This has turned into a really interesting discussion.
A few points...
- "Exhaustive" Induction is, in a way, a deductive argument. Suppose there are only 4 instances of consideration in the domain of proposition, P: a, b, c, and d. Suppose further that P(a), P(b), P(c), and P(d) are true. Then P is true on it's domain.
Here we are reasoning from specifics to the general case. However, the specifics form independent premises which fully justify the conclusion (which is, in effect, a disjunction of all the premises).
- Statistical "Sampling," seems to also be a way to reason from specifics to generalities. We get a sufficient sample, and calculate the mean of a particular observable on that sample, and use that mean as an estimate for the mean of the observable for the whole population.
But, again, we see that the power of this method actually comes from deduction--the deduction that starts from the axioms of probability theory, and a further assumption about the population distribution. Now, along with the estimate, when "done right" with deduction, we get confidence intervals for various confidence levels based on our sample size.
Similar things are true of further tools of statistics...regression, factor analysis, cluster analysis, etc. The power of the seemingly "inductive" method is actually derived from deduction--the deduction starting from the axioms of probability and the assumptions of the probability models involved.
- Although the process of deduction that leads to many things that we widely accept to be true (the laws of physics, the laws of probability, the conclusions of geometry) is painstaking and an even ingenious at points, I am more impressed with the (often error-prone) process of abductive reasoning that produced the very axioms we use.
How did Newton come to posit the basic premises of Classical Mechanics?
How did Euclid and others create the postulates of Euclidean Geometry?
Many of of us live in structures that rely on the amazing accuracy of these incredibly well chosen premises of deductive reasoning.
Certainly, they were based on on experiments and observations. But I very much doubt that these experiments and observations were as slavishly scrutinized and analyzed, before their proposition of premises, as many of the observations and experiments leading to (far less accurate) theories of modern life and social sciences. They were incredibly scrutinized after they were proposed, and withstood those tests to amazing effect.
Like, reason, I challenge the value of inductive reasoning. It somehow seems to lack the intellectual courage that abductive reasoning takes.
As a further example, consider Quantum Mechanics--Among the most accurate theories we have. The abductive leaps taken (from "photons" to "matter waves") often seemed downright audacious.
I posit that what requires intellectual rigour and care is NOT what leads to the generalities we posit, but the intellectual rigor and care that follows. If the premises we create are malformed, then the wake of deductive reasoning and empirical testing should destroy the weaker theories.