here it is a minor, and the middle term _man_ is expressly connected
with class _hero_.
A connective relation to the major, and a disjunctive relation to the
minor are impossible in negative syllogisms. The exceptions to this
are only apparent. The two most prominent are the formulæ _Camestres_
and _Camenes_, in both of which it is the minor premiss wherein the
relation is disjunctive. But this is an accident; an accident arising
out of the fact of the major and minor being convertible.
_Bokardo_ is in a different predicament. _Bokardo_, along with
_Baroko_, is the only formula containing a particular negative as a
premiss. Now the particular negatives are, for so many of the purposes
of logic, particular affirmatives, that they may be neglected for the
present; the object at present being to ascertain the rules for the
structure of truly and unquestionably negative syllogisms. Of these we
may predicate that--their minor proposition is always either actually
affirmative or capable of becoming so by transposition.
To go further into the relations between the middle term and the minor,
would be to travel beyond the field under present notice; the immediate
object of the present paper being to explain the import of the word
_distributed_. That it may, both logically and etymologically, mean
_related to two classes_ is clear--clear as a matter of fact. Whether,
however, _related to two classes_ be the meaning that the history of
logic gives us, is a point upon which I abstain from giving an opinion.
I only suggest that, in elementary treatises, the terms _universal_ and
_distributed_ should be separated more widely than they are; one series
of remarks upon--
_a._ Distribution as a condition of inference, being followed by
another on--
_b._ Universality of the middle term in one premiss as a sign of
distribution.
So much for the extent to which the present remarks suggest the purely
practical question as to how the teaching of Aristotelian logic may
be improved. There is another, however, beyond it; one of a more
theoretical, indeed of an eminently theoretical, nature. It raises
doubts as to the propriety of the word _all_ itself; doubts as to the
propriety of the term _universal_.
The existence of such a word as _all_ in the premiss, although existing
therein merely as a contrivance for reconciling the evidence of the
distribution of the middle term with a certain amount of simplicity in
the way of terminology, could scarcely fail, in conjunction with some
of its other properties, to give it what is here considered an undue
amount of importance. It made it look like the opposite to _none_.
Yet this is what it is not. The opposite to _none_ is _not-none_, or
_some_; the opposite to _all_ is _one_. In _one_ and _all_ we have
the highest and lowest numbers of the individuals that constitute a
class. In _none_ and _some_ we have the difference between existence
and non-existence. That _all_ is a mere mode of _some_, has been
insisted on by many logicians, denied by few or none. Between _all_
and _some_, there is, at best, but a difference of degree. Between
_some_ and _none_, the difference is a difference of kind. _Some_
may, by strengthening, be converted into _all_. No strengthening may
obliterate the difference between _all_ and _not-all_. From this it
follows that the logic of _none_ and _some_, the logic of connexion and
disjunction (the logic of _two_ signs), is much more widely different
from the logic of _part_ and _whole_ (the logic of _three_ signs) than
is usually admitted; the former being a logic of pure _quality_, the
latter a logic of _quality_ and _quantity_ as well.
Has the admixture done good? I doubt whether it has. The logic of pure
and simple Quality would, undoubtedly, have given but little; nothing
but negative conclusions on one side, and possible particulars on the
other. Nevertheless it would have given a logic of the Possible and
Impossible.
Again, as at present constituted, the Quantitative logic, the logic of
_all_ and _some_, embraces either too much or too little. _All_ is, as
aforesaid, only a particular form of _more than none_. So is _most_.
Now such syllogisms as--
Most men are fallible,
Most men are rational,
Some men are both frail and fallible;
_or_,
Some frail things are fallible,
are inadmissible in the Aristotelian paradigms. A claim, however,
is set up for their admission. Grant it, and you may say instead of
_most_--
Fifty-one per cent., &c.;
but this is only a particular instance. You may combine any two numbers
in any way you like, provided only that the sum be greater than unity.
Now this may be arithmetic, and it may be fact; but it is scarcely
formal logic; at any rate it is anything but general.
It is the logic of _some_ and its modifications _one_, _all_, and
_anything between one and all_, as opposed to the logic of the
simple absolute _some_ (_some_ the opposite to _none_), and a little
consideration will show that it is also the logic of the _probable_,
with its modification the _proven_, (_proven_ is _probable_, as
_all_ is _some_,) as opposed to the logic of the _possible_ and
_impossible_. Let, in such a pair of propositions as--
Some of the men of the brigade were brave,
Some of the men of the brigade were killed,
the number expressed by _some_, as well as the number of the men of the
_brigade_, be known, and the question as to whether
Some brave men were killed,
is a problem in the doctrine of chances. One per cent. of each will
make it very unlikely that the single brave man was also the single
killed one. Forty-nine per cent. of each will make it highly probable
that more than one good soldier met his fate. With fifty on one side,
and fifty-one on the other, we have _one_ at least. With _all_ (either
_killed_ or _brave_), we have the same; and that without knowing any
numbers at all.