Continuing to reason in this fashion, we could arrive at a more pessimistic state than even Hume imagined: not only is induction without logical foundation, but deduction has no scientific utility because we cannot insure the validity of all the premises. The Bayesian answer to this problem is partial, in that it makes a severe demand on the scientist and puts a severe limitation on the results. It says roughly this: If you can assign a degree of certainty or personal probability to the premises of your valid argument, you may use any and all the rules of probability theory to derive a certainty for the conclusion and this certainty will be a logically valid consequence of your original certainties. The catch is that your concluding certainty, or posterior probability, may depend heavily on what you used as initial certainties or prior probabilities. And, if those initial certainties are not the same as those of a colleague, that colleague may very well assign a different certainty to the conclusion than you derived.
Hume (1739) was the issue of causal inference and failure of induction to provide a foundation for it.
Thus not only our reason fails us in the discovery of the ultimate connexion of causes and effects, but even after experience has inform'd us of their constant conjunction, 'tis impossible for us to satisfy ourselves by our reason, why we shou'd extend that experience beyond those particular instances, which have fallen under our observation. We suppose, but are never able to prove, that there must be a resemblance betwixt those objects, of which we have had experience, and those which lie beyond the reach of our discovery. (Hume 1739)
Causal inference based on mere coincidence of events constitute a logical fallacy known as post hoc ergo propter hoc (Latin for 'after this therefore on account of this'). This fallacy is exemplified by the inference that the crowing of a rooster is necessary for the sun to rise because sunrise is always preceded by the crowing.
The post hoc fallacy is a special case of a more general logical fallacy known as the 'fallacy of affirming the consequent'. This fallacy of confirmation takes the following general form: 'We know that if H is true, B must be true and we know that B is true therefore H must be true'. This fallacy is used routinely by scientists in interpreting data.
Popper addressed Hume's problem by asserting that scientific hypotheses can never be proven or established as true in any logical sense. Instead, Popper observed that scientific statements can simply be found to be consistent with observation.
According to Popper (1968), science advances by a process of elimination that he called conjecture and refutation. Scientists form hypotheses based on intuition, conjecture, and previous experience. At any time, however, they may be refuted by further observations and replaced by other hypotheses that better explain the observations. This view of scientific inference is sometimes called refutationism or falsificationism. Refutationists consider induction to be a psychological crutch: repeated observations did not in fact induce the formulation of a natural law, but only the belief that such a law has been found. For a refutationist, only the psychological comfort that induction provides explains why it still has its advocates.
Kuhn (1962) claimed that in every branch of science the prevailing scientific viewpoint, which he termed 'normal science', occasionally undergoes major shifts that amount to scientific revolutions. These revolutions signal a decision of the scientific community to discard the scientific infrastructure rather than to falsify a new hypothesis that cannot easily be grafted onto it. Kuhn (1962) and others have argued that the consensus of the scientific community determines what is considered accepted and what is considered refuted.
Kuhn's critics characterized this description of science as one of an irrational process, 'a matter for mob psychology' (Lakatos 1970). Those who cling to a belief in a rational structure for science consider Kuhn's vision to be a regrettably real description of much of what passes for scientific activity, but not prescriptive for any good science.
Like refutationism, the modern form of this philosophy evolved from the writings of eighteenth century British philosophers, but the focal arguments first appeared in a pivotal essay by Thomas Bayes (1763) and, hence, the philosophy is usually referred to as Bayesianism (Howson and Urbach 1989). Like refutationism, it did not reach a complete expression ntil after the First World War, most notably in the writings of Ramsey (1931) and DeFinetti (1937) and, like refutationism, it did not begin to appear in epidemiology until the 1970s (Cornfield 1976).
Continuing to reason in this fashion, we could arrive at a more pessimistic state than even Hume imagined: not only is induction without logical foundation, but deduction has no scientific utility because we cannot insure the validity of all the premises. The Bayesian answer to this problem is partial, in that it makes a severe demand on the scientist and puts a severe limitation on the results. It says roughly this: If you can assign a degree of certainty or personal probability to the premises of your valid argument, you may use any and all the rules of probability theory to derive a certainty for the conclusion and this certainty will be a logically valid consequence of your original certainties. The catch is that your concluding certainty, or posterior probability, may depend heavily on what you used as initial certainties or prior probabilities. And, if those initial certainties are not the same as those of a colleague, that colleague may very well assign a different certainty to the conclusion than you derived.
Because the posterior probabilities emanating from a Bayesian inference depend on the person supplying the initial certainties and, thus, may vary across individuals, the inferences are said to be subjective. This subjectivity of Bayesian inference is often mistaken for a subjective treatment of truth. Not only is such a view of Bayesianism incorrect, but it is diametrically opposed to Bayesian philosophy. The Bayesian approach represents a constructive attempt to deal with the dilemma that scientific laws and facts should not be treated as known with certainty, yet classical deductive logic yields conclusions only when some law, fact, or connection between is asserted with 100 per cent certainty.
Perhaps the most important common thread that emerges from the debated philosophies is Hume's legacy that proof is impossible in empirical science.