The word 'proof' should strictly only be used when we are dealing with deductive inferences... Popper claimed that scientists only need to use deductive inferences... So if a scientist is only interested in demonstrating that a given theory is false, she may be able to accomplish her goal without the use of inductive inferences... When a scientist collects experimental data, her aim might be to show that a particular theory... is false. She will have to resort to inductive reasoning... So Popper's attempt to show that science can get by without induction does not succeed.
There is nothing distinctively scientific about the hypothetico-deductive process. It is not even distinctively intellectual. It is merely a scientific context for a much more general stratagem that underlies almost all regulative processes or processes of continuous control, namely feedback, the control of performance by the consequences of the act performed. In the hypothetico-deductive scheme the inferences we draw from a hypothesis are, in a sense, its logical output. If they are true, the hypothesis need not be altered, but correction is obligatory if they are false. The continuous feedback from inference to hypothesis is implicit in Whewell's account of scientific method; he would not have dissented from the view that scientific behaviour can be classified as appropriately under cybernetics as under logic.
When you are young in this world, you believe that the class of deductive truths about social matters is larger than it turns out to be. [...] I have discovered, to my infinite regret, that most of the serious debates over the basic principles of any political order have an irreducible empirical content.
Richard Allen Epstein
Without deductive logic science would be entirely useless. It is merely a barren game to ascend from the particular to the general, unless afterwards we can reverse the process and descend from the general to the particular, ascending and descending like angels on Jacob's ladder.
Alfred North Whitehead
Aristotle writes that persuasion is based on three things: the ethos, or personal character of the speaker; the pathos, or getting the audience into the right kind of emotional receptivity; and the logos, or the argument itself, carried out by abbreviated syllogisms, or something like deductive syllogisms, and by the use of example.
The mathematician starts with a few propositions, the proof of which is so obvious that they are called self-evident, and the rest of his work consists of subtle deductions from them. The teaching of languages, at any rate as ordinarily practiced, is of the same general nature authority and tradition furnish the data, and the mental operations are deductive.
Whoever builds his faith exclusively on demonstrative proofs and deductive arguments, builds a faith on which it is impossible to rely. For he is affected by the negativities of constant objections. Certainty(al-yaqin) does not derive from the evidences of the mind but pours out from the depths of the heart.
The intellectual equipment needed for the job of the future is an ability to define problems, quickly assimilate relevant data, conceptualize and reorganize the information, make deductive and inductive leaps with it, ask hard questions about it, discuss findings with colleagues, work collaboratively to find solutions and then convince others.
For the emergent process, as noted by the geneticist Theodore Dobzhansky, is neither random nor determined but creative. Just as in human order, creativity is neither a rational deductive process nor the irrational wandering of the undisciplined mind but the emergence of beauty as mysterious as the blossoming of a field of daisies out of the dark Earth.
During all those years of experimentation and research, I never once made a discovery. All my work was deductive, and the results I achieved were those of invention, pure and simple. I would construct a theory and work on its lines until I found it was untenable. Then it would be discarded at once and another theory evolved. This was the only possible way for me to work out the problem.
Thomas A. Edison
If I sound as if I'm always predicting ominous things, it's because I'm a pragmatist. I use deductive reasoning to generalize, and I suppose this sometimes ends up sounding like unlucky prophecies. You know why? Because reality's just the accumulation of ominous prophecies come to life. You have to only open a newspaper on any given day and weigh the good news versus the bad, and you'll see what I mean.
Mathematics has two faces: it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself.
Theologians, and religionists in general, start with a fantasy premise and then proceed to apply rigorous formal logic to tease out its implications. Stark himself points out that 'theology consists of formal reasoning about God.' This is admirably exact. Theologians, beginning with a wished-for creation of their own minds, analyze that creation's characteristics by rigorous application of the principles of formal-that is, deductive-logic.