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I have a problem with the distinction between induction and deduction. To me it does not make sense to talk about induction at all.

People argue that the following is induction:

A_1 is x

A_2 is x

...

A_n is x

--

All A are x;

The question whether induction is a valid way to conclude this, from my point of view, boils down and is better stated as a question concerning a deduction with an extra premise X:

A_1 is x

A_2 is x

...

A_n is x

X

--

All A are x;

The question now is: How must X look, so it allows us to use deduction?

A rough estimation of how X might look is: If we observe n instances of A to be x, we can conclude that All A are x. Please note, that there does not have to be a general answer on how X must be formulated.

So from my point of view, it follows that we should not try to answer whether induction is generally a valid way of conclusion but rather we have to find X for each case and try to verify it. Thus, we do not have to rely on the notion of induction at all.

Of course, the question is X true is very similar to the question whether inductive conclusion is valid. However, it is different in the sense that we try to verify a particular statement, a premise, rather than trying to establish a notion of drawing conclusion that is not-deductive.

From a slightly different point of view, we can say that if we can find and verify X, or if we can find good reasons on why induction is a valid way to draw conclusions, there is no induction. As soon as we justify induction, it becomes just a case of deduction, in which we can use the justification of induction as a premise, i.e. if induction is valid, there is no induction that is different from deduction.

So, what is induction other than deduction with an implicit premise? Why does it still make sense to distinguish these two notions?

I am also thankful about any pointers to literature that might provide useful answers.

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One easy, and perfectly general, such premise X is:

All A are x

From this premise one can deductively infer that all A are x, unsurprisingly.

One problem with this maneuver, of course, is that we have no better evidence for the extra premise than we have for the conclusion -- both being the same proposition. Therefore, as it is sometimes said, there is no transmission of warrant from premises to conclusion.

I suspect that the same will happen with any plausible candidate for the role of premise X. Take your own example,

If n instances of A are x, then all A are x

Suppose for a moment that all ravens are black, and that you have observed n ravens to be black. Your premise X is safisfied, but not because n ravens being black somehow makes it the case that all ravens are. Rather, the consequent of the conditional being true, the antecedent also is.

Here, too, we have no better (and no different) evidence for the X premise than we have for the conclusion: there is no transmission of warrant.

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