Wednesday, September 26, 2007

Fallacies of Distribution with Conditional Propositions

I find that Jevons, in Lesson XIX his Elementary Lessons in Logic, has a nice little discussion of the fact that conditional statements can be handled without leaving categorical syllogistics. This is the point I mentioned earlier about the fact that every conditional statement can be treated as a categorical proposition of A form. Thus, we can take the argument

If iron is impure it is brittle ;
But it is impure ;
Therefore it is brittle.

And convert it to

Impure iron is brittle ;
The iron in question is impure iron ;
Therefore the iron in question is brittle.

Which is (basically) a Barbara syllogism. He then goes on to note:

It will now be easily made apparent that the fallacy of affirming the consequent is really a breach of the 3rd rule of the syllogism, leading to an undistributed middle term.

This is something I hadn't considered before, but he's quite right. We can handle propositional logic by categorical syllogisms, and when we do so, every case of the fallacy of affirming the consequent turns out to be a case of the fallacy of the undistributed middle. Jevons's example is:

If a man is avaricious he will refuse money ;
But he does refuse money ;
Therefore he is avaricious.

Converted to categorical syllogism:

All avaricious men refuse money;
But this man refuses money ;
Therefore this man is avaricious.

This is an AAA-2 syllogism (or an AII-2, depending on how you interpret singular quantity); the middle term ('refuses money') is undistributed in both premises.

Jevons goes on to note that the fallacy of denying the antecedent turns out to be a case of the fallacy of illicit process of the major.

(One might ask if there is a fallacy in propositional logic corresponding to the fallacy of illicit process of the minor. Indeed there is, but I don't know if it has ever been given a name. It occurs in arguments of the form: p → q; p → r; therefore q → r. Arguments of this form put into categorical form are always cases of the fallacy of illicit minor.)

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