Scientific Realism and Antirealism
Who Designed the Designer? – Common Sense Atheism
Today I want to kill one of atheism’s sacred cows
It is now generally held that the core idea of Bayesian logicism is fatally flawed—that syntactic logical structure cannot be the sole determiner of the degree to which premises inductively support conclusions. A crucial facet of the problem faced by Bayesian logicism involves how the logic is supposed to apply to scientific contexts where the conclusion sentence is some hypothesis or theory, and the premises are evidence claims. The difficulty is that in any probabilistic logic that satisfies the usual axioms for probabilities, the inductive support for a hypothesis must depend in part on its prior probability. This prior probability representshow plausible the hypothesis is supposed to be based on considerationsother than the observational and experimental evidence (e.g., perhapsdue to relevant plausibility arguments). A Bayesian logicist must tellus how to assign values to these pre-evidential priorprobabilities of hypotheses, for each of the hypotheses ortheories under consideration. Furthermore, this kind of Bayesianlogicist must determine these prior probability values in away that relies only on the syntactic logical structure of thesehypotheses, perhaps based on some measure of their syntacticsimplicities. There are severe technical problems with getting thisidea to work. Moreover, various kinds of examples seem to show thatsuch an approach must assign intuitively quite unreasonable priorprobabilities to hypotheses in specific cases (see the footnote citednear the end of section 3.2 for details). Furthermore, for this ideato apply to the evidential support of real scientific theories,scientists would have to formalize theories in a way that makes theirrelevant syntactic structures apparent, and then evaluate theoriessolely on that syntactic basis (together with their syntacticrelationships to evidence statements). Are we to evaluate alternativetheories of gravitation (and alternative quantum theories) this way?This seems an extremely doubtful approach to the evaluation of realscientific theories and hypotheses. Thus, it seems that logicalstructure alone cannot suffice for the inductive evaluation ofscientific hypotheses. (This issue will be treated in more detail inSection 3, after we first see how probabilistic logics employ Bayes'theorem to represent the evidential support for hypotheses as afunction of prior probabilities together withtheir evidential likelihoods.)
Bayesian logicists like Keynes and Carnap maintained that posteriorprobabilities of hypotheses should be determined by logical formalone. The idea was that the likelihoods might reasonably be specifiedin terms of logical form; so if logical form might be made todetermine the values of prior probabilities as well, then inductivelogic would be fully “formal” in the same way thatdeductive logic is formal. Keynes and Carnap tried to implement thisidea through syntactic versions of the principle of indifference— the idea that syntactically similar hypotheses should beassigned the same prior probability values. Carnap showed how to carryout this project in detail, but only for extremely simple formallanguages. Most logicians now take the project to have failed becauseof a fatal flaw with the whole idea that reasonable priorprobabilities can be made to depend on logical form alone. Semanticcontent should matter. Goodmanian grue-predicates provide one way toillustrate the point.
What is Bayesianism? - Less Wrong
One more point about prior probabilities and Bayesian convergenceshould be mentioned. Some subjectivist versions of Bayesian inductionseem to suggest that an agent's prior plausibility assessments forhypotheses should stay fixed once and for all, and that allplausibility updating should be brought about via the likelihoods inaccord with Bayes's Theorem. Critics argue that this isunreasonable. The members of a scientific community may quitelegitimately revise their prior plausibility assessments forhypotheses from time to time as they rethink plausibility argumentsand bring new considerations to bear. This seems a natural part ofthe conceptual development of a science. It turns out that suchreassessments of priors pose no difficulty for probabilistic inductivelogic. Reassessments may sometimes come about by the addition ofexplicit statements that supplement or modify the backgroundinformation b. They may also take the form of (non-Bayesian)transitions to new vagueness sets for individual agents andto new Diversity sets for the community. The logicof Bayesian induction has nothing to say about what values the priorplausibility assessments for hypotheses should have; and it places norestrictions on how they might change. Provided that the series ofreassessments of prior plausibilities doesn't push the prior of the true hypothesisever nearer to zero, the Likelihood Ratio Convergence Theoremimplies that the evidence will very probably bring the posteriorprobabilities of empirically distinct rivals of the true hypothesis toapproach 0 via decreasing likelihood ratios; and as this happens, theposterior probability of the true hypothesis will head towards 1.
One more point about prior probabilities and Bayesian convergence should be mentioned here. Some subjectivist versions of Bayesian induction seem to suggest that an agent's prior plausibility assessments for hypotheses should stay fixed once and for all, and that all plausibility updating should be brought about via the likelihoods in accord with Bayes' Theorem. Critics argue that this is unreasonable. The members of a scientific community may quite legitimately revise their (comparative) prior plausibility assessments for hypotheses from time to time as they rethink plausibility arguments and bring new considerations to bear. This seems a natural part of the conceptual development of a science. It turns out that such reassessments of priors poses no difficulty for probabilistic inductive logic as I've described it here. Reassessments may come about by the addition of explicit statements that supplement or modify the background information b, and they may also take the form of (non-Bayesian) transitions to new vagueness sets for individual agents and to new diversity sets for the community. The logic of Bayesian induction (as described here) has nothing to say about what values the prior plausibility assessments for hypotheses should have; and it places no restrictions on how they might change. Provided that the series of reassessments of prior plausibilities doesn't push the prior of the true hypothesis ever nearer to zero, the Likelihood Ratio Convergence Theorem implies that the evidence will very probably bring the posterior probabilities of empirically distinct rivals of the true hypothesis to approach 0 via decreasing likelihood ratios; and as this happens, the posterior probability of the true hypothesis will head towards 1.
The Process of Theory Building | Asymco
1 Research | Hypothesis | Inductive Reasoning
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model of a method of scientific investigation and reasoning
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