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A hypothesis can be seriously weakened by an example that supports it. Assumption: No one can be taller than 9 feet. Example: There is a person who is 8 feet 1 inch tall". The discovery of such a person supports this hypothesis ...... but at the same time casts a long shadow of doubt on this hypothesis.

How do Popper and later philosophers of science discuss examples like this one?

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    What does it mean? Apr 28, 2022 at 9:38
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    Provided that that is not the way the science works, we have many examples of "dubious evidence". Consider the well-know example of Michelson-Morley experiment: to measure the relative speed of light vs aether. The result was: no evidence of relative speed, but there were many attempt to explain the result in the context of "classical" physics up to Einstein proposal with Relativity. Apr 28, 2022 at 9:41
  • Regarding your issue, see Vagueness. Apr 28, 2022 at 9:45

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I shall argue that a phenomenon of weakening positive evidence is coherent epistemologically and has an explanatory capacity in the matters of scientific theory confirmation.

Let us consider a similar social situation, that of being "damned by faint praise", which is quite familiar to us. Suppose X queries Y whether Z is qualified enough for some position, such as a job, a scholarship, a tenancy, a contract for an art production. Y tells her overall assessment of Z's traits with praising words, save for the key ones. For example, if a good command of Chinese is required, Y either never mentions Z's capability about it or very lightly in passing. Thus, X gets the impression that Z is not the right person according to Y, despite all the positive description. That a weak positive evidence works against the belief it is supposed to support is also a subject of field research (see Brown University's report Weak Supporting Evidence can Undermine Belief in an Outcome). An explanation of the underlying mechanism has been proposed by by A. J. L. Harris, A. Corner and U. Hahn in James is Polite and Punctual (and Useless): A Bayesian Formalisation of Faint Praise. The following discussion is benefitted from the approach of that study.

The significant effect of evidence in the present case is that it causes a belief change. Specifically, we look into an occurrence of a weak evidence that updates belief negatively. Bayes' theorem in probability theory offers a proper reference framework for an analysis of such a case. Although it is not possible in many cases to quantify the constituent probabilities to put probability theoretic formalism into quantitative work, it is useful to reason conceptually.

We consider a hypothesis H, so P(H) is the prior probability of the hypothesis being true and P(¬H) being false. We can partition the probability space associated with evidences into weak positive evidence W, strong positive evidence S, no (neutral) evidence and negative (falsifying) evidence. Thus as prior probabilities (the probabilities that we have acquired a priori with respect to the occurrence of a weak evidence), we have

P(W) + P(S) + P(No) + P(Neg) = 1
P(H) + P(¬H) = 1

By the law of total probability, we can write

P(H|W) + P(H|S) + P(H|No) + P(H|Neg) = P(H)
P(¬H|W) + P(¬H|S) + P(¬H|No) + P(¬H|Neg) = P(¬H)

Bayes' theorem tells

P(H|W) = [P(W|H).P(H)] / [P(W|H).P(H) + P(W|¬H).P(¬H)]

where P(H|W) is the posterior probability of H being true upon the occurrence of W, P(W|H) is the likelihood of the weak evidence given the hypothesis.

It should be remarked that the connections of an evidence to others: In the presence of a strong evidence, the weak evidence would have a corroborating effect.

We can draw on the law of likelihood that Anthony Edwards puts forward in his book Likelihood (2nd edition, Johns Hopkins University Press, 1992, p. 30, emphasis in the original):

Within the framework of a statistical model, a particular set of data supports one statistical hypothesis better than another if the likelihood of the first hypothesis, on the data, exceeds the likelihood of the second hypothesis.

In this formulation, we take the first hypothesis as H and the second hypothesis as ¬H. Then, ¬H is expected to be dominant over H:

P(W|¬H) > P(W|H)

Also we may expect that the prior probability of H is higher than its probability on the condition that W occurs:

P(H) > P(H|W)

Now, the question is how it is possible that W, being positive, affects the hypothesis H adversely? An answer has already been studied by Douglas Walton. Walton holds that the fallacy of argumentum ad ignorantiam (in this term, ignorance means "not having the knowledge of something that it is so and so") a cogent reasoning if the condition of epistemic closure is satisfied. He defines epistemic closure as

If one knows that it cannot be the case that A without his knowing it, then, if not-A, he can infer that not-A.

Adapting for the present case, we know that if a strong evidence has occurred, we would have the knowledge of it (for an overview of the subject, see Steven Luper's article Epistemic Closure). However, we do not know such an evidence; weak evidence(s) we have got. As it is for the law of diminishing marginal returns in economics, we input additional efforts and capacity to our research activity, however, the more we run the procedure forward, the output suggests that the less we have hope for a significant result. The research activity seems less efficient, ceteris paribus, inevitably yields a negative result.

Seen in this perspective, the significance of the Michelson-Morley experiment is that it showed all the supporting evidences, all the success of Newtonian view offers, that had been brought forth were actually just weak (circumstantial, inferential) evidences and there was no strong (direct) evidence. In this respect, it is a genuine experimentum crucis. An objection might be raised against the use of this term. A brief digression may be helpful to take a glimpse of the present issue from another aspect. Through the works by a number of philosophers and historians of science, prominently Thomas Kuhn and Paul Feyerabend, the view that observational and experimental results are theory-laden, thus, inseparable from the theory and an impartial assessment of the theory cannot be educed from them has gained wide acceptance. Admittedly, the actual scientific theory confirmation is far from the logical positivist ideal, but there is a robust objective content of evidences for and against particular scientific theories that allows us to grade the historical significance of evidences.

A recent example is the discovery of pentaquark. Existence of the particle had been predicted theoretically, however, it took decades to detect it to some level of certainty. The impact of the long and winding story of pentaquark is forceful doubts on several aspects among physicists (see, for instance, Hazel Muir's report Pentaquark Discovery Confounds Sceptics), and "the community consigned it to the dustbin of history as one of many particle “discoveries” that ultimately didn't pan out."

As an overall view, we may say that the general progress that contains the fundamental core of a hypothesis contributes to the perception of the hypothesis up to some level. Thereafter, if no or little, strong evidences occur, the subsequent weak, albeit positive, evidences bring about a negative change in belief and steer away scientists to look for alternative hypotheses.

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These never happen in science. I recommend you to read central limit theorem and normal distribution. In science, first you have to consider all the outcomes of an experiment. Then calculate the standard deviation and the mean of the experiment error. And only after that you are allowed to hypothesize. There is a difference between experience and experiment and central limit theorem for errors let scientists to make hypothesis that has considered all the possible spectrum of outcomes.

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