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1. Deductively Definable Logics of Induction

The class of logics I will investigate are "deductively definable." They are based on the inductive strengths

[A|B]

which we will read as "the strength of support that proposition A accrues from proposition B." The strength [A|B] will often be a real number, such as a probability, but it need not be.

An inductive logic is the set of rules used to assign these strengths. In a deductively definable logic of induction, these rules employ only deductive relations between propositions.

 [A|B] = some formula that mentions only deductive relations between propositions

You might think that such a logic is exceptional. They are not. Indeed one of the most common ways of specifying inductive relations in the literature on inductive inference is through deductive relations.

The idea is a natural one. In its crudest form, inductive inference is introduced as some variant of deductive inference. One way of seeing induction is as a form of partial entailment. If the truth of B necessitates the truth of A, then we have full deductive entailment. If, however, it succeeds only partially, then we have inductive support. Or, in another approach, inductive inference is seen as an inversion of deductive inference. If A deductively entails B, then B provides inductive support for A.

Elsewhere I have described how the many accounts of inductive inference can be divided into three general families. The first two are "inductive generalization" and "hypothetical induction." Both depend upon characterizing inductive support in terms of deductive relations.

Inductive generalization depends upon the notion of an instance of a generalization. The most familiar example arises in syllogistic logic. The proposition that "This A is B." is an instance of the generalization "All A's are B." Hempel famously extended this notion of an instance to first order predicate logic. All logics in the inductive generalization family depend in one form or another on the rule

Instance I confirms generalization G     if     I is an instance of G.

This rule lies within deductively definable logics since the essential clause on the right of the definition "I is an instance of G" specifies a deductive relation between propositions. This fact continues to hold for the more elaborate versions of instance confirmation, such as Glymour's "bootstrap."

Hypothetical induction depends upon the inverse notion already mentioned above and, in its simplest form, employs the rule:

Evidence E HD confirms hypothesis H     if     H deductively entails E.

Once again, the rule lies within deductively definable logics since the essential clause on the right of the definition, "H deductively entails E," specifies a deductive relation between propositions.

This simplest form of hypothetical deduction is generally found to be too simple. There are many proposals to amend it. Generally, the amendments require that H not merely entail E, but that it do it in the right way. What constitutes the "right way" is the subject of continuing investigations. One proposal requires that it do so simply. Another requires that H not just entail E but that it explain E. Yet another requires that the entailment not be ad hoc. All these amount to the addition of an extra condition "D" to form the new rule:

Evidence E confirms hypothesis H     if     H deductively entails E and condition D obtains.

This augmented rule will still lie within the deductively definable logics as long as the condition D is itself a condition expressed fully in terms of the deductive relations between the propositions. It is certainly plausible, but perhaps not necessary, that notions like simplicity and explanatory character can be specified by deductive relations. The simpler deduction, for example, might merely employ fewer steps. Or the more explanatory hypothesis might infer the evidence from some small set of propositions that have so many deductive consequences that we single them out as "laws." Ad hocness is generally characterized in terms of the order in which we infer things. If H entails an already know E, H may not get as much support as it would if it were to entail evidence not already known. This notion of ad hocness can be captured if we expand our logic to include a time indexing of propositions that indicates the order in which we learned them.

 The rules mentioned so far enable the assigning of a single value to [H|E]. Let us call that value "support." [H|E] has that value just when the particular account of hypothetico-deductive support at issue says that E supports H. Confirmational support, however, more naturally comes in degrees. We can certainly imagine augmentations of the above rules that give a quantitative measure of support. For example, in hypothetical deduction, we might reward an hypothesis H if it entails the evidence E in fewer steps. That means that H is closer to E. There might also be some reward for parsimony. What do we mean by "number of steps"? In a Boolean algebra formed by the logical closure of mutually exclusive atoms a, b, c, ..., we would define one step as the addition of one atom. So the inference from a to (a or b) is a one-step inference. The inference from (a or b) to (a or b or c or d or e or f) is a four-step inference.

This suggests the following rule

 Evidence E is confirmed to degree H with strength [H|E] = 1/`number` if H entails E in a deductive inference of `number` steps.

This rule is still rather primitive. However no particular rule is at issue here. My concern in displaying this parade of examples is merely to make plausible that deductively definable logics of induction are not exceptional. They are, in effect, the sort of thing that is discussed routinely in the inductive inference literature.

 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 What Logics of Induction are There? John D. Norton