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A principle of induction would be a statement with the help of which we could put inductive inferences into a logically acceptable form. In the eyes of the upholders of inductive logic, a principle of induction is of supreme importance for scientific method: "... this principle", says Reichenbach, "determines the truth of scientific theories. To eliminate it from science would mean nothing less than to deprive science of the power to decide the truth or falsity of its theories. Without it, clearly, science would no longer have the right to distinguish its theories from the fanciful and arbitrary creations of the poet's mind."
Now this principle of induction cannot be a purely logical truth like a tautology or an analytic statement. Indeed, if there were such a thing as a purely logical principle of induction, there would be no problem of induction; for in this case, all inductive inferences would have to be regarded as purely logical or tautological transformations, just like inferences in inductive logic. Thus the principle of induction must be a synthetic statement; that is, a statement whose negation is not self-contradictory but logically possible. So the question arises why such a principle should be accepted at all, and how we can justify its acceptance on rational grounds.
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Ch. 1 "A Survey of Some Fundamental Problems", Section I: The Problem of Induction

 

Karl Popper

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There cannot be a doubt that the method of inquiry which Mr. Darwin has adopted is not only rigorously in accordance with the canons of scientific logic, but that it is the only adequate method. Critics exclusively trained in classics or in mathematics, who have never determined a scientific fact in their lives by induction from experiment or observation, prate learnedly about Mr. Darwin's method, which is not inductive enough, not Baconian enough, forsooth, for them. But even if practical acquaintance with the process of scientific investigation is denied them, they may learn, by the perusal of Mr. Mill's admirable chapter "On the Deductive Method," that there are multitudes of scientific inquiries in which the method of pure induction helps the investigator but a very little way.

 

Charles Darwin

 

It is possible to express the laws of thermodynamics in the form of independent principles, deduced by induction from the facts of observation and experiment, without reference to any hypothesis as to the occult molecular operations with which the sensible phenomena may be conceived to be connected; and that course will be followed in the body of the present treatise. But, in giving a brief historical sketch of the progress of thermodynamics, the progress of the hypothesis of thermic molecular motions cannot be wholly separated from that of the purely inductive theory.

 

William John Macquorn Rankine

 

Now - how can we possibly examine something we use all the time and presuppose in every statement? How can we criticize the terms in which we habitually express our observations? Let us see! The first step in our criticism of commonly-used concepts is to create a measure of criticism, something with which these concepts can be compared. Of course, we shall later want to know a little more about the measuring stick itself; for example, we shall want to know whether it is better than, or perhaps not as good as, the material examined. But in order for this examination to start there must be a measuring-stick in the first place. Therefore, the first step in our criticism of customary concepts and customary reactions is to step outside the circle and either to invent a new conceptual system, for example a new theory, that clashes with the most carefully established observational results and confounds with the most plausible theoretical principles, or to import such a system from outside science, from religion, from mythology , from the ideas of incompetents, or the ramblings of madmen. This step is, again, counter-inductive, Counter-induction is thus both a fact' - science could not exist without it - and a legitimate and much needed move in the game of science.

 

Paul Karl Feyerabend

 

The logical axioms are the principle of all truth. These posit an existence towards which all cognition serves. Logic is a law which must be obeyed, and man realises himself only in so far as he is logical. He finds himself in cognition. All error must be felt to be crime. And so man must not err. He must find the truth.

 

Otto Weininger

 

The application of scientific formulations of the principle of probability statistically determined is thus a logical corollary of the principle already stated, that the subject matter of scientific findings is relational, not individual. It is for this reason that it is safe to predict the ultimate triumph of the statistical doctrine.

 

John Dewey