Sunday, April 17, 2016

Nelson Goodman on Ampliative Inference

By ampliative inference is central to many historiographical arguments, such as induction by enumeration, analogical reasoning, inference to the best explanation, and other forms that expand on what is available in the premise.

However, it is in these contexts precisely that projectibility can be lost. In my current understanding this has to do with discontinuities in the underlying space, as the bleen and grue examples show that Nelson Goodman introduced in the 1940s and revisited in his 1984 paper New Riddle of Induction in Fact, Fiction and Forecast. Thus, while
  1. All observed emeralds are green.
  2. Thus, all emeralds are green.
is valid, the form
  1. All observed emeralds are grue.
  2. Thus, all emeralds are grue.
fails, the fact that the definition of "grue" contains the notion of "observed before the year X" in its definition means that the property of "grue" is insufficiently general for projectability. 

Chris Swoyer, who wrote the SEP article used here, argues that the grue problem is "just an instance of the ubiquitous underdetermination of hypotheses by finite bodies of data". And such underdetermination is a problem for inductive inference, unless the induction is controlled in some fashion, i.e. some hypothesis are rejected from the inductive step. Restriction to projectable relations is one way to tackle the problem.

In his 3rd Lecture, Goodman looks at the way that justification and inference are related, and states a co-dependency that is virtuously circular:
A rule is amended if it yields an inference we are unwilling to accept; an inference is rejected if it violates a rule we are unwilling to amend. (p.67)
The process of justification is the delicate one of making mutual adjustments between rules and accepted inferences; and in the agreement achieved lies the only justification needed for either. (p.67)
Similarly for inductive inference.
An inductive inference, too, is justified by conformity to general rules, and a general rule by conformity to accepted inductive inferences. Predictions are justified if they conform to valid canons of induction; and the canons are valid fi they accurately codify accepted inductive practice. (p.67)
Applying this to inductive work, which is neglected as compared to deductive inference, is what Goodman calls the Constructive Task of Confirmation Theory (p.68). However, just as with defining any term, we tweak the usage and the definition to make sure that the definition picks out only the instances of usage an no others (p.69).

Goodman's discussion of the black and the raven things starting on pp.70ff is dependent on Carl Gustav Hempel's Studies in the Logic of Confirmation, from: Mind, v.54 n.213 (January 1945), pp.1-26, where Hempel (p.9ff) pulls apart Nicod's criterion for confirmation or invalidation, as Nicod termed it. The basic point is that the basic structure of a rule in its logical form as disjunct no longer separates the assumptions of the hypothesis from the predictions of the hypothesis. This has the effect that logically equivalent hypotheses (R => B and !B => !R) are supported or neutral with respect to the same observations (e.g. the observation !R & !B supports the second but not the first hypothesis). Clearly independence of formulation is an important desideratum (Hempel 1945, p.12).

[Note: I could not find an easy simplification rule that allows to rewrite S1 in the multi-variable case (Hempel 1945, p.13 Fn) into S2, i.e. ~R(x,y) => R(x,y) & ~R(y,x) into R(x,y), but the truth tables show that this is so.]
Taking a leaf from Carl Gustav Hempel, Goodman points out that the only substitutions in the hypothesis step can be the variables for the objects, not the relations (p.71), or put differently:
... the hypothesis says of all things what the evidence statement says of one thing .... (p.71)
The central idea for an improved definition is that, within certain limitations, what is asserted to be true for the narrow universe of the evidence statements is confirmed for the whole universe of discourse. (p.73) 
 (to be continued)

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