Bauhaus-Universität Weimar

Dictionary of philosophy and psychology including many of the principal conceptions of ethics, logics, aesthetics ... and giving a terminology in English, French, German and Italian, vol. 2 [lead-zwing]
Baldwin, James Mark
q, That a is b is-not-incompatible-with c being d. 
Different symbolic copulas (modifications of 
<:) may be devised for all the different 
relations of this sort, and the transformation 
from one to another may be made by 
mechanical rules. How many of these es¬ 
sentially different relations are there 1 The 
ordinary logic recognizes only four, and of 
these one is to a certain extent recalcitrant 
to rule, for the reason that it is in fact 
a member of a different scheme. Instead of 
Some a is-not b, we ought to express this 
member of the group of four as Not all a is b. 
This is the form in which it appears in 
Aristotle, and it frequently retains this form 
in the works of the schoolmen, as appears in 
the fact that the symbolic letters which stand 
for the several propositions, A, E, I, 0, are 
said to be (but upon perhaps insufficient 
authority) the characteristic vowels of nâs, 
ovàeiç (ouSA), ris, ov näs (Prantl, Gesell, d. 
Logik, xv. 277, and iv. 153-4). The pro¬ 
positions admitted into any scheme should be 
propositions which are the immediate denials 
one of another, as Some a is b, Not any a is b. 
Pairs of immediate denials are 
(a) All a is b, 
(ä) Not all a is b, 
(m) Some a is-not b, 
(m) No a is-not b. 
Either (a) and (a) should be regarded as the 
canonical forms, or else (m) and (pi) ; to mix 
them up, as is done, is a pity, for the rules 
for Tkansfobmation (q. v., in logic) apply 
very differently to the incongruous pair (a) 
and (pi), and hence much confusion arises. 
The right pair to choose is, of course, (a) and 
(«) ; All that glitters is gold is properly 
denied by Not all that glitters is gold. 
The actual number of different statements 
that are possible in terms of x and y and 
their contradictory terms x and ÿ (excluding 
double negatives) is eight. This is at once 
evident if we express everything that can be 
said in the form of propositions of existence 
and of non-existence ; thus the combinations 
of a and b and their negatives are ab, ab, ab, 
ab, and since each one of these combinations 
can be said to exist (a particular proposition, 
There is some a which is b, or Some a is b) or 
to be non-existent (a universal proposition, 
There is no a which is b, or No a is b), it is 
evident that eight different statements of fact 
are possible. These, of course, remain diffe¬ 
rent, no matter what the form in which they 
may be expressed. One reason why logic 
commonly recognizes only four out of this set 
of eight is that it has fought shy of negative 
terms, and especially of negative terms as 
subjects. This is strange, because Aristotle 
gives, in his most fundamental group of 
propositions (those in one term only), four 
with negative subjects, as Not-a exists, Not-a 
exists not, &c. It is, however, De Morgan to 
whom we owe not only the generalization of 
the copula (which, he says, he has ‘ made as 
abstract as the terms '), but also the full intro¬ 
duction into logic of negative terms as subjects 
as well as predicates, and the setting out of 
the eight propositions of a complete scheme. 
De Morgan did not, however, devise appro¬ 
priate copulas for the several statements to be 
made ; but one does not have to search far, in 
the language of real life, to find such, and 
when they are found, the eight things to be 
said can all be said by means of them, very 
simply, without the use of any negative terms 
whatever. The letters A, E, I, 0 being no 
longer adequate, we may take i and 0 and 
their negatives to stand for the symmetrical 
copulas—those in which subject and pre¬ 
dicate are simply commutable—and the 
unsymmetrical letters, a and u (u is perhaps 
sufficiently unsymmetrical), to stand for those 
copulas with which subject and predicate 
cannot be interchanged without change of 
sign. We shall then have 
(a) All x is y, (a) Hot all x is y, 
(û) Hone but x is y, (u) Something besides 
x is y, 
(Ï) Ho x is y, (i) Some x is y, 
Ip) All but x is y, (p) Hot all but x is y. 
The first two copulas in each column are non- 
symmetrical : None but x is y can only be 
inverted into None but non-y is non-x, and 
Not all x is y only into Not all non-y is non-x ; 
in the last four propositions all terms are 
simply commutable. 
Language furnishes us with perfectly ade¬ 
quate forms of expression for these eight 
modes of connection in the compound proposi¬ 
tion as well as in the simple proposition. 
Thus we have 
(a) If it rains it pours, (ä) Though it rains it 
does not always pour, 
(û) Hot unless it rains (u) Besides when it 
does it pour, rains it sometimes 
(1) Hever when it rains (i) Sometimes when 
does it pour, it rains it pours, 
(0) Unless it rains it (p) Hot always except 
pours, when it rains does 
it pour. 


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