Bauhaus-Universität Weimar

Titel:
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]
Person:
Baldwin, James Mark
PURL:
https://digitalesammlungen.uni-weimar.de/viewer/image/lit29448/673/
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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