P. HISTORY OF ' AATHEMATICAL - School of Mathematics

MATHEMATICAL LOGIC
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used circles in the same manner. In his Formal Logic, De Morgan
uses X) or (X to indicate distributiog, and X( or)X no distribution.
Expressing in his own language (p. 60): "Let the following abbreviations
be employed:
X) Y means 'every X is Y'; X. Y means 'no X is Y'
X:Y means 'some Xs are not Ys'; XY means 'some Xs are Ys'."
He lets (p. 60) A stand for the universal affirmative, I for the particular
affirmative, E for the universal negative, and 0 for the particular
negative. He lets x, y, z, be the negatives or "contrary names"
of the terms X, Y, Z. The four forms AI, E l , 1 1 , 0 1 are used when choice
is made out of X, Y, Z; the forms A', E', I', 0' are used whence choice
is made out of x, y, z. On page 61 he writes identities, one of which is
"AIX) Y =X.y=y)x," that is, every AIX is Y = no X is y = every y is
x. On page 115: "P,Q,R, being certain names, if we wish to give a.
name to everything which is all three, we may join them thus,
PQR: if we wish to give a name to everything which is either of the
three (one or more of them) we may write P,Q,R: if we want to signify
anything that is either both P and Q, or R, we have PQ,R. The contrary
of PQR is p,q,r; that of P,Q,R is pqr; that of PQ,R is (p,q)r."
In his Syllabus,! De Morgan uses some other symbols as follows:
"1. X)o)Y or both X»Y and X).)Y.
All Xs and some things besides are Ys.
2. XIIY orbothX»Y and X«Y.
All Xs are Ys, and all Ys are Xs.....
5. XI.IY or both X).(Y and X(.)Y.
Nothing both X and Y and everything one or the other."
De Morgan also takes L-l as the converse of L.
The following quotation from De Morgan is interesting: "I
end with a word on the new symbols which I have employed. Most
writers on logic strongly object to all symbols, except the venerable
Barbara, Celarent, etc., .... I should advise the reader not to make
up his mind on this point until he has. well weighed two facts which
nobody disputes, both separately and in connexion. First, logic is
the only science which has made no progress since the revival of letters;
secondly, logic is the only science which has produced no growth of
symbols.t"
1 A. de Morgan, Syllabus oj a Proposed Sys~m oj Logic (London, 1860), p.22;
C. I. Lewis, op. cit., p, 41, 42.
• A. de Morgan, op. cit., p, 72; see Monist, Vol. XXV (1915), p. 636.