Inductive reasoning

[3][4] The types of inductive reasoning include generalization, prediction, statistical syllogism, argument from analogy, and causal inference.

For example: The measure is highly reliable within a well-defined margin of error provided that the selection process was genuinely random and that the numbers of items in the sample having the properties considered are large.

[citation needed] Analogical induction requires an auxiliary examination of the relevancy of the characteristics cited as common to the pair.

It truncates "all" to a mere single instance and, by making a far weaker claim, considerably strengthens the probability of its conclusion.

This confidence is expressed as the Baconian probability i|n (read as "i out of n") where n reasons for finding a claim incompatible has been identified and i of these have been eliminated by evidence or argument.

[23] Aristotle's Posterior Analytics covers the methods of inductive proof in natural philosophy and in the social sciences.

'Epilogism' is a theory-free method that looks at history through the accumulation of facts without major generalization and with consideration of the consequences of making causal claims.

His method of inductivism required that minute and many-varied observations that uncovered the natural world's structure and causal relations needed to be coupled with enumerative induction in order to have knowledge beyond the present scope of experience.

Since Hume first wrote about the dilemma between the invalidity of deductive arguments and the circularity of inductive arguments in support of the uniformity of nature, this supposed dichotomy between merely two modes of inference, deduction and induction, has been contested with the discovery of a third mode of inference known as abduction, or abductive reasoning, which was first formulated and advanced by Charles Sanders Peirce, in 1886, where he referred to it as "reasoning by hypothesis.

Hume was also skeptical of the application of enumerative induction and reason to reach certainty about unobservables and especially the inference of causality from the fact that modifying an aspect of a relationship prevents or produces a particular outcome.

Awakened from "dogmatic slumber" by a German translation of Hume's work, Kant sought to explain the possibility of metaphysics.

Reasoning that the mind must contain its own categories for organizing sense data, making experience of objects in space and time (phenomena) possible, Kant concluded that the uniformity of nature was an a priori truth.

Positivism, developed by Henri de Saint-Simon and promulgated in the 1830s by his former student Auguste Comte, was the first late modern philosophy of science.

Human knowledge had evolved from religion to metaphysics to science, said Comte, which had flowed from mathematics to astronomy to physics to chemistry to biology to sociology—in that order—describing increasingly intricate domains.

Regarding experience as justifying enumerative induction by demonstrating the uniformity of nature,[28] the British philosopher John Stuart Mill welcomed Comte's positivism, but thought scientific laws susceptible to recall or revision and Mill also withheld from Comte's Religion of Humanity.

During the 1830s and 1840s, while Comte and Mill were the leading philosophers of science, William Whewell found enumerative induction not nearly as convincing, and, despite the dominance of inductivism, formulated "superinduction".

Having once had the phenomena bound together in their minds in virtue of the Conception, men can no longer easily restore them back to detached and incoherent condition in which they were before they were thus combined.

[30] In the 1870s, the originator of pragmatism, C S Peirce performed vast investigations that clarified the basis of deductive inference as a mathematical proof (as, independently, did Gottlob Frege).

[32] Having highlighted Hume's problem of induction, John Maynard Keynes posed logical probability as its answer, or as near a solution as he could arrive at.

[33] Bertrand Russell found Keynes's Treatise on Probability the best examination of induction, and believed that if read with Jean Nicod's Le Probleme logique de l'induction as well as R B Braithwaite's review of Keynes's work in the October 1925 issue of Mind, that would cover "most of what is known about induction", although the "subject is technical and difficult, involving a good deal of mathematics".

If this principle is not true, every attempt to arrive at general scientific laws from particular observations is fallacious, and Hume's skepticism is inescapable for an empiricist.

"[37]In a 1965 paper, Gilbert Harman explained that enumerative induction is not an autonomous phenomenon, but is simply a disguised consequence of Inference to the Best Explanation (IBE).

[38] Inductive reasoning is a form of argument that—in contrast to deductive reasoning—allows for the possibility that a conclusion can be false, even if all of the premises are true.

The conclusion for a valid deductive argument is already contained in the premises since its truth is strictly a matter of logical relations.

Inductive premises, on the other hand, draw their substance from fact and evidence, and the conclusion accordingly makes a factual claim or prediction.

Unlike deductive reasoning, it does not rely on universals holding over a closed domain of discourse to draw conclusions, so it can be applicable even in cases of epistemic uncertainty (technical issues with this may arise however; for example, the second axiom of probability is a closed-world assumption).

Recognizing this, Hume highlighted the fact that our mind often draws conclusions from relatively limited experiences that appear correct but which are actually far from certain.

[47] Bertrand Russell illustrated Hume's skepticism in a story about a chicken who, fed every morning without fail and following the laws of induction, concluded that this feeding would always continue, until his throat was eventually cut by the farmer.

[50] In Popper's schema, enumerative induction is "a kind of optical illusion" cast by the steps of conjecture and refutation during a problem shift.

We begin by considering an exhaustive list of possibilities, a definite probabilistic characterisation of each of them (in terms of likelihoods) and precise prior probabilities for them (e.g. based on logic or induction from previous experience) and, when faced with evidence, we adjust the strength of our belief in the given hypotheses in a precise manner using Bayesian logic to yield candidate 'a posteriori probabilities', taking no account of the extent to which the new evidence may happen to give us specific reasons to doubt our assumptions.