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In the book 'Introduction of the theory of Statistics' by Mood,Graybill,Boes (third edition)on page 220 (Chapter 6 on Sampling) you can read: 'Inductive inference is well known to be a hazardous process.In fact,it is a theorem of logic that in inductive inference uncertainty is present.One simply cannot make absolutely certain generalization.' What theorem of logic do they refer to? Can you give me please some reference to this fundamental result ?

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I assume you mean inductive inference in this sense; this is actually different from inductive reasoning in logic, which means that results in logic should not necessarily apply directly. Since we don't all have the book, can you briefly describe what it is meant by inductive inference in your book? –  Willie Wong Jan 25 '12 at 9:30
    
In the book when they talk about inductive inference they mean the extension from the particular to the general,the common practice in natural and physical science.The book is an introductory one and the authors don't give a formal definition of inductive inference;they only say that on the opposite side there is deductive inference in which conclusion are conclusive when in inductive inferece the conclusion are only probable.I took a look at the book of Rudolf Carnap 'Logical foundations of Probability' hoping to find an answer but it is very long and not easy to read. –  Steve0078 Jan 25 '12 at 10:09
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Instead of Carnap, first read through the second link I posted above, in particular the section "Is induction reliable?" and the references thereof. I disagree with the statement that "it is a theorem". It is more that the very definition of inductive reasoning allows for the conclusion to be false. –  Willie Wong Jan 25 '12 at 10:29
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2 Answers 2

This is an elementary but I think reasonable explanation of the difference between deductive and inductive reasoning. The authors make clear that most arguments can be framed either way, but require different types of support. In their example, one can argue that a kicked ball will fall to the ground by appeal to Newton's law (deductive) or by reference to previous instances in which balls have fallen (inductive).

This highlights the risk of induction. Absent a general rule, inference (induction) can lead one in the wrong direction. We would not want to say that a fair roulette wheel will land on black simply because it has done so several times in a row.

Hope this helps.

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The use of the term "theorem" seems rather strange. It at least doesn't feel like we can formalize this idea quite in the same way logical theorems, and mathematical theorems, can get formalized.

That said, it isn't hard to make the case that uncertainty has to come as present in inductive inference. Let us suppose that uncertainty does not exist in inductive inference (this presupposes that uncertainty either exists or does not exist in a given situation, which seems safe here). It then follows that inductive inference would qualify as certain. So, inductive inference would have to always work. But, it doesn't always work, as plenty of examples can get supplied where it leads us in the wrong direction. So, inductive inference doesn't qualify as certain, and consequently in inductive inference uncertainty is present.

"Truism of logic" or "truth of logic" might work as better here than "theorem of logic". I doubt that by "theorem" the authors meant anything more than "truth of logic", though I don't know for certain.

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