>The work "induction" has had no consistent use, either recently or
>historically. Sometimes writers have meant all inferences that arent
>deductive, sometimes they have specifically meant inductive
>generalizations, and sometimes they have meant next-case inductions as
>in the philosophical "problem of induction" raised by David Hume. C. S.
>Peirce seems to have used the term to mean: testing a hypothesis by
>generating predictions and evaluating those predictions empirically. It
>will be best, I think, for researchers, especially those who read this
>mailing list, to make appropriate distinctions to minimize confusion.
>We should write "inductive generalization" if thats what we mean.
I agree with John's claim that there is no consistent use of the word
"induction". I would like to elaborate a bit on the classes of inferences
he mentions.
1. The class of all non-deductive inferences.
In my opinion this is a meaningless class, because it is "defined" in terms
of what it is not. If people find it meaningful to distinguish a particular
reasoning type, then it is the task of logic to isolate and formalise the
essence of this reasoning type, not by saying what it is not but by saying
what it is.
There is a parallel here with the regrettable label "nonmonotonic
reasoning": virtually all non-deductive reasoning is nonmonotonic, but
people in nonmonotonic reasoning are studying a much more restricted class
of Tweety-like inferences. The logical question is not what characteristics
a particular class of inferences lacks, but which ones they enjoy. Only
recently people in nonmonotonic reasoning have started to investigate such
properties (e.g. cumulativity) for their Tweety-like inferences.
2. The class of inductive generalisations
I take inductive generalisation to be sample-to-whole inference (all
observed P's are Q's; therefore, all P's are Q's). Philosophically
speaking, this is the inference that stands most in need of justification.
Practically speaking however, this inference is rather trivial because all
the work (viz. determining P and Q) has already been done. A more realistic
pattern of inductive generalisation would be something like this:
Given: a theory Th about the sample
Find: a formula F such that
1. F is true for all observed objects
2. F is expected to be true for all objects
This pattern highlights the important fact that the hypothesis F is not
given but to be inferred.
[Note: inductive generalisation is often defined as: this P is a Q; that P
is a Q; therefore, all P's are Q's. However, here the premisses are
symmetric in P and Q, so we might just as well generalise to: all Q's are
P's. In order to break the symmetry, we must either set apart Q as the
'predicate to be learned', or observe some non-P's. Hence, the
raven-paradox is not a paradox at all, since observing some non-black
non-ravens makes us prefer 'all ravens are black' over 'everything is
black', and thus can indeed be seen as confirming evidence for the former
hypothesis.]
3. The class of inferences of inductive probabilities
In his later theory, Peirce indeed wrote that "induction does nothing but
determine a number". This is also Carnap's view in his "inductive logic":
Carnap views inductive probabilities as degrees of entailments. It is a
natural consequence of the first position (induction as non-deduction). In
my opinion it leads to a degenerated view of logic, since any pair of
premisses and conclusion will have a proof (a calculation of the associated
degree of entailment).
John continues:
>Alternatively, we might just be careful to reserve the unqualified term
>as a general category for all non-deductive inferences. However, I
>think you can make a good case that abductions, inductive
>generalizations, next-case predictions, and probably other members of
>the inference zoo, whether defined formally or informally, can be
>deductive at the same time as they are abductions, etc., etc. For
>example an inductive generalization (as sample-to-population inference)
>becomes deductive as soon as the sample grows to include the whole
>population, or, for smaller samples, if you explicitly assume a premise
>that `the sample is representative.'
>
>Abductions (as best-explanation inferences) can all be viewed as
>instances of the deductive form:
>
>{E(i)} are all of the possible explanations for D.
>Some explanation for D must be correct.
>not E (i) (for-all i not-equal j).
>Therefore, E(i).
Typo: this should read "Therefore, E(j)".
>Thus, all abductions are deductions! Moreover, any inference
>whatsoever, even irrational and stupid inferences, can be construed as
>deductive inferences with missing premises, that is they are
>"enthymemes." The stupid inference `p, therefore q,' can always be
>viewed as a deductive inference with the hidden premise `p implies q.'
This is of course the very nature of reasoning with incomplete information:
if the information were complete, the inference would deductively follow.
However, Hume showed that this will not help us if we want to infer
inductive generalisations: the missing premiss would itself be a
generalisation that is the conclusion of an inductive argument (some
assumption about the Uniformity of Nature), so we immediately get trapped
in an infinite regress.
>The lesson is, apparently, that the set of non-deductive inferences is
>empty, depriving the word "induction," of any fly poop of meaning when
>used unqualified as a name for non-deductive inferences. But wait!
>Perhaps we can mean by "induction" any inference, conceived as having
>some reasonable persuasive force, that gets its persuasive force other
>than by being a deductive inference. That is, "inductive inferences" are
>justified inferences whose justifications are other than deductive.
>"Deduction," "abduction," "inductive generalization," and so on, are
>categories of Justifications for inferences, not categories of
>inferences as such. A given instance of inferential reasoning, a single
>inferential step, might belong to more than one category at the same
>time, depending on which justifications can be given on its behalf, and
>especially, which justifications give it force. "non-deductive"
>inferences are those inferences that have non-deductive justifications.
I'm not sure if I agree here. Clearly inductive inferences stand in need of
other justifications than deductive inferences. We don't have much idea yet
of what those other justifications could be, but I think that is because
for a long time we couldn't think of any other justification than deductive
justification. What is deductive justification? An argument is deductively
justified if and only it is truth-preserving. But if one sticks to truth as
the foundation of logical semantics, one will never go beyond deduction.
For me, *preservation* is the foundation of logical semantics: just what is
the semantic quality that is preserved when moving from premisses to
conclusion in an argument of reasoning type X? In a previous mail I
suggested explanatory power as the preserved quality in explanatory
reasoning. One can try to formalise inductive generalisation as
generality-preserving (or -increasing) inference. Work in this direction
has only just started. Abduction and induction are more similar to
deduction than we think!
--Peter