Hume, Induction & Bayes's Theorem

Hume, Induction & Bayes's Theorem

Hume’s paradox is the result of an interrelationship between two distinct logical processes; namely, deductive logic and inductive logic.

David Hume 1711-1776

David Hume’s historical context was one where Descartes and other rationalists believed deductive, or a priori knowledge of the ultimate nature of reality could be attained through philosophical investigation. Empiricists such as Hume reacted by asserting deductive arguments were built into the structure of language and were strictly trivial; for example, “there are no married bachelors” or “I think, therefore I am”. The empiricists also stressed the inability of human reason to extend the deductive process to ontological concepts such as cause and effect or the uniformity of nature. According to Hume these concepts were arrived at through a process of induction.

Taking cause and effect as an example, Hume demonstrated the operation of enumerative induction. For instance, every time up until now I have wound my wristwatch it has run. I assume this pattern will hold whenever I wind my wristwatch. The justification of this process is the problem of induction.

Deductive logic is utilized by scientific reasoning to confirm or disconfirm hypotheses. This is because initial boundary conditions of hypotheses are treated as premises which are true only if observable phenomena predicted by them obtain. The paradox here is that while scientific models are confirmed deductively, they are themselves arrived at inductively before being tested for predictive accuracy. The underpinning of testability of hypotheses as laws lies in the putative uniformity of nature. However, appeal to the uniformity of nature fails to resolve the problem of induction. There is an inductive process inherent in assuming that natural processes will be the same tomorrow as yesterday.

Baye’s theorem of conditional probability states that evidence which is improbable antecedently, which obtains if the evidence is true, is most likely to raise the probability of the hypothesis. The probability of the hypothesis is boosted by the evidence. Apropos an argument stating the source of the problem of induction to be a linguistic confusion, which runs along the following lines. Probability given the evidence is the standard of rational belief. The conclusions of inductive inferences are probable when probability is viewed in this light.

The problem of induction remains despite probabilistic theories. Inductive inference determines the rules of inductive evidence. Conclusions supported by inductive evidence must derive from rules known to be correct. This can only be done inductively by inferring the probable correctness of the rules. Reformulating the problem as one concerning degrees of rational belief fails to resolve the basic problem concerning justification.