Feb 072014

I don’t just believe, I know – said Arthur Conan Doyle in his impassioned defence of Spiritualism. Beliefs are the result of an evidential tug of war, where none of the two sides manages to prevail on the other. Beliefs are measured by probabilities. Probabilities reflect the weight of evidence but, without a winner, remain dependent on prior beliefs.

Beliefs become knowledge when the tug of war has a winner. The surest way to gain a victory is to find a piece of conclusive evidence. Conclusive evidence ends the tug of war: the hypothesis is certainly true or false – it cannot possibly be any other way, irrespective of prior beliefs and of all other evidence that has been accumulated until then on either side. Conclusive evidence can be positive or negative, and can prove the hypothesis certainly true or certainly false.

Famously, Karl Popper required conclusive evidence in order to distinguish between scientific and unscientific hypotheses: a hypothesis is scientific if and only if it can designate some conclusive evidence that may refute it, i.e. prove it certainly false. Defining scientific hypotheses as universal statements (as opposed to singular or particular statements), Popper excluded that they could be proven certainly true:

It is far from obvious, from a logical point of view, that we are justified in inferring universal statements from singular ones, no matter how numerous; for any conclusion drawn in this way may always turn out to be false: no matter how many instances of white swans we may have observed, this does not justify the conclusion that all swans are white (The Logic of Scientific Discovery, p. 4).

This is the Problem of Induction, as outlined in the second century by Sextus Empiricus:

When they propose to establish the universal from the particulars by means of induction, they will effect this by a review of either all or some of the particulars. But if they review some, the induction may contravene the universal; while if they are to review all, they will be toiling at the impossible, since the particulars are infinite and indefinite.

No number of white swans can certainly prove that the hypothesis “All swans are white” is true. But just one black swan is sufficient to certainly prove that it is false. According to Popper, “All swans are white” is a scientific hypothesis because it is open to the possibility of being refuted by conclusive evidence – as indeed it did, after black swans were found in Australia at the end of the 17th century. In our framework, a black swan is a Perfect Alibi: conclusive positive evidence that proves that the hypothesis is certainly false.

But what if we invert the hypothesis, and instead of “All swans are white” we posit “Not all swans are white”, or “Some swans are non-white” or “There exist non-white swans”? Surely, a black swan would constitute conclusive evidence that the hypothesis is certainly true. As Popper specified, however, “the asymmetry between verifiability and falsifiability … results from the logical form of universal statements” (p. 19). “All swans are white” is a universal statement, a generalization valid for all swans, whereas the other posits are particular statements, valid only for some swans. In the same vein, “All psychic phenomena are due to some form of forgery” is a universal statement, whereas “Some psychic phenomena are authentic” is not. Therefore, no number of exposed frauds can prove the universal statement true – as Conan Doyle kept reminding an exasperated Houdini – but even one authentic, certified psychic experiment can prove it false.

But there is no reason why scientific hypotheses should comprise only universal statements. There exist non-white swans, or psychic phenomena, extra-terrestrials, miracles, global warming, biological evolution, Higgs bosons: these are all perfectly legitimate scientific hypotheses. What makes them scientific is that, unlike Faith, they are open to evidence – but to all kinds of evidence: conclusive as well as inconclusive, confirmative as well as disconfirmative.

Etymology often helps us to deepen our understanding of words.

Science (n.): mid-14c., “what is known, knowledge (of something) acquired by study; information;” also “assurance of knowledge, certitude, certainty,” from Old French science “knowledge, learning, application; corpus of human knowledge” (12c.), from Latin scientia “knowledge, a knowing; expertness,” from sciens (genitive scientis) “intelligent, skilled,” present participle of scire “to know,” probably originally “to separate one thing from another, to distinguish,” related to scindere “to cut, divide,” from PIE root *skei– “to cut, to split” (cf. Greek skhizein “to split, rend, cleave,” Gothic skaidan, Old English sceadan “to divide, separate;” see shed (v.)).

So Science is knowledge – a belief that has achieved certainty by means of evidence, thus allowing us to cut, separate, decide true from false. A decision may come from conclusive evidence – any of the four types, not just a Perfect Alibi. For example, a black swan is a Smoking Gun for the hypothesis “There exist non-white swans”. Likewise, a psychic experiment conducted under suitably controlled conditions would prove that the hypothesis “There exist Spirits” is true – a Smoking Gun we are still waiting for.

But certainty does not need conclusive evidence. The accumulation of inconclusive, albeit consistently confirmative (LR>1) or disconfirmative (LR<1) evidence can lead posterior probabilities to converge towards one of the two boundaries of the probability spectrum. Convergence is in the limit: we cannot be absolutely sure that the sun will rise tomorrow, but our accumulated experience is a de facto Smoking Gun – that’s why we don’t worry about sunrise.

Popper’s insistence on scratching the left ear with the right hand generated unnecessary confusion. There is no asymmetry between verifiability and falsifiability. Scientific discovery is a tug of war between confirmative and disconfirmative evidence, whose purpose is to reach certainty, but whose ethos is to be comfortable with uncertainty. As Richard Feynman brilliantly put it:

What we call scientific knowledge today is a body of statements of varying degrees of certainty. Some of them are most unsure; some of them are nearly sure; but none is absolutely certain. Scientists are used to this. We know that it is consistent to be able to live and not know. Some people say, “How can you live without knowing?” I do not know what they mean. I always live without knowing. That is easy. How you get to know is what I want to know (The Meaning of It All, p. 27).

Unlike Faith, Science is open to evidence, on both sides. When a statement is most unsure, it is natural to look for confirmative evidence and, if possible, for a conclusive proof that the statement is true. When a statement is nearly sure, it is sensible to look for disconfirmative evidence and, if possible, for a conclusive refutation.

In this sense, Popper is right. An appropriate test for a well-established hypothesis is: is there any conclusive evidence that would be able to refute it? If there is – a black swan for “All swans are white”, a certified psychic experiment for “Spiritualism is bonkers”, a Precambrian rabbit for biological evolution – passing the test, i.e. failure to find such refuting evidence, adds further strength to the hypothesis, although it does not prove it true. On the other hand, finding the evidence does imply an immediate rejection, irrespective of how much confirmative evidence had been accumulated until then.

But what if there is no possible refuting evidence? Would that render the hypothesis necessarily “unscientific”? Take global warming for example. Arguably, while there is some amount of disconfirmative evidence – not enough, it seems, to counterbalance the large and growing amount of confirmative evidence – it is virtually impossible to think of a specific piece of conclusive evidence that would lead to an outright rejection of the hypothesis. Nevertheless, no one can seriously doubt that global warming is a legitimate scientific hypothesis. What makes it scientific is not that it is open to refutation but, more simply, that it is open to evidence. In the absence of conclusive evidence on either side, we cannot be certain that the global warming hypothesis is true or false. But that does not exempt us from the need to weigh the available evidence and take a decision.

On the other hand, if a hypothesis is not well-established, but tentative and highly uncertain, the right approach is to look for conclusive evidence that it is true. Sometimes it works: a black swan for “There exist non-white swans”, a Large Hadron Collider experiment for “There exist Higgs bosons”. But failure to find a proof, while weakening the hypothesis, does not prove it is false.

To complete the symmetry (and not because Popper affirmed it – he didn’t), if finding a conclusive proof turns out to be impossible, it does not mean that the hypothesis is unscientific. Think for example of string theory: arguably, it can never be conclusively proven (or falsified) but, being open to evidence, it is also a legitimate scientific hypothesis, whose validity needs to be decided by weighing its pros and cons.

In summary, there is no asymmetry: if we are nearly sure about a hypothesis, we should try to falsify it; if we are most unsure about it, we should try to verify it. That’s nothing other than what Inspector Hubbard and the Court did with Tony and Margot.

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