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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2012
    • (edited Sep 9th 2012)

    at inductive reasoning it says

    Induction here is not to be confused with mathematical induction.

    We should point out that, however, there is a close relation:

    one can see this still in the German tem for, “induction over the natural numbers” which is not Induktion, but vollständige Induktion: meaning ” complete induction” !

    I guess the reasoning is clear, mathematical induction (at least that over the natural numbers) is a special case of inductive reasoning, namely that where we can be sure that we are inducing from a complete set of instances of the general rule.

    Does anyone feel like touching the entry accordingly to clarify this?

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeSep 9th 2012

    So another example of inducing from a complete set of instance set of instances would be checking every element of a finite set?

    And the most general case would be checking every constructor of an inductive type?

    • CommentRowNumber3.
    • CommentAuthorUlrik
    • CommentTimeSep 10th 2012

    Exactly. Finite sets are indeed inductively generated with a nullary constructor for each element. In particular, the empty set is inductively generated by no constructors, which is why you don’t have to do anything / check anything to map out of it / prove a property over it.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2012

    Okay, I have added a paragraph on inductive reasoning – mathematical induction. Feel free to improve.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2012

    I just notice that the German Wikipedia indeed amplifies this, too, at volständige Induktion:

    Der Beiname „vollständig“ signalisiert, dass es sich hier im Gegensatz zur philosophischen Induktion, die aus Spezialfällen ein allgemeines Gesetz erschließt und kein exaktes Schlussverfahren ist, um ein anerkanntes deduktives Beweisverfahren handelt.


    The qualifier “complete” indicates that, contrary to the philosophical induction, which deduces a general law from particular cases and is not an exact method of deduction, complete induction is an accepted method of deductive proof.

    (I mean, you know all this of course. :-) I am just saying that its good to make that connection between philosophical induction and mathematical induction explicit.)

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeSep 11th 2012

    I added a rule of inference for inductive reasoning, which I have referred to in the sections on mathematical induction and Bayesian induction. I moved some of the probability stuff from the Idea section to the Bayesian section.

    • CommentRowNumber7.
    • CommentAuthorTobyBartels
    • CommentTimeSep 11th 2012

    And then I moved much of that stuff to a new page on Bayesianism: Bayesian reasoning. But I don’t have time to finish that now.

    • CommentRowNumber8.
    • CommentAuthorTobyBartels
    • CommentTimeSep 11th 2012

    By the way, very nice discussion of the relationship between the two notions of ‘induction’, Urs.