1 7 5V a l i d i t y f o r C a t e g o r i c a l A r g u m e n t sNOTES
(^1) We say “not always” rather than simply “not,” because there are some strange cases— logicians
call them “degenerate cases”—for which inferences of this pattern are valid. For example, from
“Some men are not men,” we may validly infer “Some men are not men.” Here, by making the
subject term and the predicate term the same, we trivialize conversion. Keeping cases of this
kind in mind, we must say that the inference from an O proposition to its converse is usually,
but not always, invalid. In contrast, the set of valid arguments holds in all cases, including
degenerate cases.
(^2) We cannot say “only if” here because of degenerate cases of categorical syllogisms that are
valid, but not by virtue of their syllogistic form. Here is one example: “All numbers divisible by
two are even. No prime number other than two is divisible by two. Therefore, no prime number
other than two is even.” This syllogism is valid because it is not possible that its premises are
true and its conclusion is false, but other syllogisms with this same form are not valid.
(^3) We need to add “by virtue of its categorical form,” because, as we saw above, it still might
be valid on some other basis. In this particular example, however, nothing else makes this
argument valid.
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