Kuhnian ‘paradigms’). Hence, p(E) will usually be a relatively small number, and the smaller it is, the more implausible a priori the known empirical laws will be (we shall encounter this important fact again). What theorem (2) asserts is that, for being ‘eligible’ as a serious explanatory candidate, a theory H has not only to explain E, but has to be at least more plausible once we know E, than E was a priori (by the way, another formulation of theorem (2) is that p(H) must be higher than p(E)²).

My two first theorems seem to have a strongly anti-Popperian flavor, for Sir Karl apparently was famous for rejecting ‘confirmation’ as a legitimate scientific method, and for defending under all circumstances what he called ‘bold’ theories. Instead, (1) looks like a kind of ‘positivist’ sympathy for confirmationist strategies, and (2) puts a limit to the ‘boldness’ it is sensible to demand of a theory. However, I think that my two first methodological norms are consistent with a realist (though not necessarily ‘naïve falsificationist’) interpretation of the scientists’ epistemic utility function (to the extent this is appropriately represented -though in a simplified and idealized way- by our Vs): both transmit the idea that we are pursuing not just ‘truth’, but closeness to the whole truth, and hence it is good (for being a purported description of the ‘whole’ truth) to be more verisimilar than any tautology (and, of course, the further from the verisimilitude of Taut, the better); and in particular the second norm cautions scientists about the risk of being too speculative, for this can lead them further from the places where the whole truth may lay (I mean, further from the ‘real world’ that is one of the many possible worlds contained in the set of worlds not discarded by E). Hence, it is good to be a scientific realist, but it is important to be a realistic realist. But more on that in our next entries.

References

Zamora Bonilla, J., 1996, “Verisimilitude, structuralism and scientific progress”, Erkenntnis 44 (1), 25-47.