APRIL -12, 1901. ] 
superior to it in facility of manipulation. 
This superiority has its source in the fact 
that of the eight (not four, to which the 
ordinary logic has reduced the forms of 
speech) propositions necessary to a com- 
plete description of the universe, — 
a. allaisb, a. not all ais B, 
uw. none but a is b, wu. some besides a is b, 
2. no ais bd, i. some a is J, 
0 
0. all but a is}, . not all but a is }, 
it is the last four that possess the great ad- 
vantage of being symmetrical (that is, of 
having subject and predicate amenable to 
the same rules of manipulation), and of 
these it is ‘no ais 6’ and ‘someais b’ that 
possess the other advantage of naturalness. 
These last should therefore be regarded 
as the canonical form of the proposition ; 
and correspondingly, the ideal form of the 
syllogism is that in which it appears as a 
statement of the impossibility of concur- 
rence of the premises and the denial of the 
conclusion of the ordinary syllogism. The 
canonical form of the syllogism is there- 
fore this: 
noaisb, 
no c is non-b, 
and some ais c 
are inconsistent (or cannot all three be true 
together). This may be called the In- 
consistency, or the Incompatibility, or, per- 
haps, the Antilogism. But in this all three 
propositions play an exactly similar role— 
there is no distinction between premises 
and conclusion, and it is therefore the one 
single form to which every syllogism may 
be at once reduced, provided we (1) ex- 
press every universal proposition in the 
negative form, no p is q, (2) express every 
particular proposition in the affirmative 
form, some r is s, and also (3) deny the 
conclusion. When thus reduced to the form 
of an inconsistency, the rule for validity is 
this: of the three propositions, two are uni- 
versal and one is particular; each two 
propositions have one and only one term in 
SCIENCE. 
575 
common; the term common to unlike 
propositions appears with like signs and 
the term common to like propositions ap- 
pears with unlike signs. (Thus in the 
above typical form, } and non-b are com- 
mon to the two universal propositions, but 
a, or c, of like quality, are common, re- 
spectively, to the particular and either uni- 
versal. ) 
Any given statement of fact may be ex- 
pressed in terms of any one of the four dif- 
ferent copulas given above, or again, with 
the aid of the special terms the non-existent 
and the existent (0 and in Symbolic Logic), 
it may be expressed in four different ways 
with one and the same copula; thus 
all a is B, nothing is a and b, 
all } is G, everything is a or 8, 
are four different forms of one and the same 
statement of fact (expressed in the four pos- 
sible combinations of two terms and their 
negatives); but in thetwo symmetrical copu- 
las (no ais non-d, all but non-a is 6)* the 
four forms all become practically identical. 
There are therefore ten (4 + 4 + 1 + 1) es- 
sentially different ways of saying one and 
the same thing. As each proposition of 
the Inconsistency can be expressed in 
any one of these different ways, and again 
as each Inconsistency can appear in the 
form of the universal or of the particu- 
lar syllogism, the total number of pos- 
sible syllogisms (when full latitude is 
given to mode of expression) is two 
thousand (10 x 10 x 10 x 2). An example 
of one of these outlying forms is this: 
none are athletic and unhealthy, none are 
healthy and unhappy, hence all are either 
*Tt is, to a certain extent, matter of taste whether 
nothing is both a and b, 
a which is 0 is non-existent, 
no ais b, 
nod isa, 
be regarded as different forms or not,—and so ‘all 
but ais b,’ ete. If it be preferred to consider them as 
different, then the entire number of propositional 
forms is 16 instead of 10 and the number of different 
syllogisms is 16 X 16 > 16 K 2= 8192. 
