My question deals with induction in the context of scientific enquiry and with the confirmation of scientfic hypothesis.

Discussions regarding induction rapidly turn into debates regarding confirmation. How to explain this precisely?

Should one say that

(a) " evidence e confirms hypothesis H " is equivalent to " e gives some inductive support to H"?

(b) if e ( strongly) confirms H , then one can infer inductively H from e?

(c) the statement " e confirms H" contains requires itself an inductive inference?

The impression I get is that we are moving in a circle: should one say that the confirmation relation has to obtain in order one to be justified in doing an inductive inference ( in that case confirmation is a necessary condition of induction) , or should one say that one has to do an inductive inference in order to claim that a confrmation relation holds ( in which case induction would be a necessary condition of confirmation , or at least, of claims regarding confirmation)?

Does the question I am asking originate in an insufficient knwoledge of the problem, or is there actually a difficulty here?


This is a boot-strapping issue. Given N examples of a phenomena P (where N is a non-negative integer), we make an induction that P conforms to theory T. Once we've made that induction, further examples of P that happen to conform to T are confirmation; further examples of P that do not conform to T are refutations, anomalies, or disconfirmations (depending on which theory of science you subscribe to, and the magnitude of the failure to conform).

Put another way:

  • Before we make an induction to some theory, all we have is unorganized data points: phenomena that we have observed, but that we have not yet accounted for.
  • When we make an induction to some theory, we organize our data into a class of phenomena that share given properties, asserting that every data point — whether we've observed it or not — ought to conform to those properties.
  • After we've made an induction to some theory, further examples of the class of phenomena we just created either conform to our theory or fill us with nameless dreads about our intellectual legacy.

It makes no sense to talk about confirmation before we've made a theory, and after we've made a theory we mostly expect new phenomena to conform to it, so confirmation by itself isn't all that interesting from a scientific perspective.

  • Thanks for ths clear answer. May 26 '21 at 18:54
  • Scientists don't require "every data point - ought to conform to those properties", they just search for one or several reasonable unfalsified models from statistical hypothesis testing or some other Bayesian statistical approach. May 26 '21 at 22:48
  • @DoubleKnot: The essence of induction is generalizing a set of known instances to a universal principle, so yes, it has to be every data point. I mean, if the law of gravity applied to every massy object except bowling balls, we'd all be a bit flustered, wouldn't we? And not just because of the bowling balls floating past our heads... May 27 '21 at 1:26
  • Understand in theory ideally, my comment is about realistic situations in reality especially the verification/confirmation criterion for the social sciences as emphasized by positivists Like Comte, even physics experiments have various kinds of error needing statistical processing. Falsification mostly rely on null hypothesis significance testing (NHST) such as in psychology, not based on a single data point failure usually. It's also a reason why positivists had to give up their original stringent verificationism similar to what you idealized. May 27 '21 at 1:39
  • @DoubleKnot: Statistical testing is merely a tool for identifying theoretical universals within complex multidimensional contexts. It doesn't change the nature of induction; it merely factors out conflicting and confounding interactions. I don't subscribe to falsificationism, so single-point failure is irrelevant. But still, every theory aspires to be universal, because fragmented, non-universal theories aren't much use to anyone. May 27 '21 at 2:52

Questions of this nature are best seen in a Bayesian context. e confirms H, if P(H|e) > P(H). In words, evidence confirms a hypothesis if the evidence increases the posterior probability of the hypothesis.

We may interpret this as e ruling out certain possible worlds. We consider the set of all possible worlds. In some of these worlds e holds, and in some of these worlds H holds, and in some of these worlds both hold. By observing e, we rule out all the worlds where e does not hold, and the probability of H is obtained from the proportion of remaining worlds where H holds. The probability of H may increase or decrease as a result of crossing out the worlds where e doesn't hold - it depends on what worlds remain and on the relationship between e and H.


Confirmation in light of your posited evidence e of a hypothesis H of a particular scientific theory to support Baconian inductive inference is attacked heavily in modern philosophy of science such as the epistemology proposed by Quine's confirmation holism:

In philosophy of science, confirmation holism, also called epistemological holism, is the view that no individual statement can be confirmed or disconfirmed by an empirical test, but rather that only a set of statements (a whole theory) can be so. Van Orman Quine who motivated his holism through extending Pierre Duhem's problem of underdetermination in physical theory to all knowledge claims... Duhem's idea was, roughly, that no theory of any type can be tested in isolation but only when embedded in a background of other hypotheses, e.g. hypotheses about initial conditions. Quine thought that this background involved not only such hypotheses but also our whole web of belief, which, among other things, includes our mathematical and logical theories and our scientific theories. This last claim is sometimes known as the Duhem–Quine thesis.

So it is impossible to test a scientific hypothesis in isolation, because an empirical test of the hypothesis requires one or more background assumptions. If there's no background hypotheses, pretty much everything like a blue cup can confirm the hypothesis "All ravens are black" since this propositional hypothesis is logically equivalent as "If something is not black, then it is not a raven", which sounds ridiculous but actually true conforming with Bayes theorem if we don't know there're much more non-ravan objects than the number of ravens. Even one insists only observations of ravens should affect one's view as to whether all ravens are black such as the positive Nicod's criterion, I. J. Good gives an example of background knowledge with respect to which the observation of a black raven actually decreases the probability that all ravens are black:

Suppose that we know we are in one or other of two worlds, and the hypothesis, H, under consideration is that all the ravens in our world are black. We know in advance that in one world there are a hundred black ravens, no non-black ravens, and a million other birds; and that in the other world there are a thousand black ravens, one white raven, and a million other birds. A bird is selected equiprobably at random from all the birds in our world. It turns out to be a black raven. This is strong evidence ... that we are in the second world, wherein not all ravens are black... Hempel insists our background knowledge itself is a red herring, and that we should consider induction with respect to a condition of perfect ignorance.

So even you empirically confirms a hypothesis, you cannot claim such confirmation is strong enough or even relevant at all to inductively infer hypothesis H in most cases without clarification about all your other web of beliefs including other assumptions, hypotheses and knowledges.

As referenced here:

In 1936, Carnap sought a switch from verification to confirmation. Carnap's confirmationism would not require conclusive verification (thus accommodating for universal generalizations) but allow for partial testability to establish "degrees of confirmation" on a probabilistic basis. Carnap never succeeded in formalizing his thesis despite employing abundant logical and mathematical tools for this purpose... Karl Popper's The Logic of Scientific Discovery proposed falsificationism as a criterion under which scientific hypothesis would be tenable. Falsificationism would allow hypotheses expressed as universal generalizations, such as "all swans are white", to be provisionally true until falsified by evidence, in contrast to verificationism under which they would be disqualified immediately as meaningless... Though generally considered a revision of verificationism, Popper intended falsificationism as a methodological standard specific to the sciences rather than as a theory of meaning. Popper regarded scientific hypotheses to be unverifiable, as well as not "confirmable" under Rudolf Carnap's thesis.

In summary, inductive inference doesn't entail or imply the cogency of a naive confirmation of a standalone hypothesis intending as a causal mechanism of a theory. This conclusion can also be hinted from Hume's famous problem of induction as inductive inference itself is not causal but mere constant conjunction, hardly can be used as a logic foundation for any scientific theory trying to explain causality like Einstein's GR, no matter how many valid confirmations are established about the hypotheses of classic Galilean relativity and flat space, these hypotheses both turned out to be false and mere illusions which only act as convenient approximate beliefs under applicable classic contexts instead of true ontic causality of mechanics.

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