The present paper is an attempt to reconcile the logical and
etymological meanings of the word _Distributed_.
Speaking roughly, _distributed_ means _universal_: "a term is said
to be _distributed_ when it is taken universally, so as to stand for
everything it is capable of being applied to."--_Whately_, i. § 5.
Speaking more closely, it means _universal in one premiss_; it being a
rule in the ordinary logic that no conclusion is possible unless one
premiss be, either negatively or affirmatively, universal.
Assuredly there is no etymological connexion between the two words.
Hence De Morgan writes:--"By _distributed_ is here meant _universally
spoken of_. I do not use this term in the present work, because I do
not see why, in any deducible meaning of the word _distributed_, it can
be applied to universal as distinguished from particular."--_Formal
Logic_, chap. vii.
Neither can it be so applied. It is nevertheless an accurate term.
Let it mean _related to more than one class_, and the power of the
prefix _dis-_, at least, becomes intelligible.
For _all_ the purposes of logic this is not enough; inasmuch as the
particular character of the relation (all-important in the structure
of the syllogism) is not, at present, given. It is enough, however, to
give import to the syllable _dis-_.
In affirmative propositions this relation is connective on both sides,
_i. e._ the middle term forms part of _both_ the others. In negative
propositions this relation is connective on _one_ side, disjunctive on
the _other_.
In-- All men are mortal,
All heroes are men,
the middle term _men_ forms a part of the class called _mortal_, by
being connected with it in the way that certain contents are connected
with the case that contains them; whilst it also stands in connexion
with the class of _heroes_ in the way that cases are connected with
their contents. In--
No man is perfect,
Heroes are men,
the same double relation occurs. The class _man_, however, though part
of the class _hero_, is no part of the class _perfect_ but, on the
contrary, expressly excluded from it. Now this expression of exclusion
constitutes a relation--disjunctive indeed, but still a relation; and
this is all that is wanted to give an import to the prefix _dis-_ in
_distributed_.
Wherever there is distribution there is inference, no matter whether
the distributed term be universal or not. If the ordinary rules for
the structure of the syllogism tell us the contrary to this, they only
tell the truth, so far as certain assumptions on which they rest are
legitimate. These limit us to the use of three terms expressive of
quantity,--_all_, _none_, and _some_; and it is quite true that, with
this limitation, universality and distribution coincide.
Say that Some Y is X,
Some Z is Y,
and the question will arise whether the Y that is X is also the Y that
is Z. That _some_ Y belongs to both classes is clear; whether, however,
it be the same Y is doubtful. Yet unless it be so, no conclusion can
be drawn. And it may easily be different. Hence, as long as we use the
word _some_, we have no assurance that there is any distribution of the
middle term.
Instead, however, of _some_ write _all_, and it is obvious that some Y
must be both X and Z; and when such is the case--
Some X must be Z, and
Some Z must be X.
Universality, then, of the middle term in one premiss is, by no
means, the _direct_ condition that gives us an inference, but only a
_secondary_ one. The direct condition is the distribution. Of this,
the universality of the middle term is only a _sign_, and it is the
only sign we have, because _all_ and _some_ are the only words we have
to choose from. If others were allowed, the appearance which the two
words (_distributed_ and _universal_) have of being synonymous would
disappear. And so they do when we abandon the limitations imposed
upon us by the words _all_ and _some_. So they do in the numerically
definite syllogism, exemplified in--
More than half Y is X,
More than half Y is Z,
Some Z is X.
So, also, they do when it is assumed that the Y's which are X and the
Y's which are Z are identical.
Y is X,
The same Y is Z,
Some Z is X.
In each of these formulæ there is distribution without universality,
_i. e._ there is distribution with a quality other than that of
universality as its criterion. The following extract not only explains
this, but gives a fresh proof, if fresh proof be needed, that
_distributed_ and _universal_ are used synonymously. The "comparison
of each of the two terms must be equally with the whole, or with the
same part of the third term; and to secure this, (1) either the middle
term must be distributed in one premiss at least, or (2) the two terms
must be compared with the same specified part of the middle, or (3), in
the two premises taken together, the middle must be distributed, and
something more, though not distributed in either singly."--_Thompson,
Outline of the Laws of Thought_, § 39.
Here _distributed_ means _universal_; Mr. Thompson's being the ordinary
terminology. In the eyes of the present writer "distributed in one
premiss" is a contradiction in terms.
Of the two terms, _distributed_ is the more general; yet it is not the
usual one. That it has been avoided by De Morgan has been shown. It may
be added, that from the Port Royal Logic it is wholly excluded.
The statement that, in negative propositions, the relation is
connective on _one_ side, and disjunctive on the _other_, requires
further notice. It is by no means a matter of indifference on which
side the connexion or disjunction lies.
(_a._) It is the class denoted by the major, of which the middle term
of a negative syllogism is expressly stated to form _no_ part, or from
which it is disjoined. (_b._) It is the class denoted by the minor, of
which the same middle term is expressly stated to form part, or with
which it is connected.