independent from the other two, then either the uniformity of nature can be rationally justified through abduction, or Hume's dilemma is more of a trilemma Inductive reasoning refers to a variety of methods of reasoning in which the conclusion of an argument is supported not with deductive certainty, but at best with some degree of probability. Unlike deductive reasoning (such as mathematical induction), where the conclusion is certain, given the premises are correct, inductive reasoning produces conclusions that are at best probable, given the premi So… In other words, it takes for granted a uniformity of nature, an unproven principle that cannot be derived from the empirical data itself. Arguments that tacitly presuppose this uniformity are sometimes called Humean after the philosopher who was first to subject them to philosophical scrutiny. === Prediction === An inductive prediction draws a conclusion about a future, current, or past instance from a sample of other instances. Like an inductive generalization, an inductive prediction relies on a data set consisting of specific instances of a phenomenon. This form of induction was explored in detail by philosopher John Stuart Mill in his System of Logic, where he states, "[t]here can be no doubt that every resemblance [not known to be irrelevant] affords some degree of probability, beyond what would otherwise exist, in favor of the conclusion." See Mill's Methods. Some thinkers contend that analogical induction is a subcategory of inductive generalization because it assumes a pre-established uniformity governing events. Analogical induction requires an auxiliary examination of the relevancy of the characteristics cited as common to the pair. While observations, such as the motion of the sun, could be coupled with the principle of the uniformity of nature to produce conclusions that seemed to be certain, the problem of induction arose from the fact that the uniformity of nature was not a logically valid principle, therefore it could not be defended as deductively rational, but also could not be defended as inductively rational by appealing to the fact that the uniformity of nature has accurately described the past and therefore, will likely accurately describe the future because that is an inductive argument and therefore circular since induction is what needs to be justified. 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." Inference to the best explanation is often, yet arguably, treated as synonymous to abduction as it was first identified by Gilbert Harman in 1965 where he referred to it as "abductive reasoning," yet his definition of abduction slightly differs from Pierce's definition. Regardless, if abduction is in fact a third mode of inference rationally independent from the other two, then either the uniformity of nature can be rationally justified through abduction, or Hume's dilemma is more of a trilemma. 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. ==== Immanuel Kant ==== 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. A class of synthetic statements that was not contingent but true by necessity, was then synthetic a priori. Kant thus saved both metaphysics and Newton's law of universal gravitation. On the basis of the argument that what goes beyond our knowledge is "nothing to us," he discarded scientific realism. Kant's position that knowledge comes about by a cooperation of perception and our capacity to think (transcendental idealism) gave birth to the movement of German idealism. Regarding experience as justifying enumerative induction by demonstrating the uniformity of nature, 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. Comte was confident in treating scientific law as an irrefutable foundation for all knowledge, and believed that churches, honouring eminent scientists, ought to focus public mindset on altruism—a term Comte coined—to apply science for humankind's social welfare via sociology, Comte's leading science. Rather, the premises of an inductive logical argument indicate some degree of support (inductive probability) for the conclusion but do not entail it; that is, they suggest truth but do not ensure it. In this manner, there is the possibility of moving from general statements to individual instances (for example, statistical syllogisms). Note that the definition of inductive reasoning described here differs from mathematical induction, which, in fact, is a form of deductive reasoning. Mathematical induction is used to provide strict proofs of the properties of recursively defined sets. Our assumption, however, becomes invalid once it is discovered that there are white ravens. Therefore, the general rule "all ravens are black" is not the kind of statement that can ever be certain. Hume further argued that it is impossible to justify inductive reasoning: this is because it cannot be justified deductively, so