SYMBOLIC LOGIC
ordinary syllogism. Thus, the premises of
Baroko, ‘ Any M is P ' and ‘ Some S is not P,
may be written { M[P~\ f S (P). Then, as just
seen, we can write { M |F] [ B (P). Then, by
iteration, {M [P (P)] j B (P). Breaking the
line under even enclosures, we get
{[P(P)]J/j B(P). But we have already
shown that [P (P)] can be written unen¬
closed. Hence it can be struck out under
one enclosure ; and the unenclosed (P) can be
erased. Thus we get {Af\ S, or £ Some S is
not M! The great number of steps into which
syllogism is thus analysed shows the perfec¬
tion of the method for purposes of analysis.
In taking account of relations, it is neces¬
sary to distinguish between the different sides
of the letters. Thus let Z be taken in such
a sense that X—Z— Y means (X loves Y'.’
Then will mean * Y loves X.’ Then,
if m—- means ‘ Something is a man,’ and —w
means ‘ Something is a woman,’ m—Z—w
will mean ‘ Some man loves some woman
m [(J-Z-J)”Lm;] will mean ‘ Some man loves all
women ’ ; [(m.—l^ÿw] will mean ‘ Every
woman is loved by some man,’ &c.
Since enclosures signify negation, by en¬
closing a part of the line of identity, the
relation of otherness is represented. Thus,
will assert ‘Some A is not some P.’
Given the premises ‘ Some A is B ’ and
‘ Some C is not P,’ they can be written
A—BC(B). By Rule III, this can be
written A {[P]} 0(B). By iteration, this
gives A {[P (P)]} C (P). The lines of identity
are to be conceived as passing through the
space between the braces outside of the
brackets. By breaking the lines under even
enclosures, we get JTp [P (Bf\ ^}C (P). As
we have already seen, oddly enclosed [P (P)]
can be erased. This, with erasure of the
detached (P), gives A{~j} C. Joining the
lines under odd enclosures, we get J~{~| )C,
‘ Some A is not some C!
For all considerable steps in ratiocination,
the reasoner has to treat qualities, or collec¬
tions (they only differ grammatically), and
especially relations, or systems, as objects of
relation about which propositions are asserted
or
and inferences drawn. It is, therefore, neces¬
sary to make a special study of the logical
relatives ‘-is a member of the collection
-,’ and ‘- is in the relation-to
The key to all that amounts to much
in symbolical logic lies in the symbolization
of these relations. But we cannot enter into
this extensive subject in this article.
The system of which the slightest possible
sketch has been given is not so iconoidal as
the so-called Euler’s diagrams; but it is by
far the best general system which has yet
been devised. The present writer has had
it under examination for five years with
continually increasing satisfaction. However,
it is proper to notice some other systems that
are now in use. Two systems which are
merely extensions of Boole’s algebra of logic
may be mentioned. One of these is called by
no more proper designation than the ‘ general
algebra of logic.’ The other is called
‘ Peirce’s algebra of dyadic relatives.’ In the
former there are two operations—aggregation,
which Jevons (to whom its use in algebra is
due) signifies by a sign of division turned on its
side, thus • | • (I prefer to join the two dots,
in order to avoid mistaking the single character
for three) ; and composition, which is best
signified by a somewhat heavy dot, \
Thus, if A and P are propositions, A • | • P is
the proposition which is true if A is true, is true
if P is true, but is such that if A is false and
P is false, it is false. A'B is the proposition
which is true if A is true and P is true, but is
false if A is false and false if P is false.
Considered from an algebraical point of view,
which is the point of view of this system,
these expressions A • | • P and A *P are mean
functions', for a mean function is defined as
such a symmetrical function of several vari¬
ables, that when the variables have the same
value, it takes that same value. It is, there¬
fore, wrrong to consider them as addition and
multiplication, unless it be that truth and
falsity, the two possible states of a proposi¬
tion, are considered as logarithmic infinity and
zero. It is therefore well to let o represent
a false proposition and oo (meaning logarithmic
infinity, so that + oo and — oo are different)
a true proposition. A heavy line, called an
‘ obelus,’ over an expression negatives it.
The letters i,j, k, &c., written below the line
after letters signifying predicates, denote
individuals, or supposed individuals, of udiich
the predicates are true. Thus, Zÿ may mean
that i loves j. To the left of the expression
a series of letters n and 2 are written, each
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