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# What Logics of Induction are There?

John D. Norton
Center for Philosophy of Science
Department of History and Philosophy of Science
University of Pittsburgh

Most work on inductive logic is restricted to considering just one logic of induction that happens to be favored by the researcher. Since I don't think any one of these is the One True Logic, my goal is to broaden the perspective by considering a large class of inductive logics. A very common way of specifying inductive import in the literature is through an examination of the deductive relations among propositions. This suggests that we delineate a class of inductive logics that are "deductively definable." It will be shown here that no purely deductively definable logic is admissible, so that many favored approaches to induction in the literature fail. They all need some sort of inductive supplement. I show a simple way to provide this inductive supplement and display the general properties of the resulting class of inductive logics.

The ideas presented here are informal versions of material developed more precisely in my "Deductively Definable Logics of Induction." For paper referred to as "elsewhere" below, look here.

# Why Logics and Not Just Logic?

 In papers elsewhere, I have argued that our present understanding of inductive inference is in a shambles as far as fundamentals are concerned. The model that we have long employed is that of deductive inference. There we separate out the good from the bad deductive inferences by checking whether they conform to some set of universal templates that comprises our deductive logic. We have long tried to do the same with inductive logic and failed. We all know that simple inductive inferences don't work. No one is willing to endorse the universal template that says that finding an A that is a B warrants us to infer that all As are B. However we hang onto the idea some sort of template like this will eventually work. We just need to make it more complicated. After two millennia of failure in these efforts, we still hope that perhaps tomorrow the solution will come.

In the "material theory of induction," I offer an alternative. That theory urges that inductive inferences are not licensed in the same way as deductive inference. Rather they are licensed by facts. Showing that an inductive inference is licit involves a chain of justifications that terminates in facts, not in a universal schema. Since each fact is contingent, the inferences it licenses can only hold in the limited domains in which the fact is true. As a result there are many logics of inductive inference, differing from domain to domain according to the facts obtaining in each.

 Thus, contrary to Bayesian doctrine, I maintain that the probability calculus cannot be the universal logic of induction. To illustrate the point, I have displayed problems that I do not think any Bayesian can solve responsibly. I am, of course, quite happy with the idea that certain problems can be dealt with handily by the Bayesian system. Whether it is the right system to apply in some domain is determined by the facts that prevail in that domain. What I am denying is that the Bayesian system is the right way to treat all problems of inductive inference.

# The Project

What other logics are there that may prevail in these other domains? What is our menu? Answering that question is the present project. Of course it is possible to find easy answers by cooking up new logics of induction arbitrarily. That just leads to the confusion of many logics without much of a sense of how they connect with each other. The goal here is to avoid this problem by delineating a large and interesting class of inductive logics, whose general, common properties can be understood.

Here's what is coming:

 1. Deductively definable inductive logics Taking present work in inductive inference as a model, we can describe one of the simplest classes of inductive logics. In it, the strengths of inductive support are defined in terms of the deductive relations obtaining between propositions. 2. Symmetry theorem for deductively definable logics of induction The fact of deductive definability places a powerful restriction on these inductive logics: they must inherit all the symmetries of the deductive structure. That fact will drive much of what ensues in the sections following. 3. No-Go result for deductively definable, asymptotically stable inductive logics It turns out that all these logics are trivial or fail to be admissible if we require them to stabilize under expansions of the set of propositions under consideration. 4. Deductively definable inductive logics in preferred partitions The natural escape from the no-go result is to add just enough independent inductive content to escape the no-go result. This is done by introducing inductively adapted partitions of outcomes in which the symmetries of the deductive relations are also symmetries of inductive relations. 5. A specific conditioning inductive logic While there are many inductive logics in the class defined, here is one that hasn't, as far as I know, been explored in the philosophy of science literature. It illustrates how a non-Bayesian, numerically based logic of induction can be quite sensible and well-behaved. 6. Inductive independence is generic These new logics manifest some interesting properties. One is that inductive independence is generic. For most pairs of propositions, learning that one is true or false makes no difference to the inductive support accrued the other. 7. A limit theorem These logics also host a limit theorem: learning the truth of finitely many deductive consequences of an hypothesis can lead us to be arbitrarily sure of the truth of the next of its consequences that we encounter. 8. Scale-free inductive logics A subclass of these inductive logics is scale-free in the sense that their inductive strengths do not change as we enlarge the set of propositions in their domain by splitting each proposition into disjunctive parts. 9. Visualizing deductive structure An important notion here has been the that of a symmetry of the deductive relations among propositions in a Boolean algebra. Here is a way of using graphs to visualize those symmetries and other properties.

Copyright John D. Norton, November 28, 2008.