Inductive essays | Ricky Martin

Elements of the logicist conception of inductive logic live ontoday as part of the general approach called Bayesian inductivelogic. However, among philosophers and statisticians the term‘Bayesian’ is now most closely associated with thesubjectivist or personalist account of belief and decision. And theterm ‘Bayesian inductive logic’ has come to carry theconnotation of a logic that involves purely subjectiveprobabilities. This current usage is misleading since for inductivelogics the Bayesian/non-Bayesian distinction should really hang onwhether the logic gives Bayes' theorem a prominent role, or whetherthe logic largely eschews the use of Bayes' theorem in inductiveinferences (as do the classical approaches to statisticalinference developed by R. A. Fisher (1922) and by Neyman and Pearson(1967)). Indeed, any inductive logic that employs the sameprobability functions to represent both the probabilities ofevidence claims due to hypotheses and the probabilities ofhypotheses due to those evidence claims must bea Bayesian inductive logic in this broader sense; becauseBayes' theorem follows directly from the axioms that each probabilityfunction must satisfy, and Bayes' theorem expresses a necessaryconnection between the probabilities of evidence claims due tohypotheses and the probabilities of hypotheses due to thoseevidence claims.

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Thus, Bayesian inductive support for hypotheses is a form of eliminative induction, where the evidence effectively refutes false alternatives to the true hypothesis. The eliminative nature of Bayesian evidential support doesn't require precise values for prior probabilities. It only need draw on bounds on comparative plausibility ratios, and these bounds only play a significant role while evidence remains fairly sparse. If the true hypothesis is comparatively plausible (due to plausibility arguments contained in b), then plausibility assessments give it a leg-up over alternatives. If the true hypothesis is comparatively implausible, the plausibility assessments merely slow down the rate at which it comes to dominate its rivals, reflecting the idea that extraordinary hypotheses require extraordinary evidence (or an extraordinary accumulation of evidence) to overcome their initial implausibilities.


Any inductive logic that encompasses such arguments should address two challenges. (1) It should tell us which enumerative inductive arguments should count as good inductive arguments rather than as inductive fallacies. In particular, it should tell us how to determine the appropriate degree p to which such premises inductively support the conclusion, for a given margin of error q. (2) It should demonstrably satisfy the CoA. That is, it should be provable (as a metatheorem) that if a conclusion expressing the approximate proportion for an attribute in a population is true, then it is very likely that sufficiently numerous random samples of the population will provide true premises for good inductive arguments that confer degrees of support p approaching 1 for that true conclusion—where, on pain of triviality, these sufficiently numerous samples are only a tiny fraction of a large population. Later we will see how a probabilistic inductive logic may meet these two challenges.

Philosophical Dictionary: Decision-Deontology

There is a result, a kind of Bayesian Convergence Theorem, that shows that if hi (together with b·cn) is true, then the likelihood ratios / comparing evidentially distinguishable alternative hypothesis hj to hi will very probably approach 0 as evidence accumulates (i.e., as n increases). Let's call this result the Likelihood Ratio Convergence Theorem. When this theorem applies, Equation 9 shows that the posterior probability of false competitor hj will very probably approach 0 as evidence accumulates, regardless of the value of its prior probability As this happens to each of hi's false competitors, Equations 10 and 11 say that the posterior probability of the true hypothesis, hi, will approach 1 as evidence increases.[] Thus, Bayesian induction is at bottom a version of induction by elimination, where the elimination of alternatives comes by way of likelihood ratios approaching 0 as evidence accumulates. We will examine the Likelihood Ratio Convergence Theorem in detail in Section 5.[]