Deductive reasoning moves from the general to the specific, that is, from the principle to particulars.
Abductive reasoning moves from relevant evidence (specific) to the best possible explanation (principle). Abduction is inference to the best explanation – beginning from a set of accepted facts, inference proceeds to the most likely explanation for those facts. Inference is the process of deriving a conclusion that is based solely on what is already known (a posteriori).
Inductive reasoning involves coming to a conclusion that is inferred from multiple observations. Repeated testing will help to ascertain whether first inference (conclusion derived) is correct or incorrect. In contrast, valid deductive reasoning is based in formal logic and will yield a true conclusion if the premises on which it is based are themselves true. That is, the inferred conclusion of a valid deductive inference is necessarily true when the premises are true, so a formally valid deductive inference cannot be false.
The Scottish empiricist, David Hume raised the skeptical philosophical objection that inductive reasoning might fail whenever the past cannot be taken to be a reliable guide to the future. In essence, if we have just seen 43 white swans, are we justified in assuming that all swans are white or even that the next swan that we see will be white?
The philosopher of science, Karl Popper expanded Hume's ideas of the 'problem of induction' and argued that there can be no solution to the problem of induction in that empirical observations cannot provide proof for a scientific hypothesis, theory, or law. Popper argued that, since only disproof is a certainty, science should proceed by a 'deductivist' method of conjecture and refutation, employing deductively valid reasoning that does not resort to inductive confirmation.
In practice, modern scientific method employs inductive, deductive, and abductive reasoning to move from empirical data to elucidation of principles. Ideally, hypotheses are couched so that they could be falsified should they fail to correctly predict experimental observations. However, since the hard sciences involve empirical observations of natural phenomena that can reasonably be expected to operate consistently, relevant positive results are taken to provide practical support for the likelihood that a theory is accurate.
Unlike the case for swans, would we not be fully justified in believing that all apples that fall from an apple tree will fall toward Earth rather than floating toward the moon? To stipulate that the apple tree is situated on Mars would appear to put a worm in this theory, but that problem can be avoided by asking instead, 'would we not be fully justified in believing that all apples will fall toward the planet?' Whether we talk of the force of gravity, or the more modern understanding of warping of space-time, natural laws dictate that objects are most 'attracted' to the largest nearby mass.
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