Sunday, February 14, 2010

Hinton's Impossible Syllogism

In his book The Fourth Dimension, C. H. Hinton argued for the existence of an undiscovered valid syllogism, IEO-4: that is, a syllogism with all of the following characteristics:

(1) The major premise was an I proposition.
(2) The minor premise was an E proposition.
(3) The conclusion was an O proposition.
(4) The syllogism was in the fourth figure.

Hinton argued for this on the basis of the fact that you could arrange the valid syllogisms in such a way that they exhibited a perfect symmetry except for one missing space, that represented by IEO-4. I will not go into his reasoning, which is rather complex, but he is right: if there were an IEO-4, this would increase the number of symmetries in the valid syllogisms. Hinton even gave an example of the form:

Some Americans (P) are of African stock (M).
No one of African stock (M) is Aryan (S).
Therefore Aryans (S) do not include all Americans (P).

Actually, Hinton gives this premise as "No Aryan is of African stock," but this would make the premises of the argument suitable for the second figure, not fourth. However, since E propositions convert, Hinton's premise is equivalent to the above premise, which gives the right premises for a fourth figure syllogism. So it seems we can take "Some P is M," add it to "No M is S," and get a valid conclusion in mood O.

This would make a beautiful test question for a logic class, because all of the following are true (and it is in fact fairly easy to show it):

(1) Hinton's example argument is a valid argument.
(2) Hinton is right that it adds an I premise to an E premise to get an O conclusion.
(3) Hinton's proposed valid mood is logically impossible.

So here's a logic challenge for you to puzzle over: show how Hinton went wrong by showing that (1), (2), and (3) are all correct.

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