John D.
Norton


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Nontrivial calculi of inductive inference are shown to be incomplete. That is, it is impossible for a calculus of inductive inference to capture all inductive truths in some domain, no matter how large, without resorting to inductive content drawn from outside that domain. Hence inductive inference cannot be characterized merely as inference that conforms with some specified calculus. 
"A Demonstration of the Incompleteness of Calculi of Inductive
Inference" Draft "The Ideal of the Completeness of Calculi of Inductive Inference: An Introductory Guide to its Failure" Draft 

A probabilistic logic of induction is unable to separate cleanly neutral support from disfavoring evidence (or ignorance from disbelief). Thus, the use of probabilistic representations may introduce spurious results stemming from its expressive inadequacy. That such spurious results arise in the Bayesian “doomsday argument” is shown by a reanalysis that employs fragments of inductive logic able to represent evidential neutrality. Further, the improper introduction of inductive probabilities is illustrated with the “selfsampling assumption.”  "Cosmic Confusions: Not Supporting versus Supporting Not" Philosophy of Science. 77 (2010), pp. 50123. Download.  
While Bayesian analysis has enjoyed notable success with many particular problems of inductive inference, it is not the one true and universal logic of induction. I review why the Bayesian approach fails to provide this universal logic of induction. Some of the reasons arise at the global level through the existence of competing systems of inductive logic. Others emerge through an examination of the individual assumptions that, when combined, form the Bayesian system: that there is a real valued magnitude that expresses evidential support, that it is additive and that its treatment of logical conjunction is such that Bayes' theorem ensues.  "Challenges to Bayesian Confirmation Theory," Philosophy of Statistic, Vol. 7: Handbook of the Philosophy of Science. Prasanta S. Bandyopadhyay and Malcolm R. Forster (eds.) Elsevier, 2011. Download.  
The probability calculus is not the universal logic of induction; there is no such thing. An axiom system disassembles the probability calculus into distinct notions about induction, which it is urged, may be invoked independently to tailor a logic of induction to the problem at hand. The probability calculus fails as the inductive logic of certain indeterministic systems.  "Probability Disassembled," British Journal for the Philosophy of Science, 58 (2007), pp. 141171. Download.  
In a material theory of induction, inductive inferences are warranted by facts that prevail locally. This approach, it is urged, is preferable to formal theories of induction in which the good inductive inferences are delineated as those conforming to some universal schema. An inductive inference problem concerning indeterministic, nonprobabilistic systems in physics is posed and it is argued that Bayesians cannot responsibly analyze it, thereby demonstrating that the probability calculus is not the universal logic of induction.  "There are No Universal Rules for Induction,"Philosophy of Science, 77 (2010) pp. 76577. Download  
A simple indeterministic system is displayed and it is urged that we cannot responsibly infer inductively over it if we presume that the probability calculus is the appropriate logic of induction. The example illustrates the general thesis of a material theory of induction, that the logic appropriate to a particular domain is determined by the facts that prevail there.  "Induction without Probabilities." Download. See also "Induction without Probabilities" in Goodies. 

The epistemic state of complete ignorance cannot be a probability distribution. The instruments that characterize it are innocuous platitudes of evidence: the principle of indifference and certain invariance conditions. However the literature has misdiagnosed these instruments as defective since these instruments rule out the representation of complete ignorance by probability distributions.  "Ignorance and Indifference." Philosophy of Science, 75 (2008), pp. 4568. Download.  