2' Peirce's symbol is — < which he explains as meaning the same as ^ but being sim- pler to write. J29 Mermrirs of the Amer. Acad., n. s., ix (1867), 317-78. 130 "Description of a Notation for the Logic of Relatives," he. cit., pp. 334-35, 338-39, 342. 131 In this paper, not-x is symbolized by n^, "different from every x," or by a ^. 84 A Survey of Symbolic Logic (18) x + = X. (19) a;+l = 1. (20)

(fi) = 0. (23) If

p{l))a "
The chief points of difference between this modified calculus of prob-
abilities and the original calculus of Boole are as follows:
(1) Where Boole puts p, q, etc. for the "probability of a, of b, etc.",
in passing from the logical to the arithmetical interpretation of his equa-
tions, Peirce simply changes the relations involved from logical relations to
the corresponding arithmetical relations, in accordance with the foregoing,
and lets the terms a, b, etc. stand for the frequency of the a's, b's, etc.
in the system under discussion.
(2) Boole has no symbol for the frequency of the a's amongst the b's,
which Peirce represents by «&. As a result, Boole is led to treat the
probabilities of all unconditioned simple events as independent — a pro-
cedure which involved him in many difficulties and some errors.
(3) Peirce has a complete set of four logical operations, and four
analogous operations of arithmetic. This greatly facilitates the passage
from the purely logical expression of relations of classes or events to the
arithmetical expression of their relative frequencies or probabilities.
Probably there is no one piece of work which would so immediately
reward an investigator in symbolic logic as would the development of this
calculus of probabilities in such shape as to make it simple and practicable.
Except for a monograph by Poretsky and the studies of H. MacColl,"-^"
the subject has lain almost untouched since Peirce wrote the above in 1867.
Peirce's contribution to our subject is the most considerable of any up
to his time, with the doubtful exception of Boole's. His papers, however,
are brief to the point of obscurity: results are given summarily with little
or no explanation and only infrequent demonstrations. As a consequence,
the most valuable of them make tremendously tough reading, and they
have never received one-tenth the attention which their importance de-
serves."^ If Peirce had been given to the pleasantly discursive style of
De ^lorgan, or the detailed and clearly accurate manner of Schroder, his
work on symbolic logic would fill several volumes.
'^" Since the above was written, a paper by Couturat, posthumously published, gives
an unusually clear presentation of the fundamental laws of probability in terms of symbolic
logic. See BM.
I'l Any who find our report of Peirce's work unduly difficult or obscure are earnestly
requested to consult the original papers.
The Development of Symbolic Logic 107
VIII. Developments since Peiece
Contributions to symbolic logic which have been made since the time
of Peirce need be mentioned only briefly. These are all accessible and in a
form sufficiently close to current notation to be readily intelligible. Also,
they have not been superseded, as have most of the papers so far discussed;
consequently they are worth studying quite apart from any relation to
later work. And finally, much of the content and method of the most
important of them is substantially the same with what will be set forth in
later chapters, or is such that its connection with what is there set forth
will be pointed out. But for the sake of continuity and perspective, a
summary account may be given of these recent developments.
We should first mention three important pieces of work contemporary
with Peirce's later treatises.^^^
Robert Grassmann had included in his encyclopedic Wissenschaftslehre
a book entitled Die Begriffslehre oder Logik,^^^ containing (1) Lehre von den
Begriffen, (2) Lehre von den Urtheilen, and (3) Lehre von den Schliissen.
The Begriffslehre is the second book of Die Formenlehre oder Mathematik,
and as this would indicate, the development of logic is entirely mathematical.
An important character of Grassmann's procedure is the derivation of the
laws of classes, or Begriffe, as he insists upon calling them, from the laws
governing individuals. For example, the laws a + a = a and a- a = a,
where a is a class, are derived from the laws e + e = e, e-e = e, ei-e2 = 0,
where e, ei, Ci represent individuals. This method has much to commend
it, but it has one serious defect — the supposition that a class can be treated
as an aggregate of individuals and the laws of such aggregates proved
generally by mathematical induction. As Peirce has observed, this method
breaks down when the number of individuals may be infinite. Another
difference between Grassmann and others is the use throughout of the
language of intension. But the method and the laws are those of extension,
and in the later treatise, there are diagrammatic illustrations in which
"concepts" are represented by areas. Although somewhat incomplete, in
162 Alexander MacFarlane, Principles of the Algebra of Logic, 1879, gives a masterly
presentation of the Boolean algebra. There are some notable extensions of Boole's methods
and one or two emendations, but in general it is the calculus of Boole unchanged. Mac-
Farlane's paper "On a Calculus of Relationship" {Proc. Roy. Soc. Edin., x, 224-32) re-
sembles somewhat, in its method, Peirce's treatment of "elementary relatives". But
the development of it seems never to have been continued.
"3 There are two editions, 1872 and 1890. The later is much expanded, but the plan
and general character is the same.
108 A Survey of Symbolic Logic
other respects Grassmann's calculus is not notably different from others
which follow the Boolean tradition.
Hugh MacCoU's first two papers on "The Calculus of Equivalent
Statements"/^* and his first paper "On Symbolical Reasoning "/^^ printed
in 1878-80, present a calculus of propositions which has essentially the
properties of Peirce's, without n and S operators. In others words, it is
a calculus of propositions, like the Two-Valued Algebra of Logic as we know
it today. And the date of these papers indicates that their content was
arrived at independently of Peirce's studies which deal with this tonic.
In fact, MacColl writes, in 1878, that he has not seen Boole. "^
The calculus set forth in MacColl's book, Symbolic Logic and its
Applications,^'^'' is of an entirely different character. Here the funda-
mental symbols represent prepositional functions rather than propositions;
and instead of the two traditional truth values, "true" and "false", we
have "true", "false", "certain", "impossible" and "variable" (not cer-
tain and not impossible). These are indicated by the exponents r, t, e,
1], 6 respectively. The result is a highly complex system, the fundamental
ideas and procedures of which suggest somewhat the system of Strict
Implication to be set forth in Chapter V.
The calculus of Mrs. Ladd-Franklin, set forth in the paper "On the
Algebra of Logic" in the Johns Hopkins studies,"^^ differs from the other
systems based on Boole by the use of the copula v Where a and b are
classes, a v6 represents "a is-partly b", or "Some a is h", and its negative,
a V &, represents " a is-wholly-not-& ", or " No a is & ". Thus a v 6 is equiva-
lent to a 6 =t= 0, and awb to ab = Q. These two relations can, between
them, express any assertable relation in the algebra, acb will be a v -b,
and a = b IS represented by the pair, (a v-6)(-a v6). For propositions,
a V 6 denotes that a and b are consistent — a does not imply that b is false
and b does not imply that a is false. And a v 6 symbolizes " a and b are
inconsistent" — if a is true, b is false; if b is true, a is false. The use of the
terms "consistent" and "inconsistent" in this connection is possibly mis-
leading: any two true propositions or any two false propositions are con-
I" (1) Proc. London Math. Soc, ix, 9-20; (2) ibid., ix, 177-86.
'«5 Mind, V (1880), 45-60.
'«« Proc. London Math. Soc, ix, 178.
1" Longmans, 1906.
i«* The same volume contains an interesting and somewhat complicated system by
O. H. Mitchell. Peirce acknowledged this paper as having shown us how to express uni-
versal and particular propositions as II and 2 functions. B. I. GUman's study of relative
number, also in that volume, belongs to the number of those papers which are important
in connecting symbolic logic with the theory of probabilities.
The Development of Symbolic Logic 109
sistent in this sense, and any two propositions one of which is true and
the other false are inconsistent. This is not quite the usual meaning of
"consistent" and " inconsistent "—it is related to what is usually meant by
these terms exactly as the "material implication a c6 is related to what
is usually meant by "b can be inferred from a".
That a given class, x, is empty, or a given proposition, x, is false, x = 0,
may be expressed by a; v oo, where «> is "everything" — in most systems
represented by 1. That a class, y, has members, is sj-mbolized by 2/ v oo.
This last is of doubtful interpretation where y is a proposition, since Mrs.
Ladd-Franklin's system does not contain the assumption which is true
for propositions but not for classes, usually expressed, "If a; =t= 0, then
X = 1, and if a: =H 1, then a; = 0". a; v » may be abbreviated to a;v,
ab V CO to ab V , and t/vqo to 2/v,ccZvoo to cc?v, etc., since it is always
understood that if one term of a relation v or v is missing, the missing
term is «> . This convention leads to a very pretty and convenient opera-
tion: V or V may be moved past its terms in either direction. Thus,
(avb) = (abv) = {vab)
and (x'^y) = (xyv) = i^x y)
But the forms (vab) and ( v a; 2/) are never used, being redundant both
logically and psychologically.
Mrs. Ladd-Franklin's system symbolizes the relations of the traditional
logic particularly well :
All a is b. a V -b,
No ais b. avb,
Some ais b. avb,
Some a is not b. ■ a v-b,
Thus V characterizes a universal, v a particular proposition. And any
pair of contradictories will differ from one another simply by the difference
between v and v . The syllogism, " If all o is & and all b is c, then all
a is c, " will be represented by
{a V -b) (b V -c) V (avc)
where v , or v , within the parentheses is interpreted for classes, and v
between the parentheses takes the propositional interpretation. This ex-
pression may also be read, "'All a is 6 and all 6 is c' is inconsistent with
the negative (contradictory) of 'Some a is not c'". It is equivalent to
(a V -b) (b V -c) (a v -c) v
or
a-bv
or
abv
or
abv
or
a-bv
110 A Survey of Symbolic Logic
"The three propositions, 'All a is b', 'All 6 is c, ' and 'Some a is not c',
are inconsistent — they cannot all three be true". This expresses at once
three syllogisms:
(1) (a V -6) (6 V -c) V (a V -c)
"If all a is 6 and all b is c, then all a is c";
(2) (a V -b) (a v -c) v (& v -c)
"If all a is 6 and some a is not c, then some b is not c";
(3) (6v-c)(av-c) v(av-6)
" If all 6 is c and some a is not c, then some a is not b ".
Also, this method gives a perfectly general formula for the syllogism
(a V -b) (b '^c)(avc) v
where the order of the parentheses, and their position relative to the sign v
which stands outside the parentheses, may be altered at will. This single
rule covers all the modes and figures of the syllogism, except the illicit
particular conclusion drawn from universal premises. We shall revert to
this matter in Chapter III.^^^
The copulas v and v have several advantages over their equivalents,
= and =t= 0, or c and its negative: (1) v and v are symmetrical rela-
tions whose terms can always be interchanged; (2) the operation, mentioned
above, of moving v and v with respect to their terms, accomplishes trans-
formations which are less simply performed with other modes of expressing
the copula; (.3) for various reasons, it is psychologically simpler and more
natural to think of logical relations in terms of v and v than in terms
of = and =t= 0. But v and v have one disadvantage as against =, 4=,
and c , — they do not so readily suggest their mathematical analogues in
other algebras. For better or for worse, symbolic logicians have not
generally adopted v and v .
Of the major contributions since Peirce, the first is that of Ernst Schroder.
In his Operationskreis des Logikkalkuls (1877), Schr5der pointed out that
the logical relations expressed in Boole's calculus by subtraction and divi-
sion were all otherwise expressible, as Peirce had already noted. The
meaning of + given by Boole is abandoned in favor of that which it now
has, first introduced by Jevons. And the "law of duality", which con-
nects theorems which involve the relation + , or + and 1, with corresponding
theorems in terms of the logical product x, or x and 0, is emphasized.
i6» See below, pp. 188 ff.
The Development of Symbolic Logic 111
(This parallelism of formulae had been noted by Peirce, in his first paper,
but not emphasized or made use of.)
The resulting system is the algebra of logic as we know it today. This
system is perfected- and elaborated in Vorlesungen iiber die Algebra der
Logik (1890-95). Volume I of this work covers the algebra of classes;
A^olume II the algebra of propositions; and Volume III is devoted to the
calculus of relations.
The algebra of classes, or as we shall call it, the Boole-Schroder algebra,
is the system developed in the next chapter."" We have somewhat elabo-
rated the theory of functions, but in all essential respects, we give the algebra
as it appears in Schroder. There are two differences of some importance
between Schroder's procedure and the one we have adopted. Schroder's
assumptions are in terms of the relation of subsumption, c , instead of the
relations of logical product and =, which appear in our postulates. And,
second, Schroder gives and discusses the various methods of his predecessors,
as well as those characteristically his own.
The calculus of propositions (Aussagenkalkid) is the extension of the
Boole-Schroder algebra to propositions by a method which differs little
from that adopted in Chapter IV, Section I, of this book.
The discussion of relations is based upon the work of Peirce. But
Peirce 's methods are much more precisely formulated by Schroder, and
the scope of the calculus is much extended. We summarize the funda-
mental propositions which Schroder gives for the sake of comparison both
with Peirce and with the procedure we shall adopt in Sections II and III
of Chapter IV.
1) A, B, C, D, E . . . symboHze "elements" or individuals."' These
are distinct from one another and from 0.
2) V = A+B + C + D+ ..
P symboHzes the universe of individuals or the universe of discourse of
the first order.
3) i, j, k, I, m, n, p, q represent any one of the elements A, B, C, D, ...
of P.
4) P = Si i
"o For an excellent summary by Schroder, see Abriss der Algebra der Logik ; ed. Dr.
Eugen Muller, 1909-10. Parts i and ii, covering Vols, i and ii of Schroder's Vorlesungen,
have so far appeared.
1" The propositions here noted will be found in Vorlesungen iiber die Algebra der Logik,
III 3-42. Many others, and much discussion of theory, have been omitted.
112 A Survey of Symbolic Logic
5) i : j represents any two elements, i and j, of P in a determined order.
6) {i = j) = (i -.j = j : i), (^ 4= j) = (i : i 4= J : i)
for every i and j.
7) i : i +
Pairs of elements of 1^ may be arranged in a "block":
A:A,A:B,A:C,A:D, ...
B : A, B : B, B : C, B : D, ...
8)
C -.A, C -.B, C -.C, C -.D, ...
D : A, D : B, D : C, D : D, ...
These are the "individual binary relatives".
P = {A ■.A) + iA ■.B) + {A :€) + ...
+ {B •.A) + {B ■B) + {B -.0) + ...
9)
+ {C ■.A) + {C ■.B) + {C -.0) + ...
+ .
P represents the universe of binary relatives.
10) P = S,S.- (i : i) = SiS, [i : j) = S^,- (i : j)
9) and 10) may be summarized in a simpler notation:
1 = Si,- i : j = yl : ^ + ^ : 5 + ^ : C +
+ B:A + B -.B + B : C + .
11)
+ C:^ + C:5 + C: C+ .
12) i : j : h will symbolize an "individual ternary relative".
13) P = 2aS,S,- (i : i : /i) = S^-, i : j : /»
Various types of ternary relatives are
14) A: A: A, B : A : A, A : B : A, A : A : B, A : B : C
It is obvious that we may similarly define individual relatives of the
fourth, fifth, ... or any thinkable order.
The Development of Symbolic Logic 113
The general form of a binary relative, a, is
a = Si,- an {i : j)
where a,,- is a coefficient whose value is 1 for those {i : j) pairs in which i has
the relation a to j, and is otherwise 0.
1 = Si,- i : j
= the null class of individual binary relatives.
/= Siy(z =j){i:j) = Si(t:z)i'2
N= ^iiii=¥j){i:j)
{ah)ij = aijbii {a + h)ij = an + bij
-an = {-a)ij = -(a,,-)
(a\b)ij = Sa flift b/,j (a t b)ii = lih (am + b^,)
The general laws which govern propositional functions, or Atissngen-
schemata, such as {_ab)i,; Zhttihbhi, Uh (aih + bhj), Uattn, l^a an, etc., are as
follows :
Au symbolizes any statement about u; UuAu will have the value 1 in
case, and only in case, Au = 1 for every u; XuAu will have the value 1 if
there is at least one u such that Au = 1. That is to say, UuAu means
" Au for every u", and S,jy4„ means " Au for some u".
a) UuAu cA^c ZuAu, -[liuAu] c -A^ c -[UuAu]
(The subscript u, in a and 13, represents any value of the variable u.)
y) -[UuAu] = S„ -Au, -[S„4„] = n„ -Au
8) If Au is independent of u, then UuAu = A, and S„^„ = A.
e) UuiA cBu) = (AcUuBu), UuiAucB) = (S„^„ cB)
f) n„,„ or UuU,{AucB;) = CSuAuCU^B^)
r,) l^uiAucB) = (UuAucB), 2uiAcBu) = U c S,B„)
e) S„,„ or S„S„(^„c5„) = (UuAuCXyB,)
f n„(^„ = 1) = (n,^„ = 1), Uu{Au = 0) = (s„^„ = 0)
1 -EuiAu = 0) = (UuAu = 0), S„(^„ = 1) = i^uAu = 1)
"2 We write I -where Schroder has 1'; N where he has 0'; {a\ b) for (a; 6); (a t &)
for (a ^ 6); -a for a; ^a for a.
*(a;i, X2, . . Xk, Xk+i) be any function of k + 1 variables.
By 6-11, for some/ and some/',
$(a;i, X2, .i-k, Xk+i) = f{xi, X2, . . Xk)-Xk+i
+ /'(a-i, X2, ... Xk)--Xk+i (1)
Since this last expression may be regarded as a function of Xk+i in
which the coefficients are the functions / and / ', [6-3]
fixu X2, ... Xk) x/ '{xi, X2, . . . Xk) c $(.ri, X2, . . Xk, Xk+i)
The Classic, or Boole-Schroder, Algebra of Logic 141
Let Ai{$}, A^l^}, A4$}, etc., be here the coefficients in $; Ai{f},
Mf}, Mf\, etc., the coefficients in/; and Ai{f'}, A,{f'], As{f'},
etc., the coefficients in / '.
If IlA{f} c/ and n^{/'} c/', then [6-3]
n^{/!xn^{/')c/x/'
and, by (1), IJ^l/} > \ Xl Xo . . . Xn
Ak ] f*Bki
I -Xj -l-o . . -X„ J I -Xi -X-i . . . -Xn
, , { Xi X2 . . . Xn
= (Ak + Bk) -\
L ~X\ -X2 -Xn
And since addition is associative and commutative, the sum of the
two functions is equivalent to the sum of the sums of such corre-
sponding terms, pair by pair.
6-43 The product of two functions of the same variables, ^{xi, x^, ... Xn)
and ■^(a-'i, X2, ... x„), is another function of these same variables,
F{Xi, Xi, . . Xn),
such that the coefficient of any term in F is the product of the coeffi-
cients of the corresponding terms in $ and ^.
Let Ajci ' ' ' " r and Bkl ' ' " r be any two
L -Xi -X2 . . . -Xn J L -Xi -X2 . . -Xn J
corresponding terms in $ and ^.
Xl X2 . . . Xn\ „ \ Xi X2 . . . Xn
f ^Bk]
-Xl -Xi . . . -Xn J L -Xl -X2 . . -Xn
= {Ak X Bk)
Xn
I -Xl -;
X2 ■ —Xn
By 6-15, $ and ^ do not differ except in the coefficients, and by
6-17, whatever the coefficients in the normal form of a function, the
product of any two terms is null. Hence all the cro^s-products of
terms in $ and ^ will be null, and the product of the functions will
144 A Survey of Symbolic Logic
be equivalent to the sum of the products of their corresponding terms,
pair by pair.
Since in this algebra two functions in which the variables are not the
same may be so expanded as to become functions of the same variables,
these theorems concerning functions of functions are very useful.
IV. Fundamental Laws of the Theory of Equations
We have now to consider the methods by which any given element
may be eliminated from an equation, and the methods by which the value
of an " unknown" may be derived from a given equation or equations. The
most convenient form of equation for eliminations and solutions is the
equation with one member 0.
Equivalent Equations of Different Forms. — If an equation be not in the
form in which one member is 0, it may be given that form by multiplying
each side into the negative of the other and adding these two products.
7-1 a = b is equivalent to a -b + -a b = 0.
[2-2] a = b is equivalent to the pair, acb and b ca.
[4-9] a cb is equivalent to a-b = 0, and b ca to -ab =0.
And [5 • 72] a -6 = and -a 6 = are together equivalent to a -b
+ -ab = 0.
The transformation of an equation with one member 1 is obvious:
7-12 a = 1 is equivalent to -a = 0.
[.3-2]
By 6-41, any equation of the form f(xi, .T2, . . . x„) = 1 is reduced to the
form in which one member is simply by replacing each of the coefficients
in / by its negative.
Of especial interest is the transformation of equations in which both
members are functions of the same variables.
7-13 If $(a-i, a-2, . . . x„) and SE'Ccci, x^, . . . x„) be any two functions of the
same variables, then
$(a-i, X2, . . . Xn) = ^(.x'l, X2, . . x„)
is equivalent to F{xi, X2, . . . x„) =0, where f is a function such that
if Ai, Ai, Ai, etc., be the coefficients in $, and Bi, B^, B3, etc., be the coef-
ficients of the corresponding terms in S^, then the coefficients of the corre-
sponding terms in F will be (^1 -Bi + -Ai Bi), {A2 -B2 + -A2 B2}, (A^ -B3
+ -.4.353), etc.
The Classic, or Boole-Schroder, Algebra of Logic 145
By 7- 1, $ = ^ is equivalent to ($ x-'i') + (-$ x ^) =0.
By 6-41, -$ and -^ are functions of the same variables as $ and SI'.
Hence, by 6-43, $ x-^ and -$ x ^ will each be functions of these
same variables, and by 6-42, ($ x-^) + (-$ x ^) will also be a
function of these same variables.
Hence $, ^, -$, -^, $ x-^, -$ x ^, and (* x-^) + (-$ x ^) are all
functions of the same variables and, by 6-15, will not differ except
in the coefficients of the terms.
If Ak be any coefficient in $, and Bk the corresponding coefficient
in SI>, then by 6-41, the corresponding coefficient in -$ will be -Ak
and the corresponding coefficient in -^ will be -5/,.
Hence, by 6-4,3, the corresponding coefficient in $x-^ will be
Ai -Bk, and the corresponding coefficient in -$ x N[' will be -AkBk-
Hence, by 6-42, the corresponding coefficient in ($ x-^) + (-$ x^)
will be Ak -Bk + -AkBk.
Thus ($ x--*^) + (-$ X -^if) is the function F, as described above, and
the theorem holds.
By 7-1, for every equation in the algebra there is an equivalent equation
in the form in which one member is 0, and by 7 • 13 the reduction can usually
be made by inspection.
One of the most important additions to the general methods of the
algebra which has become current since the publication of Schroder's work
is Poretsky's Law of Forms. ^ By this law, given any equation, an equiva-
lent equation of which one member may be chosen at will can be derived.
7-15 a = is equivalent to t = a-t + -at.
If a = 0, a-t + -at = 0--t+l-t = t.
And U t = a~t+-at, then [7 • 1]
{a-t + -at) -t+ {at + -a-t) t = = a-t + at = a
Since t may here be any function in the algebra, this proves that every
equation has an unlimited number of equivalents. The more general form
of the law is :
7-16 a = & is equivalent to if = {ab + -a-b) t+ (a -b + -a b) -t.
[7-1] a = b is equivalent to a-b + -ab = 0.
And [6 • 4] -{a -h + -ab) = ab + -a -b.
Hence [7-15] Q.E.D.
The number of equations equivalent to a given equation and expressible
' See Se-pt lois fandamentales de la theorie des egaliles logiques, Chap. i.
11
146 A Surrey of Symbolic Logic
in terms of n elements will be half the number of distinct fmictions which
can be formed from n elements and their negatives, that is, 2"72-
The sixteen distinct functions expressible in terms of two elements,
a and b, are:
a, -a, b, -b, (i. e., a -a, h-b, etc.), 1 (i- e., a + -a, b + -b, etc.), ab,
a -b, -a b, -a -b, a + b, a + -b, -a + h, -a + -b, ab + -a -b, and a-b + -a b.
In terms of these, the eight equivalent forms of the equation a = b are:
a = b; -a = -b; = a-b + -ab; I = ab + -a-b; ab = a + b; a-b
= -ab; -a -b = -a + -b; and a + -b = -a + b.
Each of the sixteen functions here appears on one or the other side of an
equation, and none appears twice.
For any equation, there is such a set of equivalents in terms of the
elements which appear in the given equation. And every such set has
what may be called its "zero member" (in the above, = a-b + -ab)
and its "whole member" (in the above, 1 = ab + -a-b). If we observe
the form of 7-16, we shall note that the functions in the "zero member"
and "whole member" are the functions in terms of which the arbitrarily
chosen t is determined. Any t = the t which contains the function ( = 0}
and is contained in the function { = 1 } . The validity of the law depends
simply upon the fact that, for any t, ct cl, i. e., t = 1-t + 0--i. It is
rather surprising that a principle so simple can yield a law so powerful.
Solution of Equations in One Unknown. — Every equation which is pos-
sible according to the laws of the system has a solution for each of the un-
knowns involved. This is a peculiarity of the algebra. We turn first to
equations in one unknown. Every equation in x, if it be possible in the
algebra, has a solution in terms of the relation c .
7-2 A X + B -X = is equivalent to B ex c-A.
[5 -72] A X + B -X = is equivalent to the pair, A x = and
B -X = 0.
[4 • 9] 5 -a; = is equivalent to S c x.
And A X = Q is equivalent to x -(-.4) = 0, hence to x c -A.
7-21 A solution in the form H ex cK is indeterminate whenever the equa-
tion which gives the solution is symmetrical with respect to x and -x.
First, if the equation be of the form A x + A -x = 0.
The solution then is, A ex c-A.
But if A X + A -x = 0, then A = A {x + -x) = A x + A -x = 0, and
-A = 1.
The Classic, or Boole-Schroder, Algebra of Logic 147
Hence the solution is equivalent to Ocxcl, which [5 -61 -63] is
satisfied by every value of x.
In general, any equation symmetrical with respect to x and -x
which gives the solution, H cxcK, will give also H c-xc K.
But U Hex and H c-x, then [i-Q] H x = H and H-x = H.
Hence [1-62] H = 0.
And if a;c/v and -xcK, then [5-33] x + -xcK, and [4-8, 5-63]
K = 1.
Hence H ex c K will be equivalent to c a; c 1.
It follows directly from 7-21 that if neither x nor -x appear in an equa-
tion, then although they may be introduced by expansion of the functions
involved, the equation remains indeterminate with respect to x.
7 ■ 22 An equation of the form Ax + B -x =0 determines x uniquely when-
ever A = -B, B = -A.
[3-22] A = -B and -A = B are equivalent; hence either of
these conditions is equivalent to both.
[7-21 A X + B -X = is equivalent to B ex c -A.
Hence if 5 = -.1, it is equivalent to B ex cB and to -A ex e-A,
and hence [2-2] to a; = B = -A.
In general, an equation of the form A x + B -x = determines x be-
tween the limits B and -A. Obviously, the solution is unique if, and only
if, these limits coincide; and the solution is wholly indeterminate only
when they are respectively and 1, the limiting values of variables generally.
7-221 The condition that an equation of the form A x + B -x = he pos-
sible in the algebra, and hence that its solution be possible, is A B = 0.
By 6-3, ABeAx + B-x. Hence [5-65] ii A x + B -x = 0, then
A B = 0.
Hence if .4 5 =1= 0, then A x + B -x = must be false for all values
of x.
And Ax + B -x = and the solution B ex c-A are equivalent.
A B = is called the "equation of condition" ol A x + B -x = 0: it is
a necessary, not a sufficient condition. To call it the condition that A x
+ B -X = have a solution seems inappropriate : the solution B ex e -A
is equivalent to Ax + B -x = 0, whether ^ a; + 5 -a; = be true, false, or
impossible. The sense in which ^45=0 conditions other forms of the
solution of ^4 a; + jB -a; = will be made clear in what follows.
The equation of condition is frequently useful in simplifying the solution.
148 A Survey of Symbolic Logic
(In this connection, it should be borne in mind that A B = follows from
Ax + B -X = 0.) For example, if
ab X + {a + b) -X =
then (a + b) ex c -{a b). But the equation of condition is
ab (a + b) = ab = 0, or, -(a b) = 1
Hence the second half of the solution is indeterminate, and the complete
solution may be written
a + b ex
However, this simplified form of the solution is equivalent to the original
equation only on the assumption that the equation of condition is satisfied
and ab = 0.
Again suppose a x + b -x + c =
Expanding c with reference to x, and collecting coefiicients, we have
(a + c) x + {b + c) -X —
and the equation of condition is
(a + c){b + c) = ab + ac + bc + c = ab + c =
The solution is b + c cx c-a-c
But, by 5-72, the equation of condition gives c = 0, and hence -c = 1.
Hence the complete solution may be written
b cx c-a
But here again, the solution b cx c -a is equivalent to the original equation
only on the assumption, contained in the equation of condition, that c = 0.
This example may also serve to illustrate the fact that in any equation
one member of which is 0, any terms which do not involve x or -x may be
dropped without afFecting the solution for x. If a a; + & -t + c = 0, then
by 5-72, a a; + 6 -.r = 0, and any addition to the solution by retaining c will
be indeterminate. All terms which involve neither the unlmown nor its
negative belong to the "symmetrical constituent" of the equation — to be
explained shortly.
Poretsky's Law of Forms gives immediately a determination of x which
is equivalent to the given equation, whether that equation involve x or not.
7-23 A x + B -X = Q'ls equivalent to x = -A x-*- B -x.
[7-15] Ax + B -X = is equivalent to
x = (A x + B -x) -X + {-A X + -B -,!■) X = B -X + -A X
The Classic, or Boole-Schroder, Algebra of Logic 149
This form of solution is also the one given by the method of Jevons."
Although it is mathematically objectionable that the expression which
gives the value of x should involve x and -x, this is in reality a useful and
logically simple form of the solution. It follows from 7-2 and 7-23 that
X = -A X + B -X is equivalent to 5 c a; c -A.
IMany writers on the subject have preferred the form of solution in
which the value of the unknown is given in terms of the coefficients and an
undetermined (arbitrary) parameter. This is the most "mathematical"
form.
7-24 If A B = 0, as the equation A x + B -x = requires, then A x
+ 5 -,r = is satisfied by x = B -u + -A u, ov x = B + u -A, where u is
arbitrary. And this solution is complete because, for any x such that
A x + B -X = there is some value of u such that x = B -u + -A u = B
+ u -A .
(a) By Q-4:, U X = B -11 + -A u, then -x = -B -u + A u.
Hence if a; = B -u + -A ii, then
Ax + B-x = A iB-u + -Bu) + B i-B-u + Au)
= AB-u + ABu = AB
Hence if ^-15=0 and x = B -u + -A u, then whatever the value
of u, Ax + B-x = 0.
(6) Suppose x known and such that Ax + B-x = 0.
Then if x = B -u + -A u, we have, by 7-1,
{B -u + -A u) -X + {-B -u + A u) X
= {Ax + -A -x) u+ {B -X + -B x) -u =
The condition that this equation hold for some value of tt is, by 7 • 221,
{A X + -A -x){B -x + -Bx) = A-Bx + -A B -x =
This condition is satisfied if Ax + B -x = 0, for then
A{B + -B) X + (A+-A) B-x = AB + A-BX + -A B -x =
and by 5-72, A -B x + -A B -x = 0.
(c) If ^ 5 = 0, then B-zi + -Au = B + u -A, for:
If .4 5 = 0, then A B u = 0.
Hence B-u + -Au = B-u + -A {B + -B) u + .4 B u
= B-u+{A+-A) Bu + -A-Bu = B {-u + u) + -A -B u
= B + -A-BU.
But [5-85] B + -A-BU = B+u -A.
1" See above, p. 77.
150 ,1 Survey of Symbolic Logic
Only the simpler form of this solution, x = B + u -A, will be used hereafter.
The above solution can also be verified by substituting the value given
for .V in the original equation. We then have
A {B -u + -A u) + B i-B -u + Au) = AB-u + ABu = A B
And U A B = 0, the solution is verified for every value of u.
That the solution, .t = B-i) + -A u — B-^-u-A, means the same as
B *

p{x, y) is equivalent to "Whatever value of (x, y), in (p{x, y), {x, y)n may be, ^{x, y)n". [10-23] 11-24 Ii.Jlyip{x, y) is equivalent to "Whatever values of x and y, in ,p{x, y), X, and y, may be, ,p{xry,)". [10-23] HJlytpix, y) is equivalent to "Whatever value of x, in Ilyip{x,y), Xr may be, 'ilyip{Xry)" . And Ilyipi^Xry) is equivalent to "Whatever value of y, in

{x,y)n" is equivalent to "AYhatever values of x and y, in

< { fi^iVs) + 'pixiys) + ipixsys) + . . . }
X . . . Etc., etc.
Since x is distributive with reference to + , this expression is equal
to the sum of the products of each column separately, plus the sum
of all the cross-products, that is, to
A + { p{x, y, z) = SxS(j,, z)v{x, y, z)
Since UJliy, z) < ni/-?/) x . .
[5-98]
= Uy{ (iySz)]} = Uy -[(x Ry)x{yS z)]
= Uy[-{x Ry) + -iy S z)]
= Uy[{x -R y) + {y -S z)]
276 A Survey of Symbolic Logic
The negative of "friend of a colleague of" is "non-friend of all colleagues
(non-non-colleagues) of ".
Similarly, -(ii t S) = -R\-S
The negative of "friend of all non-colleagues of" is "non-friend of a non-
colleague of ".
Converses of relative sums and products are as follows ;
^{R\S) = v.S|"i?
for X ^{R \S)z = z{R\ S)x = 2j,[(2 Ry)x(yS x)]
= Zy[{y Sx)x{zR y)]
= Sj(a- -S y) X {y ^R z)]
= .t(-S I ^R)z
If X is employer of a benefactor of z, then the relation of 2 to x is "bene-
fitted by an employee of".
Similarly, -(fl t S) = "S t -i?
If X is hater of all non-helpers of z, the relation of 2 to a; is "helped by all
who are not hated by".
The relation of relative product is distributive with reference to non-
relative addition.
R\{S+ T) = {RS) + {R T)
for x[R I {S + T)]z = 2,{ (.T Ry)x[y{S^ T)z] ]
= i:y{{xRy)x[{ySz)^{yTz)]\
= ^y{[{x Ry)xiyS z)] + [(.r Ry)x{yT z)]]
= [x{R\S)z\ + [x{R\T)z\
Similarly, {R-i- S)\T = {R\T) + (S\ T)
"Either friend or colleague of a teacher of" is the same as "either friend
of a teacher of or colleague of a teacher of".
A somewhat curious formula is the following:
R\{SxT) B.
3 We write A ^ B where the text has A non oo B.
373
374 A Survey of Symbolic Logic
coincide, and that one is called the container. And conversely, if any term be contained in
another, then it will be one of a plurality which taken together coincide with that other.
For example, if A and B taken together coincide with L, then A, or B, will be called the
inexislent (inexistens) or the conlainbd; and L will be called the container. However, it
can happen that the container ai;)d the contained coincide, as for example, if {A and B) = L,
apd A and L coincide, for in that case B will contain nothing which is different from A. . . .*
Scholium. Not every inexistent thing is a part, nor is every container a whole — e. g.,
an inscribed square and a diameter are both in a circle, and the square, to be sure, is a certain
part of the circle, but the diameter is not a part of it. We must, then, add something for
the accurate explanation of the concept of whole and part, but this is not the place for it.
And not only can those things which are not parts be contained in, but also they can be
subtracted (or "abstracted", detrahi); e. g., the center can be subtracted from a circle
so that all points except the center shall be in the remainder; for this remainder is the locus
'Of all points within the circle whose distance from the circumference is less than the radius,
and the difference of this locus from the circle is a point, namely the center. Similarly the
locus of all points which are moved, in a sphere in which two distinct points on a diameter
remain unmoved, is as if you should subtract from the sphere the axis or diameter passing
through the two unmoved points.
On the same supposition [that A and B together coincide with L], A and B taken
together are called constituents (constitucntia) , and L is called thai which is constituted
{constituium) .
Charad. 3. A + B = L signifies that A is in or is contained in L.
Scholium. Although A and B maj' have something in common, so that the two taken
together are greater than L itself, nevertheless what we have here stated, or now state, will
:still hold. It will be well to make this clear by an example: Let L denote the straight
line RX, and A denote a part of it, say the line RS, and B denote
.another part, say the line XY. Let either of these parts, RS or R Y S X
XY, be greater than half the whole line, RX; then certainly it
cannot be said that A + B equals L, or RS -\- XY equals RX. For inasmuch as YS is a
common part of RS and XY, RS + XY will be equal to RX + SY. And yet it can truly
be said that the lines RS and XY together coincide with the line RS.^
P J£ N
/§l ^\'Y ^\''S ''\ ^7- ^'^f- 4- If some term M" is in A and also in B, it
» \ ? J is said to be common to them, and they are said to be
Y^., ■'^ ^^ ^y.' cominunicating {communicantia) .^ But if they have
\""~-.^ ^'■''^'~~~-s^'''' / nothing in common, as A and 'N (the lines RS and
^^^^ \ ^ ^^'' ^S, for example), they are said to be non-communi-
"^-^ y'" eating {incommimicantia) .
L
Def. 5. If A is in L in such wise that there is another term, N, in which belongs
everything in L except what is in A, and of this last nothing belongs in N, then A is said
to be subtracted {detrahi) or taken away {removeri), and A^ is called the remainder {residuum).
Charact. 4. L — A = N signifies that L is the container from which if A be sub-
tracted the remainder is A''.
Def. 6. If some one term is supposed to coincide with a plurality of terms which
are added (positis) or subtracted (remotis), then the plurality of terms are called the con-
stituents, and the one term is called the thing constituted.'
^ Lacuna in the text, followed by "significet A, significabit Nihil".
* Italics ours.
^ The text here has "communicatia", clearly a misprint.
' Leibniz's idea seems to be that ii A + N = L then L is "constituted" by A and A'',
and also liL — A = A then L and A " constitute" N. But it may mean that HL — A = N,
then A and N " constitute " L.
i\
Two Fragments from Leibniz 375
Scholium. Thus all terms which are in anything are constituents, but the reverse
does not hold; for exaniple, L — A = N,m which case L is not in A.
Def. 7. Constitution (that is, addition or subtraction) is either tacit or expressed, —
N or — M the tacit constitution of M itself, as A or —Am which A'' is. The expressed
constitution of A?" is obvious.'
Def. 8. Compensation is the operation of adding and subtracting the same thing in
the same expression, both the addition and the subtraction being expressed [as A -i. M
— M]. Destruction is the operation of dropping something on account of compensation,
so that it is no longer expressed, and for M — M putting Nothing.
Axiom 1. If a term be added to itself, nothing new is constituted or A + A = A.
Scholium. With numbers, to be sure, 2 + 2 makes 4, or two coins added to two
coins make four coins, but in that case the two added are not identical with the former two;
if they were, nothing new would arise, and it would be as if we should attempt in jest to
make six eggs out of three by first counting 3 eggs, then taking away one and counting
the remaining 2, and then taking away one more and counting the remaining 1.
Axiom 2. If the same thing be added and subtracted, then however it enter into the
constitution of another term, the result coincides with Nothing. Or A (however many
times it is added in constituting any expression) — A (however many times it is subtracted
from that same expression) = Nothing.
Scholium. Hence A — A or (A + A —) — A or A {A + A), ete. = Nothing. For
by axiom 1, the expression in each case reduces to A —A.
Postulate 1. Any plurality of terms whatever can be added to constitute a single
term; as for example, if we have A and B, we can write A + B, and call this L.
Post. 2. Any term. A, can be subtracted from that in which it is, namely A + B
or L, if the remainder be given as B, which added to A constitutes the container L — that
is, on this supposition [that A -\- B = L] the remainder L — A can be found.
Scholium. In accordance with this postulate, we shall give, later on, a method for
finding the difference between two terms, one of which. A, is contained in the other, L,
even though the remainder, which together with A constitutes L, should not be given — •
that is, a method for finding L — A, or A + B — A, although A and L only are given,
and B is not.
Theorem 1
Terms which are the same loilh a third, are the same with each other.
1{ A = B and B = C, then A = C. For if in the proposition A = B (true by hyp.)
C be substituted for B (which can be done by def. 1, since, by hyp., B = C), the result
"is A =C. Q.E.D.
Theorem 2
// one of two terms which are the same be different from a third term, then the other of the
two will be different from it also.
li A = B and B ^ C, then A =^ C. For if in the proposition B =^ C (true by hyp.)
A be substituted for B (which can be done by def. 1, since, by hyp., A = B), the result is
A + C. Q.E.D.
[Theorem in the margin of the manuscript.]
Here might be inserted the following theorem: Whatever is in one of two coincident
terms, is in the other also.
If A. is in i? and B = C, then also A is in C. For in the proposition A is in fi (true
by hyp.) let C be substituted for B.
Theorem 3
// terms which coincide be added to the same term, the results will coincide.
li A = B, then A + C = B + C. For if in the proposition A +C = A +C (true
* This translation is literal : the meaning is obscure, but see the diagram above.
376 ,-1 Survey of Symbolic Logic
per se) you substitute B for A in one place (which can be done by def. 1, since A = B),
it gives A +C = B + C. Q.E.D.
CoHOLLARY. If terms which coincide be added to terms which coincide, the results will
coincide. HA = B and L = Af , then A + L = B + M. For (by the present theorem)
since L = M, A + L = A + M, a,nd in this assertion putting B for A in one place (since
by hyp. A = B) gives A + L = B + M. Q.E.D.
Theorem 4
A container of the container is a container of the contained; or if that in which some-
thing is, be itself in a thu-d thing, then that which is in it will be in that same third thing—
that is, if yl is in B and B is in C, then also A is in C.
For A is in S (by hyp.), hence (by def. 3 or charact. 3) there is some term, which we
may call L, such that A + L = B. Similarly, since B is in C (by hyp.), B + M = C,
and in this assertion putting A + L ior B (since we show that these coincide) we have
A + L + M = C. But putting N ior L + M (by post. 1) we have A + N = C. Hence
(by def. 3) A is in C. Q.E.D.
Theorem 5
Whatever contains terms individually contains also that which is constituted of them.
If A is in C and B is in C, then A + B (constituted of A and B, def. 4) is in C. For
since A is in C, there will be some term M such that A + M = C (hy def. 3). Similarly,
since B ism C, B + N = C. Putting these together (by the corollary to th. 3), we have
A+M+B+N = C + C. But C + C = C (by ax. 1), hence A+M + B + N = C.
And therefore (by def. 3) A + B is in C. Q.E.D."
Theorem 6
Whatever is constituted of terms which are contained, is in that which is constituted of the
containers.
If A is in M and B is in N, then A + B is in M + A^. For A is in M (by hyp.) and M
isin M + N (by def. 3), hence A is in M + N (by th. 4). Similarly, S is in Af (by hyp.)
and N iain M + N (by def. 3), hence B is in ilf + A^ (by th. 4). But ii A is in M + N
and B is in M + N, then also (by th. 5) A + B isin M + N. Q.E.D.
Theorem 7
If any term be added to that in which it is, then nothing new is constituted; or if B is in A,
then A + B = A.
For if B is in A, then [for some C] B + C = A (def. 3). Hence (by th. 3) A + B
= B + C + B = B + C (by ax. 1) = A (by the above). Q.E.D.
Converse of the Preceding Theorem
If by the addition of any term to another nothing new is constituted, then the term added
is in the other.
If A + B = A, then B is in A; for B is in A + B (def. 3), and A + B = A (by hyp.).
Hence B is in A (by the principle which is inserted between ths. 2 and 3). Q.E.D.
Theorem 8
// terms which coincide be subtracted from terms which coincide, the remainders will
coincide.
If A = L and B = M, then A - B = L - M. For A-B=A-B (true per se),
' In the margin of the manuscript at this point Leibniz has an untranslatable note,
the sense of which is to remind him that he must insert illustrations of these propositions in
common language.
Ttvo Fragments from Leibniz 377
and the substitution, on one or the other side, of L for A and M for B, gives A - B = L
- M. Q.E.D.
[Note in the margin of the manuscript.] In dealing with concepts, suhiradion {de-
tmctio) is one thing, negation another. For example, "non-rational man" is absurd or
impossible. But we may say; An ape is a man except that it is not rational. [They
are] men except in those respects in which man differs from the beasts, as in the case of
Grotius's Jumbo" {Homines nisi qua bestiis differt homo, ut in Janibo Grotii). "Man"
- "rational" is something different from "non-rational man". For "man" - "rational"
= "brute". But "non-rational man" is impossible. "Man" - "animal" - "rational"
IS Nothing. Thus subtractions can give Nothing or simple non-existence— even less than
nothing — but negations can give the impossible."
Theorem 9
(1) From an expressed compensation, the destruction of the term compensated follows,
provided nothing be destroyed in the compensation which, being tacitly repeated, enters
into a constitution outside the compensation [that is, + A^ — A'' appearing in an expression
may be dropped, unless A^ be tacitly involved in some other term of the expression];
(2) The same holds true if whatever is thus repeated occur both in what is added and
in what is subtracted outside the compensation;
(3) If neither of these two obtain, then the substitution of destruction for compensa-
tion [that is, the dropping of the expression of the form -\- N ~ N] is impossible.
Case 1. liA+N-M-N=A-M, and A, N, and M be non-communicating.
For here there is nothing in the compensation to be destroyed, -|- A'' — A'', which is also
outside it in A or M — that is, whatever is added in -f- A'", however many times it is added,
is in -f A'', and whatever is subtracted in — A'^, however many times it is subtracted, is in
- N. Therefore (by ax. 2) for 4- A^ — A'' we can put Nothing.
Case 2. liA+B— B — G = F, and whatever is common both to A + B [i. e.,
to A and B] and to G and B, is M, then F = A — G. In the first place, let us suppose
that whatever A and G have in common, if they have anything in common, is E, so that
if they have nothing in common, then E = Nothing. Thus [to exhibit the hypothesis
of the case more fully] A=E + Q+M, B=N + M, and G=E + H + M, so that
F = E + Q + M+N-N-M-H-M, where all the terms E, Q, M, N, and H are
non-communicating. Hence (by the preceding case) F = Q~H = E + Q+M — E
- H ~ M = A - G.
Case 3. \i A + B — B — D = C, and that which is common to A and B does not
coincide with that which is common to B -\- D [i. e., to B and Z)], then we shall not have
C = A - D. Yor let B = E + F + G, laiA A = H -{- E, and D = K -[- F, so that these
constituents are no longer communicating and there is no need for further resolution.
Then C = H+E + F + G-E-F~G~K-F, that is (by case 1) C = H - K,
which is not = A — D (since A— D = H + E — K — F), unless we suppose, contrary
to hypothesis, that E = F — that is that B and A have something in common which is also
common to B and D. This same demonstration would hold even if A and D had
something in common.
" Apparently an allusion to some description of an ape by Grotius.
" This is not an unnecessary and hair-splitting distinction, but on the contrary, per-
haps the best evidence of Leibniz's accurate comprehension of the logical calculus which
appears in the manuscripts. It has been generally misjudged by the commentators, because
the commentators have not understood the logic of intension. The distinction of the
merely non-existent and the impossible (self-contradictory or absurd) is absolutely essential
to any calculus of relations in intension. And this distinction of subtraction (or in the more
usual notation, division) from negation, is equally necessary. It is by the confusion of
these two that the calculuses of Lambert and Castillon break down.
378 A Survey of Synibolic Logic
Theorem 10
A sitbiracled term and the remainder are iion-communicating .
liL - A = N,l affirm that A and A^ have nothing in common. For by the definition
of "subtraction" and of "remainder", everything in L remains in A'' except that which is
in A, and of this last nothing remains in N.
Theorem 11
Of that which is in two communicating terms, whatever part is common to both and the
two exclusive parts are three non-communicating terms.
If A and B be communicating terms, and A = P + M and B = N ■{- M, so that
whatever is in A and B both is in M, and nothing of that is in P or N, then P, M, and A''
are non-communicating. For P, as well as A'', is non-communicating with M, since what-
ever is in M is in A and B both, and nothing of this description is in P or A''. Then P and
A'" are non-communicating, otherwise what is common to them would penetrate into A
and B both.
Problem
To add non-coincident terms to given coincident terms so that the resulting terms shall
coincide.
li A = A, 1 affirm that it is possible to find two terms, B and A'", such that B Jp N
and yet A + B = A + A^.
Solution. Choose some term M which shall be contained in A and such that, N
being chosen arbitrarily, M is not contained in A^ nor A'' in M, and let i? = Af + A''. And
this will satisfy the requirements. Because B = M -\- N (by hyp.) and M and A^ are
neither of them contained in the other (by hyp.), and yet A -{- B = A -\- N, since A + B
= A -\- M + N and (by th. 7, since, by hyp., M is in A) this is = A + A^.
Theorem 12
Where non-communicating terms only are involved, whatever terms added to coincident
terms give coincident terms will he themselves coincide?it.
That is, a A -\- B = C -\- D a,nd A = C, then B = D, provided that A and B, as
well as C and D, are non-communicating. For A-^B — C = C-]-D — C (by th. 8) ;
but A+B-C = A+B~A (by hyp. that A = C), and A + fi - A = B (by th. 9,
case 1, since A and B are non-communicating), and (for the same reason) C -{- D — C = D.
Hence B = D. Q.E.D.
Theorem 13
In general; if other terms added to coincident terms give coincident terms, then the terms
added are communicating.
If A and A coincide or are the same, and A-|-B=A-fA'", I affirm that B and A''
are communicating. For if A and B are non-communicating, and A and N also, then
B = N (by the preceding theorem) . Hence B and iV" are communicating. But if A and B
are communicating, let A = P -{- M and B — Q -\- M, putting M for that which is common
to A and B and nothing of this description in P or Q. Then (by ax. l)A-^B = P-\-Q
-\-M = P-hM-\-N. But P, Q, and M are non-communicating (by th. 11). Therefore,
if A'' is non-communicating with A — that is, with P -|- M — then (by the preceding theorem)
it results from P -^ Q -{- M = P + M + N that Q = N. Hence A'' is in B; hence N and
B are communicating. But if, on the same assumption (namely, that P -\- Q -{- M
= P -\- M -{- N, OT A is communicating with B) N also be communicating with P -{- M
or A, then either A'" will be communicating with M, from which it follows that it will be
communicating with B (which contains M) and the theorem will hold, or. A'' will be com-
municating with P, and in that case we shall in similar fashion let P = G -|- i? and N = F
-j- H, so that G, F, and H are non-communicating (according to th. 11), and from P -{- Q
Two Fragments from Leibniz 379
+ M = P + M +N we get G+H + Q+M=G + H + M + F + H. Hence (by
the preceding theorem) Q = F. Hence N {= F + H) and B (= Q + M) have something
in common. Q.E.D.
Corollary. From this demonstration we learn the following: If any terms be added
to the same or coincident terms, and the results coincide, and if the terms added are each
non-communicating with that to which it is added, then the terms added [to the same or
coincidents] coincide with each other (as appears also from th. 12). But if one of the
terms added be communicating with that to which it is added, and the other not, then [of
these two added terms] the non-communicating one will be contained in the communicating
one. Finally, if each of the terms is communicating with that to which it is added, then at
least they will be communicating with each other (although in another connection it would
not follow that terms which communicate with a third communicate with each other).
To put it in symbols: A + B = A + N. If A and B are non-communicating, and A
and A'' likewise, then B = N. li A and B are communicating but A and A'" are non-com-
municating, then N is in B. And finally, if B communicates with A, and likewise N com-
municates with A, then B and N at least communicate with each other.
XX
Def. 1. Terms which can be substituted for one another wherever we please without
altering the truth of any statement (salva verilate), are the same [eadem) or coincident
(coincidentia) . For example, "triangle" and "trilateral", for in every proposition demon-
strated by Euclid concerning "triangle", "trilateral" can be substituted without loss of
truth.
A = iJ'2 signifies that A and B are the ^,- -~,^
same, or as we say of the straight line XY ^-'' ^"^^
and the straight line YX, XY = YX, or the x4^ ^Y
shortest path of a [point] moving from X to '-., ,-''
Y coincides mth that from Y to X. "~- — M--''"
Def. 2. Terms which are not the same, that is, terms which cannot always be sub-
stituted for one another, are different (diversa). Such are "circle" and "triangle", or
"square" (supposed perfect, as it always is in Geometry) and "equilateral quadrangle",
for we can predicate this last of a rhombus, of which "square" cannot be predicated.
A + B^^ signifies that A and B are different, as for example, R Y *? ^
the straight lines XY and RS.
Prop. 1. If A = B, then also B = A. If anything be the same vrith another, then
that other will he the same with it. For since A = B (by hyp.), it follows (by def. 1) that
in the statement A = B (true by hyp.) B can be substituted for A and A for B; hence we
have B = A.
Prop. 2. If A ^ B, then also fi =|= A. // any term he different from another, then that
other vMl he different from it. Otherwise we should have B = A, and in consequence (by
the preceding prop.) A = B, which is contrary to hypothesis.
Prop. 3. If A = B and B = C, then A = C. Terms which coincide with a third term
coincide vxith each other. For if in the statement A = B (true by hyp.) C be substituted
for B (by def. 1, since A = B), the resulting proposition will be true.
Coroll. UA = S and B = C and C = i3, then A = D; and so on. For A = B = C,
hence A = C (by the above prop.). Again, A = C = D; hence (by the above prop.)
A =D.
Thus since equal things are the same in magnitude, the consequence is that things
equal to a third are equal to each other. The Euclidean construction of an equilateral
triangle makes each side equal to the base, whence it results that they are equal to each
12 ^ = S for A