Closer to the truth (5): Reconstructing ‘the scientific method’

By Jesús Zamora Bonilla March 24, 2025 0 comments Print

theorems — Photo: Maryna Nikolaieva / Unsplash

I shall end this series on the problem of verisimilitude by sketching the main methodological norms that can be derived from our favorite definition of “empirical truthlikeness” –remember: the verisimilitude of a hypothesis H on the light of the empirical data E, or Vs(H,E), would be equivalent to p(H,E)/p(HvE). Remember as well that by ‘methodological norms’ we are understanding criteria that allow to compare the epistemic value of one or several scientific hypotheses. Besides the two most basic principles presented in our last entry (that, amongst hypotheses confirmed by the data, the more ‘contentful’, the better; and amongst hypotheses that correctly explain the data, the less ‘speculative’, the better), here are some of the most interesting results (I will dispense the reader of the mathematical proofs of the theorems, what can be found in the paper mentioned in the References, and which are in any case rather trivial):

The maximum verisimilitude that the empirical data E can confer to any theory is 1/p(E), and hence, the only way of getting a still higher level of empirical verisimilitude is by discovering new empirical data, not derivable from E.
A succesful new prediction increases the epistemic value of a hypothesis:
1. If H entails F, and E didn’t entail F, then Vs(H,E) ≤ Vs(H,E&F)
2. If p(F,E) ≤ p(F,H&E), then Vs(H,E) ≤ Vs(H,E&F) (this result applies when the prediction is ‘statistical’, i.e., when the hypothesis does not logically entail it, but makes it more probable).
Showing that theory H, which is empirically successful i.e., such that H entails E, is mathematically reducible to theory G which, before the proof, was not to known to entail E), increases the verisimilitude of G. [This is a trivial corollary of (1), for it amounts to ‘import’ all the successful predictions of H to G].
If the hypothesis H is closer to E than G in the Popperian sense of ‘closeness’ (i.e., if all possible worlds compatible with E and G are also compatible with H, and all possible worlds compatible with H and incompatible with E are compatible with G; or, more graphically, if G&E entails H, and H&¬E entails G), then Vs(G,E) ≤ Vs(H,E).
Since usually a scientific theory T is a conjunction of different independent hypotheses (H₁, H₂, …), and since the empirical verisimilitude of T, when T successfully explains E, depends on the prior probability of p(T), it is rational to expect that scientists trying to find the ‘best’ theory out of a given set of possible hypotheses H_i (or a ‘research program’) start with those combinations of hypotheses that are more mutually coherent (that is, whose conjunction has the highest prior probability).
The statistical expected value of the verisimilitude of H given E equals p(E,H) (what is usually known as the likelihood of H given E). [This is the only non trivial result; to prove it, it suffices to consider that the expected verisimilitude of H given E is the weighted sum of the verisimilitude of H given W_i, for every complete description W_i of the world compatible with E, pondered by p(W_i,E), and that Vs(H,W_i) equals 1/p(H) if Wi entails H, and 0 if not].

The first three of these rules offer a nice mix of the ‘Popperian’ and ‘Logical Positivist’ views of the scientific method: the state of scientific knowledge is always provisional and improvable, science progresses by new theories explaining the known data and making new predictions, and mathematical reduction of the ‘old’ successful theories to the ‘new’ ones is one common way of achieving scientific progress. The fourth methodological norm is surely not very useful, because I doubt that theories H and G exist very often that have the precise logical relation to E stated in its premise, but it is a lovely reminding of how Popper’s initial formal intuition at the beginning of the verisimilitude debate may, in spite of its initial logical failure, nevertheless have some rationale when interpreted a little bit differently.

Theorems with a Lakatosian taste

The last two theorems are perhaps the most interesting ones and have what I would call a strong ‘Lakatosian’ taste. Imre Lakatos was a disciple of Popper that defended that scientific debates are not structure around single hypotheses or theories, but that these were organized within what he called scientific research programs: a ‘blueprint’ of how to arrange and combine several alternative hypotheses in order to construct theories more and more empirically successful, so that, in the first place, the falsification of one theory does not amount to the ‘falsification’ of the program itself, and that, in the second place, the theories within the program are not elaborated and developed in a random order, but according to a kind of logical plan. What theorem 5 asserts is that, when successful, the defenders of the scientific program will prefer those theories that have the highest prior probability (i.e., that look less speculative, as we explained in our last entry), and it is this criterion the one that can be mainly used in the ‘ordering’ of the hypotheses that are going to be used in the construction of the theories of the program. This methodological rule can be phrased as ‘try first the most likely’.

Provisionality of our epistemic assessments

Theorem 6 allows adding an interesting turn of the screw to this Lakatosian predicament, and (besides offering a nice explanation of why statistical likelihood is very often used as a measure of the epistemic value of a hypothesis, a measure easily applicable in cases where H does not completely explain E) connects itself with the basic idea of the first theorem, that about the intrinsic limitation of our epistemic knowledge, and the consequent provisionality of our epistemic assessments. We can reasonably assume that, in a process of research around a particular scientific problem, scientists will expect that the empirical data they have are more limited at the beginning, whereas as the research proceeds, it is less and less difficult to expect that new important data are discovered that add much more of interest to what was already known. This means that it is reasonable to assume that, at the beginning of the research, scientists will be more predisposed to judge the competing theories by their expected empirical value, whereas when the available empirical evidence ‘solidifies’, they will turn to assess theories by their ‘direct’ level of empirical verisimilitude. This puts a limit to the conclusions of the previous paragraph, because, at the beginning and the most intense phases of discoveries, scientists would not be that much fixated with the prior plausibility of the considered hypotheses (i.e., (H)), and only will take into account how well these explain the available data (i.e., p(E,H)), no matter how implausible H is. For example, they can start testing very simple (or simplistic) theories or models, just to see if they manage to make sense of the data, or even very speculative ideas that happen to have wonderful empirical results. All this means that the norms with which scientists evaluate their theories in the short run are more ‘instrumentalist’ (i.e., only concerned about empirical success) than the norm they apply in the long run, which would be more ‘realist’ (not only concerned about empirical success, but also about conceptual plausibility; and, by the way, when they are engaging in that kind of discussions, they will usually resort to ‘philosophical’ arguments trying to show why certain ideas or assumptions are more ‘intrinsically plausible’).

Margin of error

One major flaw that might be imputed to our model is that it automatically makes the empirical verisimilitude of a refuted theory equal to zero. This can be circumvented by allowing scientists to judge theories not merely against the last and most precise corpus of available empirical data, but, more benevolently, against the most favorable subset of that corpus, or even allowing for the fact that empirical facts or laws are usually expressed with a margin of error. Perhaps H is logically incompatible with E strictly speaking, but happens to entail a ‘weakened’ version of E (one containing a few less facts, and others that are expressed only in an approximate way). For example, Kepler’s laws are literally falsified by the actual measured trajectories of the planets, but ‘perfectly’ explain a very close approximation of them. It is interesting to note that this move of mine is in itself an application of the ideas underlying my two last theorems: what I am offering here is not so much a ‘rational theory’ of how science should work, but just a ‘(mini) scientific model’ of how real scientists evaluate their own hypotheses, and this model can be considered in itself as simplified, provisional, and improvable according to some intuitive guidelines, as scientific theories themselves are, even if the most initial versions of it are ‘refuted’ by the relevant data. Hence, please take all this as just a minimal model, not a profound philosophical theory.