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Symbols
13.
of
x.
function of
the symbols
x,
(p(a),
(j)(P),
&c, respectively
denote what
denote 5a; 2 denotes 5a — 3 a + 1
ample, (p(u)
— Sx +
2
let (p(x)
can see that <£(4)
=
Then, by definition, 1 tyro in mathematics any and
;
69, that cp(l)
.
=
3,
that <£(0)
=
that
1,
As an example in symbolic logic, 1) = 9, and so on. let (p(x) denote the complex implication (A B) (A* B*). Then 0(e) will denote (A B) (A e B ), which is easily while (f)(6) will denote seen to be a valid # formula :
:
:
e
:
:
:
;
(A B B ), which is not valid. 14. Symbols of the forms F(x, y), (f)(x, y), &c, are Any of the forms may be called functions of x and y. employed to represent any expression that contains both the Let
:
B)
:
:
;
*
Any formula
is
values (or meanings),
ar x
,
called valid
x2 ,
x 3 &c ,
,
when
of x.
it
is
true for
all
admissible
SYMBOLIC LOGIC
10
1416
[§§
becomes when a is put for x and /3 for y. Hence,
"
The
swallowed the rabbit." B) will assert that
boaconstrictor
symbol
that the
(p(R,
follows
It
The
"
rabbit
swallowed the boaconstrictor." 15. As another example, let t (as usual) denote true, and let p denote probable. Also let
A
:
(AP B P ) T which
asserts that " If
,
B
and
are both true,
it
is
A
true that
it
is
and
B are both probable." Then
A
and B are both probable, it is probable that A and B are both true." A little consideration will show that <£(t, p) is always true, but not always
The symbol
(p(x, y),
&c.
that
true for
its
is,
constituents
sible,
that
is,
all
e
admissible values (or meanings) of
the symbol
;
asserts that (p is certain,
(p
v
asserts that
of its constituents
;
the symbol
e
71
,
asserts that
= conclusion.
that
"
A
;
a meaningless statement which
is
For example,
true nor false. c
impos
is
meanings) means cp^cp' which
false for all admissible values (or
Also
let
w = whale, denote the
(p(w, h)
let
while is
h
(p°
neither
= herring, statement
small whale can swallow a large herring."
We
get e
(p
(w, h)
a threefactor statement
which
•
(p\w,
c),
asserts (1) that
it is
certain
that a small whale can swallow a large herring, (2) that it is impossible that a small herring can swallow a large
whale, and (3) that
unmeaning
to say that a small
whale can swallow a large conclusion.
Thus we see that more or less
it
is
F(x, y), &c, are really blank forms of
§§
APPLICATION TO
16, 17]
GRAMMAR
11
complicated expressions or statements, the blanks being represented by the symbols x, y, &c, and the symbols or words to be substituted for or in the blanks being u, /3,
&c, as the case may be. containing the
a
17.
+
/3,
Let
;
;
;
languages alike, is hardly unit of all possible. is the reasoning the manner in which the separate words are combined to construct a proposition varies according to
classification applicable to all
The
complete
proposition
;
In no Consider the two languages is it exactly the same. following example. Let S = His son, let A = in Africa, let K = has been killed, and let (p(S, K, A) denote the By proposition " His son has been killed in Africa!' our symbolic conventions, the symbol (p(S, A, K), in which the symbols K and A have interchanged places, denotes the proposition " His son in Africa has been Do these two propositions differ in meaning? killed." Clearly they do. Let S A denote his son in Africa (to distinguish him, say, from S c his son in China), and let KA denote has been killed in Africa (as distinguished from K c has been killed in China). It follows that
the particular bent of the language employed.
,
,
SYMBOLIC LOGIC
12
[§§17,18
referring to the
noun
S,
whereas, in the
the force of an adverb referring to
as regards symbols of the
in general,
latter,
the verb K.
form
A
has
And
Ax Ay A ,
,
2
,
A
denotes the leading or class idea, the point of resemblance, while the subscripta x, y, z, &c., denote the points of difference which distinguish the &c., the
letter
members of the general or class idea. Hence when A denotes a noun, the subscripta denote adjectives, or adjectiveequivalents; whereas when A
separate it
is
that
denotes a verb, the subscripta denote adverbs, or adverbWhen we look into the matter closely, the equivalents. inflections
of verbs,
to
indicate
really the force of adverbs, and,
of view,
or tenses, have logical point
For
be regarded as adverbequivalents.
S denote the word speak, & x may denote may denote will speak, and so on just as when Sy
example, spoke,
may
moods
from the
if
;
S denotes He spoke French,
may denote He spoke well, may denote He spoke slowly, and S y spoke,
S^
or or
He He
So in the Greek expression ol totc avOpunroi (the then men, or the men of that time), the adverb Tore has really the force of an adjective, and may be considered an adjective equivalent.
spoke
Dutch,
and so
on.
CHAPTER
III
cause of symbolic paradoxes is the 8. The main Take, for ambiguity of words in current language. When we say, " If example, the words if and implies. in the centigrade thermometer the mercury falls below zero, water will freeze," we evidently assert a general law which is true in all cases within the limits of tacitly This is the sense in which the understood conditions. It word if is used throughout this book (see § 10). is understood to refer to a general law rather than to 1
§§
PARADOXES AND AMBIGUITIES
18, 19]
13
So with the word implies. Let M z F denote " The mercury will fall below zero',' and let The preceding conditional denote " Water will freeze." z F which statement will then be expressed by M a particular case.
W
:
asserts
W
F
that the proposition
But
.
convention
this
M
z
forces
us
W
,
the proposition
implies
accept some
to
x and x e, which hold good whether the statement x be true or false. The former asserts that if an impossibility be true any statement x is true, or that an impossibility implies any The latter asserts that the statement x statement. (whether true or false) implies any certainty e, or (in The paradox other words) that if x is true e is true. will appear still more curious when we change x into e in the second. We in the first formula, or x into e, which asserts that any imthen get the formula The reason why the possibility implies any certainty. last formula appears paradoxical to some persons is e to probably this, that they erroneously understand mean Q^ Q e and to assert that if any statement Q is But impossible it is also certain, which would be absurd. paradoxicallooking formulas, such as
>/
:
:
rj
r\
:
>/
:
:
,
by definition it e does not mean this (see § 74) / simply means (>/e )'', which asserts that the statement Similarly, t]e is an impossibility, as it evidently is. x means {qx'J*, and asserts that nx is an impossibility, which is true, since the statement r\x' contains the im}}
:
r\
:
;
We prove x e as follows x\e — (xe'y = (x>i) = if —
possible factor
n.
:
ri
e.
For
e
=>],
since
implication for
Q"
e
:
is
19.
Q = rp e
:
is
some
That, on the other hand, the not a valid formula is evident
§ 20).
clearly fails in the case Q?.
it
sign
1
Q Q
denial of any certainty
the
impossibility (see
:
>f
=
Other paradoxes
of equivalence
(
=
*
:
'/
Q=
= («/)" = («0" =
»/,
we get
>/.
from the ambiguity of the In this book the statement
arise ).
Taking
SYMBOLIC LOGIC
14
[§
1
20
9,
=
not necessarily assert that a and /3 are /3) does synonymous, that they have the same meaning, but only that they are equivalent in the sense that each implies (a
the other, using the word 'implies' as denned in
In this sense any two
however
meaning
different in
and n 2 62 and variables, B x definition, we have
possibilities, n l
= e = (e 2
l
)
;
x
and
and
e
2
,
so are
§
10.
are equivalent,
any two im
but not necessarily two different We prove this as follows. By
\
.
(e
e
certainties,
= (e/ ne/ y = »w=V4= e«;
:e )(e :e l
2
2
,
,
=(vi) (Vt) for the denial of any certainty Again we have, by definition,
1)
2
i
,
ex
some
is
impossibility
r\
y
.
= *K = Vi= But we
,
equivalent.
necessarily
are
any two variables, 6 X and # 2 For example, 6 2 might be
cannot assert that
the denial of 6 V in which case (
e l
=
2
)
= (0 = e\) = (0 X
The symbol used a and
/3,
to
:
we should
e\)(6\
.
ej
l
assert
that
get (
0A)Wi)"
any two
are not only equivalent (in
statements,
the sense of each
implying the other) but also synonymous, is (a = /3); but this being an awkward symbol to employ, the symbol (a = /3), though it asserts less, is generally used instead. 20. Let the symbol it temporarily denote the word possible, let p denote probable, let q denote improbable, and have 6, t, let u denote uncertain, while the symbols e, r\,
We (A u = A'
i
by definition, and A 5 will A = and A" ) ), have (A respectively assert that the chance of A is greater than These conventions give us the \, that it is less than \. ninefactor formula their usual significations. 7r
e
)
shall then, p while
(*V)W(^WV)W
§
PARADOXES AND AMBIGUITIES
20]
15
2) that the denial of a truth is an and conversely; (3, 4) that the denial of a probability is an improbability, and conversely; (5, G) that the denial of a certainty * is an impossibility, and con
which
asserts (1,
untruth,
versely
(7
;
that the denial of a variable
)
that the denial of a possibility
(8, 9)
The
and conversely.
first
the other five are less
a variable
is
an uncertainty,
is
four factors are pretty evident
Some
so.
for example, that instead of
(tt')
that the denial of a possibility *
persons might reason, w
we should have
is
(n'y
;
not merely an uncer
A single concrete example but an impossibility. show that the reasoning is not correct. The statement " It will rain tomorrow " may be considered a
tainty will
possibility
;
but
its
denial " It will not rain tomorrow,"
though an uncertainty may be proved
u {tt')
is
not an impossibility.
Let
Q
equivalent to proving
Q
The formula
denote any statement taken at random out of a collection of statements containing certainties, impossibilities, and variables. To
prove
{ir')
u
is
as follows
:
77 :
(Q')
M
Thus we
.
get (Tr'y
= Q
(Q'f = Q
Q.
(Q y = Q
e
:
e
e
for
e
+ Q (Q'r + (Q7 = Q + Q :Q + Q = e; :
e
e
and (Q f = Q, whatever be the statement To prove that (^y, on the other hand, is not valid, ,
/
e
l
we have only
e
,
,
to instance a single case of failure.
Q the same meaning as before, a case for we then get, putting Q = 6 V
=e
;r 1
1i
= (e/ y = (e € y = i
1
2
of failure
,
Giving is
Q
8 ;
l2
* By the " denial of a certainty " is not meant (A e )', or its synonym A*, which denies that a particular statement A is certain, but (A e )' or its synonym A' e the denial of the admittedly certain statement A e This statement Ae (since a suffix or subscriptum is adjectival and not predicative) assumes A to be certain for both A x and its denial A'x assume the truth of A* (see §§ 4, 5). Similarly, "the denial of a possibility" does not mean A"' but AV, or its synonym (Att)', the denial of the admittedly possible .
,
;
statement
An.
SYMBOLIC LOGIC
16
may seem
[§21
the proA' with nor position A is not quite synonymous with A' = Then A yet such is the fact. Let A It rains. 21. It
paradoxical
say that
to
A
T
,
=
1
;
A T = it
It does not rain ;
is
true that
it
rains
;
A
and A' = it and AT are
The two propositions false that it rains. equivalent in the sense that each implies the other
is
;
but
they are not synonymous, for we cannot always substitute In other words, the equivalence the one for the other. (A A T ) does not necessarily imply the equivalence e then For example, let (p(A) denote A (p(A T ). (p(A)
=
=
;
T € Sup
,
true, or it is
happens
be true in the case considered, though
to
all cases.
not true in
ct>(A_)
We
get
= A< = e;=(e y = r T
,
never a certainty, though
for a variable is
it
may
turn out
true in a particular case.
Again,
we
get
^(AT ) = (AT )e = (^) = e« = e; e
T
for 6 T
means
T
which is a formal certainty. In this T though we have A = A yet (p(A) is not
(0 T )
case, therefore,
,
,
T equivalent to (p(A ).
A
suppose
Next,
denotes
t
,
a
variable that happens to be false in the case considered, get though it is not false always.
We
0(A') for
no
= (A') = A* = 0? = e
(though
variable
it
may
turn out false in a parti
cular case) can be an impossibility.
we
»7;
On
the other hand,
get
= (A') = A" = 6[ = (d[y = e = e; e
€
for 6[
e
means (Oy, which is a formal certainty. In this though we have A' = A\ yet <£(A') is not
case, therefore,
equivalent to
(A
l
).
It is a
remarkable fact that nearly
all civilised languages, in the course of their evolution, as if impelled by some unconscious instinct, have drawn
DEGREES OF STATEMENTS
21, 22]
§§
17
between a simple affirmation A and the statement A T that A is true ; and also between a simple denial A' and the statement A that A is false. It is the first step in the classification of statements, and marks a this distinction ,
1
,
faculty which
man
alone of
all terrestrial
animals appears
to possess (see §§ 22, 99).
22. As already remarked, my system of logic takes account not only of statements of the second degree, such a/3y as A" but of statements of higher degrees, such as A 13
,
A
afiyS
,
may
be asked, what
meant by statements of the second, third, &c, degrees, when the primary subject is itself a statement ? The statement A a/iy or its a/3 7 synonym (A ) is a statement of the first degree as regards its immediate subject A a/3 but as it is synonymous with (A a ) Py it is a statement of the second degree as regards A and a statement of the third degree as regards A, the root statement of the series. Viewed from another ,
But,
&c.
it
is
,
,
;
,
tt
,
standpoint,
A a may
be called^a revision of the
A, which (though here
judgment, of the
series)
it
is
may
judgment
the root statement, or root itself laave
been a revision
some previous judgment here unexpressed. Similarly, a (A")* may be called a revision of the judgment A and so on. To take the most general case, let A denote any complex statement (or judgment) of the ntb degree. of
3
,
be
neither a formal certainty (see § 109), like nor a formal impossibility, like (a/3 af, it may be a material certainty, impossibility, or variable, according If
it
(a/3
:
a)
e
:
,
If it follows data on which it is founded. from these data, it is a certainty, and we write A* if it is incompatible with these data, it is an impossibility, and we write A'' if it neither follows from nor is incompatible with our data, it is a variable, and we write A". But whether this new or revised judgment be A or A^ or A", it must necessarily be a judgment (or statement) of the (w+l) th degree, since, by hypothesis, the statement A is of the w th degree. Suppose, for ex
to the special
necessarily ;
;
e
ample,
A
denotes a functional statement
y,
B
z)
of
SYMBOLIC LOGIC
18
[§§
2224
which may have m different meanings &c, depending upon the different meanings x., x2 x y &c, y v yz y y &c, zv z 2 zy &c., of x, y, z. Of these m different meanings of A, or its synonym
th
degree,
(or values)
,
,
,
,
/
/
e
.
.
11
,
eT,e
,
,
degree as regards \Ja 23. It may be remarked that any statement A and T its denial A' are always of the same degree, whereas A synonyms (see and A', their respective equivalents but not The statement SS 19, 21), are of one degree higher.
A
and confirmation of the judgment A; and reversal of the judgment A. We suppose two incompatible alternatives, A and A' to be placed before us with fresh data, and we are to decide which is true. If we pronounce in favour of A, we conT if we profirm the previous judgment A and write A nounce in favour of A', we reverse the previous judgment A and write A\ 24. Some logicians say that it is not correct to speak of any statement as " sometimes true and sometimes that if true, it must be true always and if false, false " T
is
while
a revision
A
1
is
a revision
',
;
;
;
it
must be
false always.
To
this I reply, as I did in
my
seventh paper published in the Proceedings of the London
§
VARIABLE STATEMENTS
24]
Mathematical
that
Society,
when
false," or "
true and sometimes
"
say
I
A
is
A
19 is
sometimes
a variable," I merely
mean that the symbol, word, or collection of words, denoted by A sometimes represents a truth and someFor example, suppose the symbol A times an untruth. denotes the statement " Mrs. Brown is not at home." This is not a formal certainty, like 3 > 2, nor a formal impossibility, like 3<2, so that when we have no data out the mere arrangement of words, " Mrs. Brown is not at home," we are justified in calling this proposition, that is to say, this intelligible arrangement of words, a variable, and in asserting
A
6
If at the
.
Brown
moment the servant tells me home " I happen to see
that
"
Mrs.
Brown walking away in the distance, then / have and form the judgment A which, of course,
Mrs.
is
not at
e
fresh data
,
In this case I say that " A is certain" because its denial A' (" Mrs. Brown is at home ") would But if, contradict my data, the evidence of my eyes. instead of seeing Mrs. Brown walking away in the distance, I see her face peeping cautiously behind a curtain through a corner of a window, I obtain fresh
A
implies
T
.
data of an opposite kind, and form the judgment Av which implies A'. In this case I say that " A is im
,
because
possible,"
dicts
my
medium
A
is
what is
a
a it is
Mrs.
different
when
different
when she
the
Brown
statement
represented
by A,
not at home," this time contradata, which, as before, I obtain through the of my two eyes. To say that the proposition
"
namely,
'proposition
it is true, is
person
is out.
is
when
when
it
is
false
like saying that Mrs.
she
is
in from
from
Brown
what she
is
SYMBOLIC LOGIC
20
[§25
CHAPTER IV The
25.
following three rules are often useful:
= A'tf>(e). A"4>(A) = A*0(>;). A
(1) A'(A) (2)
In the
e
e
(3)
).
denotes the
last of these formulae, 6 X
first
variable
&c, that comes after the lastnamed For example, if the last variable that in our argument. has entered into our argument be 6 then X will denote 6 In the first two formulae it is not necessary to state which of the series e e2 e y &c, is represented by the e in (p(e), nor which of the series &c, is represented y 2 in (p(>i); for, as proved in § 19, we have always by the e ), and (t] x = (ex whatever be the certainties ex and y y ), e and whatever the impossibilities x ana %• Suppose, of the series 6
6
,
6
.
,
,
rj
,
rj
,
r\
=
r]
"
rj
for
example, that
\j/
denotes
A'B'C^C AB + CA). :
We
get
=AB £
T,
C
fl
e
= A B"C e
9 ;
may
be omitted In this without altering the value or meaning of \f/. operation we assumed the formulas
so that the fourth or bracket factor of
(1)
(ariz=r]);
(2) (ae
= a);
(3)
\j/
(*i
+ a==a).
Other formulae frequently required are (4) (AB)' (6)
(9)
e
= A' + B';
+ A = e; / = *,;
(5) (A + B)' = A'B'; = >7; (8) A + A' = e; AA' (7) = (10) >/ (11) A + AB = A; *;
(12) (A
+ B)(A + C) = A + BC.
§§
FORMULAE OF OPERATION
26, 27]
we
26. For the rest of this chapter
consideration of variables, so that A,
21
shall exclude the
A
T
A*
,
this
be con
will
sidered mutually equivalent, as will also A', A',
A''.
On
understanding we get the formulas (1)
A<£(A) = A
(3) A<£(A')
From
;
= A<^);
these formulas
(2)
Aty( A) = Aty(i)
(4)
A^(A') = A'(/)(e).
we derive
(5) AB'(/)(A, B)
;
others, such as
= AB'<£(e,
rj) ;
= AB'^)(»;, n)\ B') = AB'<£(>/,
(6) AB'<J>(A', B) (7) AB'<£(A',
e),
and so on; like signs, as in A(p(A) or A / ^)(A ), in the same letter, producing (p(e) and unlike signs, as in / B'(p(B) or B^>(B ), producing <jf>(>/)The following examples will show the working of these formulas /
;
:
Let
B) = AB'C
+ A'BC'.
Then we
get
= AB'(AB'C + A'BC) = AB («C + wC') = AB'(C + >/)=AB'C. A'B0'(A, B) = A B(AB C + A BC / = A B( w C + eeC y = A'B(C')' = A'BC. let cp(B, D) = (CD' + CD + B C B'D'0(B, D) = B'D'(CD' + CD + B'C')' = B D (Ce + C + eC'/ = B'D'(C + C)' = B D'e = B'D'>i = AB'<£( A, B)
/
/
/
/
/
Next,
Then,
/
/
/
/
/
)
/
,
.
/
>;
f
,
>
The
].
application of Formulas (4), (5), (11) of § 25 would, have obtained the same result, but in a more
of course,
troublesome manner. 27. If in any product
ABC
any statementfactor
is
implied in any other factor, or combination of factors, If in any sum (i.e., the implied factor may be omitted.
SYMBOLIC LOGIC
2%
[§§
27, 28
A + B + C, any term implies any other, or any others, the implying term may be omitted. rules are expressed symbolically by the two
alternative)
the
sum
These
of
formulae
(A:B):(AB = A);
(1)
By
(2)
(A:B):(A + B = B).
virtue of the formula (x a)(x :
may
formulae
:
/3)
=x
:
these two
a/3,
be combined into the single formula
(A:B):(AB = A)(A + B = B).
(3)
As the converse of each holds good, we get
of
these three formula?
also
A:B = (AB = A) = (A + B = B). Hence, we get A + AB = A, omitting the term AB, because and we also get A(A + B) = A, it implies the term A; omitting the factor A + B, because it is implied by the (4)
factor A.
A B
28. Since
:
AB,
factor of
it
is
(AB = A), and B
equivalent to
called a factor of the antecedent A, in
and
same
that, for the
any implication
reason, the antecedent
called a multiple of the consequent B.
of
and (A = AB)
A B :
may
The equivalence as follows
(A
:
of
A B :
A
:
A may
a
be B,
be
The equivalence
be proved as follows
(A = AB) = (A AB)(AB A) = ( A AB)e = A:AB = (A:A)(A:B) = e(A:B) :
is
B may
follows that the consequent
:
:
and (A
= A:B.
+ B = B) may
be proved
:
+ B = B) = (A + B:B)(B:A + B) = (A + B:B)e = A + B:B = (A:B)(B:B) = A:B.
The formula? assumed (x
:
aft)
= (x
"
If
a)(x
may
both of which assert that
:
x
is
in these :
/3),
two proofs are
and a
+ /3
:
x = (a
:
x)(fi
:
x),
For to be considered axiomatic. then a and /3 are both true " is
true,
equivalent to asserting that
" If
x
is
true a
is
true,
and
x
if
or
a
REDUNDANT TERMS
28, 29]
§§
true
is
8 is
is
true x
is
Also, to assert that " If either a
true."
/5 is
true x
is
" is
true
true,
23
and
equivalent to asserting that
if /3 is
true x
" If
is true."
29. To discover the redundant terms of any logical sum, or alternative statement. These redundant terms are easily detected by mere inspection when they evidently imply (or are multiples of) single coterms, as in the case of the terms underlined in
the expression
a fty
+ a'y + aft/ + ft/,
which therefore reduces to a!y + fiy'. But when they do not imply single coterms, but the sum of two or more coterms, they cannot generally be thus detected by inspection. They can always, however, be discovered by the following rule, which includes all cases. Any term of a logical sum or alternative may be omitted as redundant when this term multiplied by the denial of the sum of all its coterms gives an impossible product the term must not but if the product is not be omitted. Take, for example, the alternative statement rj
rj,
;
CD' + C'D Beginning with the
CD'(C'D
first
+ B'C' + B'D'.
term we get
+ B'C + B'D')' = CD'(w + B'» + B'e)' 7
= CD (B = BCD /
/
)'
/ .
Hence, the first term CD' must not be omitted. next the second term CD, we get
CTKCD' +
B'C/
Taking
+ B'D'/ = C'D( w + B'e + B',,)'
= C D(BY=BC D. /
/
CD
must not be omitted. Hence, the second term next take the third term B'C, getting
B^CD
7
I
C/D
+ B'D'/ = B'C'(>iI)' + eD + eD'/ = B'C'(D + D / = B C'>/ = ,
This shows that the third term
B'C
We
/
>/.
can be omitted as
SYMBOLIC LOGIC
24
2931
[§§
Omitting the third term, we try the
redundant.
last
term B'D', thus
B'D'(CD'
+ CD)' = B'D'(Ce + C>,)' = B'D'C.
This shows that the fourth term B'D' cannot be omitted But if we retain as redundant if we omit the third term. term B'D', fourth the third term B'C, we may omit the
we then get
for
B'D'(CD'
+ CD + B'C7 = B'D'(Ce + C'n + eC')' = B D (C + C ) =B D )'
/
/ ,
,
/
/
J?
=
i7.
Thus, we may omit either the third term B'C, or else the fourth term B'D', as redundant, but not both. 30. A complex alternative may be said to be in its simplest form* when it contains no redundant terms, and none of its terms (or of the terms left) contains any redundant factor. For example, a + ab + m + m'n is reduced
form when we omit the redundant term ab, term strike out the unnecessary factor m' last of the out and m + m'n m + n, so that the simplest and a, ab For a + to its simplest
=
= +m+
(See § 31.) n. form of the expression is a to its simplest alternative complex a reduce 31. To /3)' a'/3' to the denial of (a formula form, apply the +
=
the alternative. Then apply the formula (a/3/ = a' + ft' to the negative compound factors of the result, and omit Then develop the redundant terms in this new result. and go formulae, same the by product the denial of this result final The before. as process through the same
be the simplest equivalent of the original alternative. Take, for example, the alternative given in § 30, and We get denote it by (p.
will
= a + ab + m + m'n = a + m + m!n. = (a + m + m'n)' = a'm'(m'n)' = a'm'(m + nf) = a'm'n'. = (cp')' = (a' m'n')' = a + m + n.
(p'
here call its " simplest form " I called its " primitive form in my third paper in the Proceedings of the London Mathematical Society but the word " primitive" is hardly appropriate. *
What
"
I
;
METHODS OF SIMPLIFICATION
§§31,32]
As another example take the
25
alternative
AB'C + ABD + A'B'D' + ABD' + A'B'D, and denote it by (p. Then, omitting, as we go along, all terms which mere inspection will show to be redundant, we get r/>
= AB'C + AB(D + D') + A'B'(D' + D) = AB'C + ABe + A'B'e = AB'C' + AB + A'B'.
=(AB C ) (AB)'(A B ) = (A' + B + C)(A' + B')(A + B) = (A' + B'C)( A + B) = A'B + AB'C. p = «p')' = (A'B + AB'C)' = (A + B')(A' + B + C) = AB + AC' + A'B' + B'C. /
/
/ ,
/
, /
<£
i
Applying terms, B'C')
we
may
the
find
test
of
§
29
to
redundant term (AC or
discover
that the second or fourth
be omitted as redundant, but not both.
We
thus get (p
either
of
of
= AB + A'B' + B'C = AB + AC + A'B', which may be
taken
as
the simplest
form
(p.
32. We will now apply the preceding principles to an interesting problem given by Dr. Venn in his " S} mbolic Logic" (see the edition of 1894, page 331). Suppose we were asked to discuss the following set of rules, in respect to their mutual consistency and T
brevity.
Financial Committee a. The amongst the General Committee.
shall
be
chosen
from
No one shall be a member both of the General /3. and Library Committees unless he be also on the Financial Committee. y. No member of the Library Committee shall be on the Financial Committee.
SYMBOLIC LOGIC
26
[§32
Solution.
Speaking of a member taken at random, let the symbols F, G, L, respectively denote the statements " He will be on the Financial Committee," " He will be on the General Committee," " He will be on the Library as usual, for any statement that Putting Committee." contradicts our data, we have >;,
a
= (F:G);
/3
= (GLF'
>,)
:
7 = (LF:#,);
;
so that a/3 .
7 = (F:G)(GLF
/
:i?)(FL:i7)
= (FG':*i)(GLF':T,)(FL:r,) = FG' + GLF + FL:>/. /
Putting
+ GLF' + FL, ^(F' + GXG' + L' + FXF' + I/)
for the antecedent
(J>
(See
we get
FG'
25, Formulae (4) and (5))
§
= (F' + GL')(G' + L' + F) = F'G' + F'L' + GL' + FGL' = F'G' + GL'; term FGL', being a multiple of the term GL', is / / redundant by inspection, and F L is also redundant, because, by § 29, for the
F L (F G + GL')' = F'L'(eG' + /
/
,
Hence,
/
finally, <£
G<)'
= F'L'(G' + G)' = n
.
we get (omitting the redundant term FL)
= (<£')' = (F'G' + GL')' = FG' + GL,
and there ore I
a/3y
That
is
= (f>:ri= (FG' + GL
to say,
:
rf)
= (F
:
the three club rules,
replaced by the two
simple rules
F G :
G)(G u,
:
(3,
and
G
L')
7 :
,
may
L',
be
which
any member is on the Financial Committee, he must be also on the General Committee," which is rule a in other words and, secondly, that " If any member is on the General Committee, he is not to be on the Library Committee." assert, firstly, that
"
If
;
SOLUTIONS, ELIMINATIONS, LIMITS
§33]
27
CHAPTER V 33.
From
the formula (a
b)(c
:
:
d)
= ah' + cd'
:
»/
number of implications can always be expressed in the form of a single implication, the product of any
«
+ fi + 7 + &c
of which the antecedent
:
i],
a logical
is
sum
(or alternative),
and the consequent an impossibility. Suppose the implications forming the data of any problem that contains the statement x among its constituents to be thus reduced to the form
Ax + B,v' +
A
which
in
is
efficient of x',
the coefficient or cofactor of
and C the term,
contain neither x nor
data
may
C:tj,
or
sum
is
B
the co
It is easy to see that the
x'.
also be expressed in the
which above
form
(B:asX«:A')(C!:9)
which
x,
of the terms,
J
equivalent to the form
(B^iA'XCm).
When
the data have been reduced to this form, the ^iven
implication, or product of implications,
is
said to be solved
x ; and the statements B and A' (which are generally more or less complex) are called the limits of x; the antecedent B being the strong * or superior limit and Since the the consequent A', the weak or inferior limit. with respect
to
;
*
When from
:
u,
A+B.
we can
our data
we say that AB:A:A + B, we say
8
el
is
infer a:
that
AB
is
/3,
but have no data for inferring
For example, since we have stronger than A, and A stronger than
stronger than
/3.
SYMBOLIC LOGIC
28 factor
33,
[§§
34
(B x A') implies (B A'), and our data also imply >j), it follows that our data imply :
:
:
the factor (C
:
(B:A')(C:>,),
which
is
AB + C
equivalent to
Thus we get the
: *).
formula of elimination
+ B,/ + C:>,):(AB + C:>;),
(Ac
which asserts that the strongest conclusion deducible from our data, and making no mention of x, is the implication
AB + C
:
>/.
As
the twofactor statement
this conclusion
C^ABy,
ment C and the combination
it
is
equivalent to
asserts that the state
of statements
AB
arc both
impossible. 34.
From
this
we deduce the
solution of the follow
Let the functional symbol the symbol (p, denote data simply or a, b), z, <J)(x, y, which refer to any number of constituent statements x, y, z, a, b, and which may be expressed (as in the
ing more general problem.
33) in the form of a single implication + y + &c. rj, the terms a, /3, y, &c, being more or complex, and involving more or less the statements
problem of a
+
§
18
less x, y,
z,
:
a,
firstly, to find
It is required,
b.
successively in
the weakest antecedent
any and strongest consequent) of x, y, z; secondly, to eliminate x, y, z in the same order and, thirdly, to find the strongest implicational statement (involving a or b, but neither x nor y nor z) that remains after this desired order the limits
(i.e.,
;
elimination.
Let the assigned order of limits and elimination be Let A denote the sum of the terms containing the factor z let B denote the sum of the terms containing the factor z and let C denote the sum of the terms Our data being (p, we get containing neither z nor z
z, y, x.
;
,
.
(j,
= kz + B/ + C = (B = (B z A')(C = (B :
:
:
:
17
>;)
:
:
z)(z
z
:
:
A
,
)(C
:
r,)
A')(B A')(C :
:
>/).
§
SOLUTIONS, ELIMINATIONS, LIMITS
34]
The expression represented by Az + to
Bz'
to its simplest
have been reduced
+G
29
understood
is
form
(see §§
30,
The 31), before we collected the coefficients of z and z'. and the result after limits of z are therefore B and A' ;
the elimination of
z is
(B A')(C
:
:
= AB + C
which
>/),
n
:
.
To find the limits of y from the implication AB + C we reduce AB + C to its simplest form (see §§ 30, 31), We thus get, which we will suppose to be By + Ey' + F. >/,
:
as in the previous expression in
AB + C The
:
= By + E/ + F
r\
:
r,
limits of y are therefore
z,
= (E
y D')(E D')(F
:
:
E and
:
:
>,),
is
= ED + F
which
>/)•
:
and the result
D',
and y
after the successive elimination of z
(E D')(F
:
:
>/.
To find the limits of x from the implication ED + F we proceed exactly as before. We reduce ED + F to its simplest form, which we will suppose to be Gx + Hx + K, and get :
ED + F The
:
n
= Gx + Ha/ + K
:
r,
limits of x are therefore
= (H
H
after the successive elimination of
(H G0(K :
:
>/),
which
:
x G')(H G')(K :
:
and
G',
z,
x
y,
and the
:
>/,
>;).
result
is
= HG + K
:
>,.
x having thus been successively +K eliminated, there remains the implication the connecting which indicates the relation (if any) b. Thus, we remaining constituent statements a and
The statements
z,
y,
GH
:
}j,
finally get (/)
=
(B
:
z
:
which A and mention of z)
in
A')(E
:
//
:
D')(H x G')(GH :
:
:
,,).
B
do not contain z (that is, they make no D and E contain neither z nor y G and and the expression K contain neither z nor y nor x ;
;
H
+K
;
SYMBOLIC LOGIC
30
[§§
be destitute of
in the last factor will also
(i.e.,
34, 35
will
make
no mention of) the constitutents x, y, z, though, like G and H, it may contain the constituent statements a and b. a and a e are In the course of this process, since >)
:
:
whatever the statement a may be (see § 18), we can supply for any missing antecedent, and e for any missing consequent. certainties
>/
of the general prob
35. To give a concrete example lem and solution discussed in § 34, e
We
:
+ xyb +xy z +y
xyza
denote the data
let (p
a
z
.
get, putting (p for these data,
= {xyza + xyb' + xy'z' + y'z'a')' — x'y + + y'z + abz + ax bijz
when
r\
:
»/,
:
the antecedent of this last implication has been its simplest form by the process explained in
reduced to §
Hence we
31.
(j>
get
= (y'+ ab)z + (]jy)z + {x'y + ax')
putting
A
in § 34,
we get
for y'
+ ab, B
n
+ ax'.
As
and the result
after
and C
for by,
:
for x'y
(B:s:A')(AB + C:>7), so that the limits of z are
the
elimination of
is
z
B and
AB + C
A', :
»/.
Substituting their
values for A, B, C, this last implication becomes {ab
which we ab
+ x, E
will for
n,
(f>
+ ,c)y + ax'
denote by *Dy
and F :
z
:
=
:z
:
(B
?/,
+ Ey' + F
:
n,
A')(Dy
+ E/ + F
A0(E
y D')(ED
Having thus found the
putting
J)
for
Thus we get
for ax.
= (B
:
:
limits
:
{ix.,
:
>/)
+F
:
»;).
the weakest ante
SOLUTIONS, ELIMINATIONS, LIMITS
§§35,36]
cedents and strongest consequents) of z and y, to find the limits of x from the implication
31
we proceed
ED + F
:
n,
the strongest implication that remains after the Substituting for D, E, F the elimination of z and y.
which
is
we
values which they represent,
DE + F in
=
n
:
which G, H,
get
= Gx + BJ + K
{ah
+ J)n + «J
K
respectively denote
:
n
>/,
a,
n,
:
We
n
thus
get
DE + F
:
= (H
>i
x G')(HG
:
:
+K
tj)
:
;
so that our final result is <$>
= (B = =
To obtain
:
(by (/>//
z
A')(E
:
:
:
//
D')(H
+ b y){n :z:a'y + b'y)(y
:
z
:
f
a'y
y
:
:
:
:
a;
; rt ;«
a'x
G0(HG + K e)(>/ + b'x){a :
:
:
+ b'x)(a
:
£C
:
i,)
:
»/)
x).
we first substituted for A, B, D, E, then we the values we had assigned to them this result
G, H, K in the second factor, omitted the redundant antecedent the redundant consequent e in the third factor, and the ;
>/
redundant certainty
(»/
:
»/),
which constituted the fourth
the fourth factor (HG + K:>/) reduces to the form (n rj), which is a formal certainty (see § 18), indicates that, in this particular problem, nothing can be implicationally affirmed in terms of a or
factor.
The
fact
that
:
z) except formal f &c, which such as (ab a), (aa >;), ab(a + b') are true always and independently of our data (p. 36. If in the preceding problem we had not reduced the alternative represented by As + Bz' + C to its simplest form (see §§ 30, 31), we should have found for the not a'y + b'y, but inferior limit or consequent of z, supposed that the might be it this From b'y). x(a'y + strongest conclusion deducible from z (in conjunction with, or within the limits of, our data) was not A' but But though xh! is formally stronger than A', that xk'.
b
(without mentioning either x or y or
certainties
:
:
:
>i,
SYMBOLIC LOGIC
32 is
3638
than A' token we have no data but our here we have other data, namely,
say, stronger
to
definitions,
;
we
implies (as lent
[§§
shall prove) that A'
in this case equiva
is
to xA', so that materially (that
to say, within the
is
limits of our particular data
This
we prove
:
(z
as follows
:
A
7
y
:
:
:
D' x) :
:
(A'
:
x)
:
(A'
= x A')
;
a proof which becomes evident when for A' and D' we substitute their respective values a!y + b'y and a'x + b'x for it is clear that y is a factor of the former, and x a ;
factor of the latter.
37. In the problem solved in § 35, in which our data, namely, the implication e
:
xyza'
(p
denoted
+ xyb' + xy'z' + y'z'a',
y, x as the order of limits and of elimination. taken the order y, x, z, our final result would have
we took
z,
Had we been
(j>
38.
= (z:y:
b'x
+ xz)(z + a
The preceding method
" limits "
my method
x){z
:
a'
of finding
of logical statements
was suggested by,
:
is
+ b').
what
I call
closely allied
to,
the
and
(published in 1877, in the
Lond. Math. Soc.) for successively finding the for the variables in a multiple integration limits of In the next chapter the method integral (see § 138). Proc. of the
will be applied to the solution (so far as solution is possible) of Professor Jevons's socalled
which has given
among
rise
to
logicians but also
"
Inverse Problem,"
much discussion, not among mathematicians.
so
only
PROBLEM"
JEVONS'S "INVERSE
§39]
33
CHAPTER VI Briefly stated, the socalled "inverse problem" of Professor Jevons is this. Let tp denote any alternative, It is required to find an imsuch as abc + a'bc + aVV 39.
'.
plication,
or product of implications,* that implies this
alternative.
Now, any implication whatever implications) that e
of
or
f :
b)((f>
:
alternative
e
or
cp,
:
:
»y,
is
of
a multiple
or (abc
:
ab)(e
:
&c, must necessarily imply the given
rj),
number
that the
so
cp,
any product
(or
equivalent to
example,
as, for
,
(a
is
of possible solutions
But though the problem
as enunthus indeterminate, the number of possible solutions may be restricted, and the really unlimited.
is
ciated
by Professor Jevons
is
problem rendered far more interesting, as well as more and instructive, by stating it in a more modified form as follows Let cp denote any alternative involving any number of
useful
:
constituents,
implication
a,
e
:
c,
b,
cp
&c.
It
required to resolve the
is
that
into factors, so
it
will
take the
form
(M a N)(P :
:
:
b
:
Q)(R
:
c
:
S),
&c,
which the limits M and N (see § 33) may contain &c, but not a; the limits P and Q may contain the limits R and S may neither a nor b c, d, &c, but contain d, e, &c, but neither a nor b nor c and so on When no nearer limits of a conto the last constituent. and e stituent can be found we give it the limits the former being its antecedent, and the latter its conin
b,
c,
;
;
>;
sequent (see * Professor
at
§§ 18, 34).
Jevons
calls these implications
tific
"laws," because he arrives
by which scien" investigators have often discovered the socalled " laws of nature
them by a long tentative inductive
process, like that
(see§ 112).
C
SYMBOLIC LOGIC
34
[§39 *
As a simple example, suppose we have (p
= abc + a'bc + ab'c',
the terms of which are mutually exclusive. form (see §§ 30, 31), we get
to its simplest
Reducing
= be + ab'c',
and therefore e
:
= (f/
<£
= (be)' {ab' )' n = (b' + c')(a' + b + c):r = a!b' + J'c + aV + be' f
:
,,
:
1
>/.
:
This alternative equivalent of § 31) by omitting either the not both so that we get
cp'
first
may
be simplified (see
or the third term, but
;
e
:
= b'c + a'c' + be'
(p
Taking the
rj
= a'b' + b'c + be
equivalent of
first
the limits of a) arranging
we
:
e
:
in the
it
:
17.
and (in order form Aa + Ba'
to find
+C
:
tj,
get (see §§ 33, 34) e
:
(p
V+
= tja + c = (c a
(6'c
+ W)
»/
:
7
:
:
e)(c
:
b
c)(t]
:
:
c
e).
:
Thus, we have successively found the limits of
But
34, 35).
§§
since (a
formal certainties, they that
:
e),
may
(>;
:
c),
and
(c
:
(see
a, b, c e)
are
all
be omitted as factors, so
we get e
:
= (c'
:
«)(c
:
6
c)
:
= (c'
a)(c
:
=
b).
two factors asserts that any term of the given alternative (p which contains c' must also contain a. The second asserts that any term which contains c must also contain b, and, conversely, that any term which con
The
first
of these
tains b
must
native
(p will
also contain
c.
A
glance at the given alter
verify these assertions.
denotes an Observe that here and in what follows the symbol denotes a given implication, which In §§ 34, 35 the symbol may take either such a form ase:a + /3 + 7 + &c. or as a + /3 + 7 + &c. 7/. *
alternative.
<j>
<j>
,
:
We
now take
will
the second equivalent of a'b'
and resolve
it
the limits of
a, b,
+ b'e + be'
first
sight e
:
it
/
:rt )( c
(6
tj,
by successively rinding
= &). different
a) in the former result
the factor
(c
factor (b'
a) in the latter.
:
namely,
<jj }
:
might be supposed that the two ways of into factors gave
e
35
Proceeding as before, we get
c.
:^ =
At
:
into three factors
t
resolving
PROBLEM"
JEVONS'S "INVERSE
§§39,40]
since
results,
replaced by the
is
But
since the second factor informs us that b and c are equivalent, it follows that the two implications c a and b' a are equivalent also. :
= b), common
(c
to
both
results,
:
:
If we had taken the alternative equivalent of
e:(p in
= (p':>] = (b' + c': a)(c = b) = {1/
which either the factor
(b'
:
:
a)(c'
a) or the factor
be omitted as redundant, but not both. the factor yet
= b) alone neither implies = b) implies a), and
(c
(b'
= b),
a)(c
:
{c
:
a)
may
For though :
a) nor
(
:
a),
= b)
(c' implies This redundancy of factors in the result is a necessary consequence of the redundancy of terms in the alternative equivalent of
(b
{b' :a)(c
(c'
:
:a)(c
r
:
a).
omission
of
term
a'c'
the
implicational
factor
(a'b'
:
>/),
or
its
and the omission of the in the alternative leads, in like manner, to the
equivalent
(b'
:
a),
in the result
omission of the factor
(a'c'
:
rf),
;
or
its
equivalent
(c'
:
a),
in
the result. 40. I take the following alternative from Jevons's "Studies in Deductive Logic" (edition of 1880, p. 254, No. XII.), slightly changing the notation, abed
Let
(p
+ abe'd + ab'cd' + a'bed' + a'b'c'd'.
denote this alternative, and
let it
be required to
SYMBOLIC LOGIC
36
find successively the limits of a, b
we
are required to express
(M a N)(P :
:
in
M
which
and
N
:
b
e
Q)(R
:
n
and M.
By
e.
=d +
b
c
;
y
c
:
:
S)(T
d
U),
:
P and Q
;
;
r,,
we
get
V = d, Q = c + d, R =
b'c,
e,
U=
:
A
:
(p
= (d + be' + b'c
:
a
:
bd
glance at the given alternative
+ b'c){d <£>
>,,
6.
Omitting the last two factors R c S and because they are formal certainties, we get e
are
and S are neither to conand T and U must be respectively
N = bd + S= T=
b'c,
:
R
the process of §§ 34, 35,
+
In other words, form
d.
c,
in the
are not to contain a
neither to contain a nor b tain a nor b nor
:
[§40
T d :
:
:b:c
U
:
+ d).
will verify this result,
we have either d or be' or ( 1 that whenever we have a, then we b'c, then we have a (2) have either bd or b'c (3) that whenever we have d, then we have b (4) that whenever we have b, then we have either c or d; and (5) that from the implication e (p we can infer no relation connecting c with c£ without making which
asserts
)
that whenever ;
;
;
.
mention of a or b or, in other words, that c cannot be e is a expressed in terms of d alone, since the factor c formal certainty and therefore true from our definitions The final factor is alone apart from any special data. for only added for form's sake, for it must always have In other words, when antecedent and e for consequent. we have n constituents, if x be the n th or last in the ;
>/
:
:
>/
must
order taken, the last factor
necessarily be
may
and therefore a formal certainty which understood. of n
:
c
e
:
Others of the factors
may
(as in
taken successively in alphabetic order. reverse order d, c, b, a, our result will be :
:
x
:
e,
left
the case
here) turn out to be formal certainties also, but
not necessarily. We have found the limits of the constituents
e
>;
be
(p
= (ab + ac' + bd
:
d
:
ab)(ab'
+ a'b
:
c
a, b,
c,
d,
we take the
If
:
a
+ b),
§§
ALTERNATIVES
40, 41]
37
b e and a e omitting the third and fourth factors There is one point because they are formal certainties. Since every double in this result which deserves notice. >)
implication a
:
x
:
always implies a
(3
(in the first bracket) ab
+ ac' + he
:
/3,
:
:
>;
it
follows that
:
:
Now, the
implies ab.
formally stronger than the former, since any statement x is formally stronger than the alternative latter
is
x + y. But the formally stronger statement x, though it can never be weaker, either formally or materially, than x + y, may be materially equivalent to x + y; and it must be so whenever y materially (i.e., by the special data of Let us see the problem) implies x, but not otherwise. whether our special data, in the present case, justifies the inferred implication ab tion (/3
:
and
By
\J/.
x)(y be
:
we
x),
+ ac + be
Call this implica
ab.
:
virtue of the formula a
+ (3 + y
get (putting ab for a and for
x
:
= (a
ac for
x,
:
x) (3,
for y)
\z
= (ab al)){ac' = (ac a)(ac = e(ac' b)(bc'
:
:
:
:
ab)(bc b)(bc
:
:
a)e
:
:
ab)
a)(bc
= (ac
:
= e(ac :
:
ab)(bc
:
ab)
b)
b)(bc
:
a).
This asserts that (within the limits of our data in this
problem) whenever we have ac we have also b, and that whenever we have be we have also a. A glance at the given fully developed alternative
+ ac + be
:
ab
the fact that
is,
its
in this problem, legitimate, in spite of
antecedent
is
formally weaker than
its
consequent. 41.
An
alternative
and only when,
it
is
said to be fully developed when,
satisfies
the
conditions
following
Firstly, every singleletter constituent, or its denial,
must
be a factor of every term secondly, no term must be a formal certainty nor a formal impossibility thirdly, all the terms must be mutually incompatible, which means that no two terms can be true at the same time. This last condition implies that no term is redundant or repeated. ;
;
SYMBOLIC LOGIC
38
For example, the
developed form of a+ft is multiply the two
fully
+ aft' + aft. To obtain this we and strike factors a + a and ft + because it is equivalent to (a + As another given alternative a + aft
out the term
/3',
example, let it be developed form of a + ft'y.
ft.
Here we ft +
fully
find the product of the three factors a
first
7 + 7'.
and
ft',
the
find
to
aft',
the denial of the
ft)',
required
41, 42
[§§
equivalent to
a' (ft'y)',
We
next
which
is
that
find
+ a,
{a {ft'y)'
equivalent to
is
+ y'),
a'(ft
Then, out of the therefore, finally, to aft + ay'. eight terms forming the product we strike out the three terms a'fty, a'fty, a'/S^', because each of these contains
and
either aft or a'7',
which are the two terms
of aft
+ ay',
+ ft'y.
The
result
the denial of the given alternative a will be
aft'y
which
+ a'fty + a fty + a ft'y' + a fiy'i form of the given
therefore, the fully developed
is,
+ ft'y.
alternative a
42. Let
denote
(p
a'cclc
+ Veil + cd'e + a (Ye.
have 5 elementary constituents
a,
b,
d, e
c,
+ a), (b +
;
Here we so that the
&c, will contain 5 11 terms will terms, Of these 32 (or 32) terms. 2 the reof constitute the fully developed form
Then the
five factors (a
alternatives
and
\J/
will,
b'),
of course, only differ
Suppose the they will be logically equivalent. alternative \f/ to be given us (as in Jevons's " inverse problem "), and we are required to find the limits of the
in
form
5
constituents in the alphabetic order
;
the data \Jr
e
When we
\^.
:
to its simplest form,
a, b,
c,
d,
e,
from
have reduced the alternative shall find the result to be
we
(p.
Thus we get e:ylr
= e:
= This
is
(>7
a
=
; b d' + c)(d c e)(e d e)(r) :e:e). :
:
:
:
:
:
:
»/
:
the final result with every limit expressed.
Omit
UNRESTRICTED FUNCTIONS
4244]
§§
ting the superior limit
and the
>/
:
\Jr
= (a
:
&'c
+ ce')(b
:
wherever
inferior limit e
they occur, and also the final factor formal certainty (see § 18), we get e
39
ri'
>j
c
:
+ e)(d
:
because
e
:
c)(e
:
a
it is
rf).
Suppose next we arc required to find the limits in the order e
:
y$r
d,
e,
= (e = (e
d
:
:
&'c
+
:
b'c
+ ce)(e
d
:
Our
a. b.
c,
final result in this case will :
:
e
:
a'c
a'c
+ b'c){a
+ b'c)(a
:
:
c
e)(>7
:
:
a
be :b:e)
e)(>/
:
c).
When
an alternative
number
of possible
solution
in
the order
virtually the
same
d,
e,
c,
a,
(the
b
given),
last
the only difference being that the last first case are (as given), n a e and r\ :
:
:
is
a two factors in the while in the b e
as the solution in the order d,
e, c, b,
;
:
that is to say, a e second case they are tj:b:e and the order changes, and both, being certainties, may be It will be observed that when the order of omitted. >/
limits
is
:
:
prescribed, the exact solution
;
prescribed also
is
no two persons can (without error) give different solutions, though they may sometimes appear different in
form
(see
§§39,
40).
CHAPTER 44.
Let
~F u (x, y, z),
or
values or meanings of ;
,
l
)
y; z)
example,
its
while the symbol
Fr
synonym F( i
abbreviated
synonym F„, rey, z), when the
the functional proposition F(x,
present
stricted
its
VII
,
represents
when the values if
constituents
F r (x, the of x,
y, z),
x, y, z
or its abbreviated
functional y, z
are unre
proposition
are restricted.
x can have only four values. xy
x,
2
x.
A
,
For x4 y ;
SYMBOLIC LOGIC
40
the four values y y2 z. then we write v s ,
z„ z
;
the three symbols
x,
yz
,
y
,
;
and
44, 45
[§§
the three values
z
F r and not FM But if each of y, z may have any value (or meaning) .
,
whatever out of the infinite series x v x2 x3 &c, y v y 2 y 3 &c., z «„, z &c. then we write F M and not F r The suffix v r is intended to suggest the adjective restricted, and the The symbols F F n F e suffix u the adjective unrestricted. ,
,
;
,
,
,
,
e
,
F
as usual, assert respectively that impossible, that is .<•,
F
z
y,
means
is
variable
mean
understood to
but here the word
;
admissible value of
every z);
y,
and
Thus F e
nor impossible.
asserts that Fix,
,
F
is
certain
;
impossible
in
y, z
the
neither certain
neither
y, z) is
synonymous with
is
;
formulae
x,
means
variable
always true nor always false it F _e F~", which is synonymous with
From
,
that
true fur all the admissible values of
in the functional statement F(x, y, z) false for
statement F(x,
45.
is certain,
(F^F"/.
these symbolic conventions
we get the three
:
(1)(FF<); (2)(F?
(3)(F?:F? );
:F?.);
t
f
but the converse (or inverse) implications are not necessarily true, so that the three formulae would lose their validity if we substituted the sign of equivalence ( The first two formulae for the sign of implication (:). need no proof; the third is less evident, so we will prove
=
it
as
denote the above three two being selfevident, to be a certainty, so that we get the
Let
follows.
we assume
(f> 2
formulae respectively.
,
The
(p 3
first
deductive sorites e:
:
:
(F;
e :
F)(F7 1?) :
(FF7 FfFJ) :
(F*: F*) [for
[for a
[for
:
/3
= /3'
(A a)(B :
A'A^ = A e by ,
:
b)
:
:
«']
(AB
:
ah)]
definition].
This proves the third formula
two
,
.
,
§§
SYLLOGISTIC REASONING
45, 46]
represent the word horse, and
ment
"
The
F(H) denote the
let
Then F (H) l
has been caught."
horse
41
H H
state
asserts
&c., has been r 2 the symbol F' (H) asserts that not one horse of and the symbol the series &c., has been caught r 2 e F*(H) denies both the statements F (H) and F"(H), and
every horse of the series
that
caught
H H
is
,
)
;
;
,
therefore equivalent to
F _e (H)
F" (H), which r,
.
may
be
6
expressed by F~ E^, the symbol (H) being left This &c. ? understood. But what is the series H^ 2 universe of horses may mean, for example, all the horses owned by the horsedealer ; or it may mean a portion only of these horses, as, for example, all the horses that had
more
briefly
H
If
escaped.
by
we
write F*
{
we
assert that every horse
has been caught;
the horsedealer
,
if
we
write
owned F*
we
only assert that every horse of his that escaped lias been Now, it is clear that the first statement implies caught.
the second, but that the second does not necessarily imply the first so that we have F' F*, but not necessarily F;:F;. The last implication F;:F; is not :
;
t
all the horses that necessarily imply not had escaped were caught would had been horsedealer that all the horses owned by the and escaped, caught, since some of them may not have had of these it would not be correct to say that they
necessarily true
;
The symbol F M may
been caught.
V v F2 F3 F 60 i\, F F F 2 8 10 make evident the F* F*
:
:
,
,
.
.
.,
,
,
,
.
.
.,
.
while
may
F,.
refer
to
the series
refer only to the series
The same concrete illustration will truth of the implications F^:F? and
F* and also that the converse implications F? ,
Ff.
:
F? and t
are not necessarily true.
46. Let us called is
that
the fact
for
now examine
syllogistic.
my
a particular case of (a
or, as it
may
the special kind of reasoning will be shown,
Every valid syllogism, as general formula
(3)((3
:
:
y)
:
(a
:
y),
be more briefly expressed, (a
:
/3
:
y)
:
(a
:
y).
SYMBOLIC LOGIC
42
Let S denote our Symbolic
[§§ "
or
Universe,
46, 47
Universe
of
the things S v S 2 &c, real, or nonexistent, expressly mentioned or
Discourse," consisting of unreal, existent,
all
,
our argument or discourse. Let denote any class of individuals X X 2 &c, forming a portion of the Symbolic Universe S then 'X (with a grave accent) denotes the class of individuals 'X 'Xg, &c, that do not belong to the class X so that the individuals tacitly understood, in
X
,
,
;
,
;
X
&c, of the class X, plus the individuals X 'X 2 2 &c, of the class X, always make up the total Symbolic Universe S S 2 &c. The class 'X is called the complement of the class X, and vice versa. Thus, any class A and its complement 'A make up together the whole Symbolic Universe S each forming a portion only, and both forming the whole. 47. Now, there are two mutually complementary classes which are so often spoken of in logic that it is convenient to designate them by special symbols these are the class of individuals which, in the given circumstances, have a real existence, and the class of individuals which, in the given circumstances, have not a real existXj,
X
,
.
,
X
,
,
;
;
The
ence.
individuals
class
first e
v
e„,
the class
is
To
&c.
made up
e,
of the
this class belongs every indi
vidual of which, in the given circumstances, one can "
truly say
"
It exists
— that To
bolically but really.
town, triangle, virtue,
horse,
and
in the class
vice
exists "
or
"
Vice
e,
The second
exists "
class
We may
vice.
merely symmay belong place
because the statement really
persons, or vicious persons, exist
one would accept as
to say, not
is
this class therefore
;
asserts
that
"
virtue
Virtue
virtuous
a statement which every
true. is
the
class
0,
made up
of
the
individuals 0^ To this class belongs every in&c. 2 dividual of which, in the given circumstances, we can ,
truly say not exist exists
" It
does not exist
"
—
that
is
to say, " It does
though (like everything else named) it symbolically." To this class necessarily belong really,
REALITIES AND UNREALITIES
§§4749]
48
mermaid, round square, fiat sphere. The Symbolic Universe (like any class A) may consist wholly of realil ies or wholly of unrealities Oj, e &c, or it may 2 v e 2 &c. centaur,
;
,
,
When
be a mixed universe containing both.
Av A 2
,
&c, of any
A
class
wholly of unrealities, the class class least
;
when A
the
members
consist wholly of realities, or
A
said to be a pure
is
contains at least one reality and also at
one unreality,
it
mixed
a
called
is
class.
Since
and are mutually complementary, it is clear that V is synonymous with 0, and with e. 48. In no case, however, in fixing the limits of the class e, must the context or given circumstances be overlooked. For example, when the symbol H! is read " The horse caught does not exist," or " No horse has been caught" (see §§ 6, 47), the understood universe of realities, e v e 2 &c, may be a limited number of horses, H H 2 &c, that had escaped,, and in that case the statement Hj! merely asserts that to the classes
e
v
,
,
,
that limited universe the individual or a horse caught, does not belong;
H
it
c
,
the horse cauyht,
does not deny the
caught at some other time, Symmetry and conor in some other •circumstances. venience require that the admission of any class A into our symbolic universe must be always understood to imply the existence also in the same universe of the complementary class *A. Let A and B be any two classes that are not mutually complementary (see § 46) if A and B are mutually exclusive, their respective complements, A and 'B, overlap; and, conversely, if 'A and 'B are mutually exclusive, A and B overlap. 49. Every statement that enters into a syllogism of the traditional logic has one or other of the following four forms possibility of a horse being
;
V
X
(1) Every (3) It is
Some
evident that (3)
X is
is
is
Y
Y ;
;
(2)
(4)
No
Some
X is Y X is not
simply the denial of
;
Y.
(2),
and (4)
SYMBOLIC LOGIC
44
From
the denial of (1). get
the conventions of §§
(1)
X° Y = Every
(3)
XT = Xy = Some X is Y X! = X:° = Some X is not
X
[§§ 49,
is
Y
X°Y
(2)
;
= No X
G,
is
47,
50
we
Y
°
;
(4)
Y.
Y
The
first two are, in the traditional logic, called universals ; the last two are called particulars ; and the four are respectively denoted by the letters A, E, I, 0, for reasons
which need not be here explained, as they have now only
The following is, however, a simpler symmetrical way of expressing the above four more and of the traditional logic and it has propositions standard historical interest.
;
the further advantage,
how
appear
as will
of
later,
showing
all the syllogisms of the traditional logic are only
particular cases of
more general formulae
in the logic of
pure statements. 50. Let S be any individual taken at random out of our Symbolic Universe, or Universe of Discourse, and let respectively denote the three propositions
x, y, z
S
z
S~
Then
.
z
By
.
y',
x',
must
z'
x, y, z,
like their denials x'
certain
;
that
,
,
y', z
(x\
z\
e
f
tions (x
:
iff,
(y
:
>/)',
{y'y, (z)e
:
>/)'
,
Y ,
Hence, we
and never x
71
;
Hence, when
e
(z
S~
are all possible but un
,
nor y nor z nor x nor y nor z\ respectively denote the propositions v
,
46, the three propositions
§
to say, all six are variables.
is
must always have xe y e v
,
,
denote S~ x
respectively
the conventions of
Sx SY
x, y, z
S x S Y S z the proposi,
,
,
(which are respectively synony
must always be considered to form and their part of our data, whether expressed or not denials, (x »/), (y n), (« »?), must be considered impossible. With these conventions we get
mous with x*
1
,
y'1*, z"
)
;
:
:
:
X is Y = S x S Y = (x y) = {xy'f x S Y / = (x y)' = (xy'y (0) Some X is not Y = (S Y x S = x y = (xyY (E) No X is Y = S x T S" )' = (x y')' = {xyj*. (1) Some X is Y = (S
(A) Every (or
all)
:
:
:
:
:
:
:
:
§
GENERAL AND TRADITIONAL LOGIC
50]
In
this
way we can
express
every syllogism
of
45 the
terms of x, y, z, which represent three propositions having the same subject S, but different predicates X, Y, Z. Since none of the propositions x, y, z (as already shown) can in this case belong to the class or e, the values (or meanings) of x, y, z are restricted. Hence, every traditional syllogism expressed in terms of x, y, z must belong to the class of restricted functional statements Fr (x, ?/, z), or its abbreviated synonym Fr) and not to the class of unrestricted functional statements traditional
logic
in
r\
FJx, y, z), or its abbreviated synonym F w as this last statement assumes that the values (or meanings) of the propositions x, y, z are wholly unrestricted (see § 44). ,
The proposition Fw
assumes not only that each
(x, y, z)
statement
may
belong to the class but also that the three statements x, y, z need not even have the same subject. For example, let F (x, y, z), or its abbreviation F, denote the formula constituent >/
or
e,
x,
(x
:
y)(y
then x implies z." be the statements
z)
:
(x
:
z).
x implies
y,
and y implies
The formula holds good whatever
z,
in
:
9,
" If
This formula asserts that
(as
z
y,
as well as to the class
x,
y,
z
;
whether or not they have same subject S and
the traditional logic) the
;
whether or not they are certainties, impossibilities, or variables. Hence, with reference to the above formula, 6 it is always correct to assert F whether F denotes F M When x, y, z have a common subject S, then or F r F e will mean F^. and will denote the syllogism of the traditional logic called Barbara ;* whereas when x, y, z are wholly unrestricted, F will mean F^ and will therefore be a more general formula, of which the traditional Barbara will be a particular case. .
e
*
Barbara asserts that " If every
X is Z,"
which
is
X
is
equivalent to (S x S v ) (S v :
Y, and every :
Sz)
:
(S x
:
S z ).
Y
is Z,
then every
SYMBOLIC LOGIC
46
But now
let F, or Y(x, y,
(y
z)(y
:
denote the implication
z),
x)
:
[§§50,51
(x
:
:
z')'.
suppose the propositions x, y, z to be limited by It' we the conventions of §§46, 50, the traditional syllogism called Darapti will be represented by F r and not by 6
formula of § 45, we have F,' F, e e e but not necessarily F~ F; and, consequently, F; F~ Thus, if F u be valid, the traditional Darapti must be We find that F w is not valid, for the above valid also. implication represented by F fails in the case f(xzy, as it
FM
Now, by the
.
first
:
.,
(
6
:
:
,
.
then becomes (>1
:
z){ri
x)
:
:
(xz)~ v ,
which is equivalent to ee if, and consequently to e But since (as just shown) F; which = {er/f = (ee) = 6 does not necessarily imply F; this discovery docs not justify :
:
»/,
6
7
rj.
'
,
us in concluding that the traditional Darapti
F
is
not valid.
y\xz)n and this case cannot occur in the limited formula Fr (which here represents the traditional Darapti), because in Fr the pro
The only
case in which
fails
is
,
x, y, z are always variable and therefore possible. In the general and nontraditional implication F M the case x yv zr since it implies [piiczf, is also a case of failure; but it is not a case of failure in the traditional logic. 51. The traditional Darapti, namely, "If every Y is Z, and every Y is also X, then some X is Z," is thought by
positions
,
yi
',
some real
Y
is
nonexistent, while the classes
But
but mutually exclusive.
Y = (0
1(
2
Let P denote the Q the second, and
P = Every
),
,
Y R = Some X
;i
Z = (e v
first
R is
is
e
2
,
X
and Z are
this is a mistake, as the
following concrete example will show.
and
when
logicians (I formerly thought so myself) to fail
the class
e
3 ),
Suppose we have
X = («
4>
e
a,
e
6 ).
premise of the given syllogism,
We Q = Every Y
the conclusion.
Z= h Z= 3 >
>/
;
;
get is
X=
three statements,
>/
>;
r
2
;
»/
2,
»/
3
,
TRADITIONAL SYLLOGISMS
§§51,52]
17
each of which contradicts our data, since, by our data in this case, the three classes X, Y, Z arc mutually Hence in this case we have exclusive.
PQ R = :
that,
so
fail
52. Startling
demonstrable
—
/
:
2
>i,)
when presented
Darapti does not
logic
V
(
=
(>i,
in
:
*1
3
)
= {%n^ = e
form of an
the
;
1
implication,
(But see however, it
in the case supposed.
as
it
may
sound,
§
52.) is
a
fact that not one syllogism of the traditional
—
is neither Darapti, nor Barbara, nor any other which it is usually presented in our
valid in the form in
textbooks, and in which, I believe,
it
has been always
In this form,
presented ever since the time of Aristotle.
every syllogism makes four positive assertions it asserts it asserts the it asserts the second the first premise :
;
;
conclusion
i.e.
;
and, by the
conclusion
the
follows
word
'
therefore,'
necessarily
from
it
asserts
the
that
premises,
that if the premises be true, the conclusion must be Of these four assertions the first three may be, also.
true
and often
are, false
the fourth, and the fourth alone, is Take the standard syllogism Barbara.
;
a formal certainty.
textbook form) says this B is C therefore every A is C." every Every A is If valid it this syllogism. denote Let \f/(A, B, C) meanings) we give to (or values must be true whatever
Barbara
(in the usual
B
"
;
;
A—
=
=
camel. bear, and let C ass, let B Let syllogism must following the If \J/(A, B, C) be valid, " bear every bear is is a a Every ass ; therefore be true Is this camel." concrete a camel; therefore, every ass is it contains three Clearly not syllogism really true ?
A, B, C.
:
;
Hence, in the above form, Barbara (here denoted by \/) is not valid for have we not just adduced a case of failure ? And if we give random values to A, B, C out of a large number of classes taken false
statements.
;
haphazard
(lings, queens, sailors, doctors, stones, cities, horses,
French, Europeans, white things, black things, &c, &c), we shall find that the cases in which this syllogism will
SYMBOLIC LOGIC
48
53
[§§ 52,
turn out false enormously outnumber the cases in which it
But
will turn out true.
it is
always true in the following
form, whatever values we give to A, B, C " If every A is B, and every B is C, then every :
A
C."
is
Suppose as before that A = ass, that B = bear, and that C = camel. Let P denote the combined premises, " Every ass is a bear, and every bear is a camel," and let Q denote the conclusion, " Every ass is a camel." Also, let the symbol denote the word therefore. as is customary The first or therefore form asserts P Q, which is .'.
,
,
.".
equivalent* to the twofactor statement P(P:Q); the second or ifform asserts only the second factor P Q. The thereforeform vouches for the truth of P and Q, which are both false the ifform vouches only for the :
;
truth
of
P Q, which, by definition, (See § 10.) a formal certainty.
implication
the
means (PQ'y. and 53. Logicians
is
may
:
say (as some have said), in answer
to the preceding criticism, that
my
objection to the usual
form of presenting a syllogism is purely verbal that the premises are always understood to be merely hypothetical, and that therefore the syllogism, in its general form, is not supposed to guarantee either the truth of the ;
premises or the truth of the conclusion. This is virtually an admission that though (P •'• Q) is asserted, the weaker
statement (P
:
Q)
is
P But why
logicians assert "
the one really meant therefore Q,"
—
that though
they only mean
"
If
P
commonIn ordinary speech, when sense linguistic convention ? we say " P is true, therefore Q is true," we vouch for the truth of P but when we say " If P is true, then Q is true," we do not. As I said in the Athenmum, No. 3989 then Q."
depart from the ordinary
;
:
"
Why
should the linguistic convention be different in logic ? ? Where is the advantage 1 Suppose a general, whose mind, during his past university days, had been overimbued with the traditional logic, were in war time to say, in speaking of an
Where
is
.
.
.
the necessity
untried and possibly innocent prisoner, * I pointed out this equivalence in
'
He
is
a spy
;
therefore
Mind, January 1880.
he
§§ 53,
TRADITIONAL SYLLOGISMS
54]
49
must be shot,' and that this order were carried out to the letter. Could he afterwards exculpate himself by saying that it was all an unfortunate mistake, due to the deplorable ignorance of his subordinates that if these had, like him, received the inestimable advantages of a logical education, they would have known at once that what he really meant was If he is a spy, he must be shot'? The argument in defence of the traditional wording of the syllogism is exactly parallel." ;
'
It
is
no exaggeration
to
are due to neglect of the
say that nearly
hypotheses are accepted as
if
§
If.
Mere
they were certainties.
CHAPTER 54. In the notation of
all fallacies
conjunction,
little
VIII
50, the following are the nine
teen syllogisms of the traditional logic, in their usual As is customary, they are arranged into four order. divisions, called Figures, according to the position of the
middle term " (or middle constituent), here denoted by y. This constituent y always appears in both pre"
The constituent
mises, but not in the conclusion.
the traditional phraseology,
is
z,
in
the " major term,"
called
Similarly, minor term." " major premise," and the premise containing x the " minor premise." Also, since the conclusion is always of the form " All
and the constituent x the the premise containing
X X
is
Z," or "
Some
is
not Z,"
it
is
X
z is
is
Z
"
called the
" or "
No X
usual to speak of
X
and of Z as the predicate.' As usual major premise precedes the minor.
Barbara
=(y
Celarent
= (y = (y = (y
Darii
Ferio
z)(x
as the
'
1
:y):(x:z)
z'){x
:
y)
(x
:
z)
:
1
z)(x z')(,
y')'
:
:
y')'
(x
:
:
Some
subject
in textbooks, the
'
Figure
Z," or "
is
(x
z
:
:
)'
z)
f
D
SYMBOLIC LOGIC
50
Figure
[§
54
2
= (z y'){x y) (x z*) y\x y') (x z) Camestres = Festino = («:/)(« :/)':(*: z)' z)' = (a y)(x y)' Baroko Cesare
:
(:
:
:
:
:
:
Figure
= (y Disamis = (y = (y Datisi Felapton = (y Bokardo = y Ferison = (y
Darapti
(
z)(y
:
:
:
:
(a:
:
:
3 x)
:
,
:
:
z )\y
:
(x
:
x)
z)(y
:
z')(y
:
:
z)\y
:
x)
:
z'){y
:
x')'
a/)'
:
«)
(x
:
:
z')'
:
:
(x
:
{x
z'f
:
:
z)'
:
z)'
(x
:
z')'
:
(a;
:
z)'
:
Figure 4 Bramantip = (z y)(y :
Camenes Dismaris
Fesapo Fresison
= (z = {z = (z = (z
:
x)
:
x')
y)(y
:
y')\y
:
y')(y
:
x)
:
y')(y
:
x')'
:
x)
z
:
(x
:
:
1
(x
:
:
:
(x (x :
)'
z')
:
:
:
(x
z)' z)' :
z)'
the symbols (Barbara),,, (Celarent) M &c. denote, in conformity with the convention of § 44, these nineteen functional statements respectively, when the values of
Now,
let
,
their constituent statements
x. y, z
;
are unrestricted
;
while
the symbols (Barbara),., (Celarent),., &c, denote the same functional statements when the values of x, y, z are restricted The syllogisms (Barbara),., (Celarent),., &c, as in § 50. with the suffix r, indicating restriction of values, are the real
syllogisms
of
the traditional logic
;
and
all
these,
within the limits of the without exception, are valid The nineteen syllogisms of general understood restriction*. logic, that is to say, of the pure logic of statements,
GENERAL LOGIC
545 0]
§§
namely, (Barbara),,,
which
x, y, z
are
more general than and imply nineteen in which x, y, z are restricted as
in values, are
a n restricted
the traditional in § 5
(Celarent),,, &c., in
51
and four of these unrestricted syllogisms, namely, and (Fesapo),,, fail
;
(Darapti),,, (Felapton),,, (Bramantip),,,
certain
in
(Darapti) w
cases.
the
in
fails
7
case
y '(".:)\ /
and (Fesapo) w fail in the case y%ez ) and (Bramantip u fails in the case &(x'yf. 55. It thus appears that there are two Barbaras, two Celarents, two Dai'ii, &c, of which, in each case, the one
(Felapton),,
TI
,
)
belongs to the traditional logic, with restricted values its constituents x, y, z; while the other is a more
of
general syllogism, of which the traditional syllogism
Now,
particular case.
Fw
law
,
as
shown
in § 45,
when
is
a
a general
with unrestricted values of its constituents, implies F,., with restricted values of its constituents,
a general law
the former
if
may
is
true absolutely and never
be said of the
latter.
This
is
fails,
the same
expressed by the
formula F„ F*. But an exceptional case of failure in F„ does not necessarily imply a corresponding case of failure :
in
F,.
FM
e :
for
;
F;
e
though
(which
F r F ,) e
e
e
F,
is
a valid formula, the implication
F;. is
:
,
equivalent to
the converse implica
For example, the general and nontraditional syllogism (Darapti),, implies the less general and traditional syllogism (Darapti),.. tion
:
The former
is
not necessarily valid.
but y\xzj in the traditional syllogism this case cannot occur because of the restrictions which limit the statement Hence, though this case of y to the class 6 (see § 50). fails
the exceptional
in
case
failure necessitates the conclusion (Darapti);;*,
from
this
conclusion,
conclusion
(Darapti);
infer
6 .
the
i
;
we
cannot,
but incorrect, reasoning applies to
further,
Similar
the unrestricted nontraditional and restricted traditional
forms of Felapton, Bramantip, and Fesapo. 56. All the preceding syllogisms, with many others not recognised in the traditional logic may. by means of the formulae of transposition a j3 = /3 r a! and a/3' \y' ay:f$, :
:
=
SYMBOLIC LOGIC
52
57
[§§ 56,
be shown to be only particular cases of the formula Two or which expresses Barbara.
(x'.y)(y:z):(x:z),
examples
three
make
will
this
§
54,
Lut
clear.
<j)(x, y,
Referring to the
denote this standard formula.
z)
in
list
we get
Baroko = (z
=
(.«
y)(x
:
z){z
:
//)'
:
y)
:
:
(x
:
(x
z)'
:
which, by transposition,
;
y) =
:
(f)(x, z, //).
obtained from the general standard is formula
Thus Baroko
we
get
= (y z)(x z) (x if) x). z){z x) (y x') = (p(y, We get (see § 54) Next, take (Darapti) = (y:zx): (xz n)' z)(y x) (x (Darapti),. = = = xz)(xz n (y (y xz){xz = since, by the for, in the traditional logic, (y:rf) = (y = (y
Darii
:
z)(x
:
yj
:
:
(x
:
:
z)'
:
:
:
:
:
z,
:
r.
(//
(//
:
:
:
:
:
:
>j)
z')'
:
:
:
:
>/)
»/)
:
:
con
»/,
S 5 0, y must always be a variable, and, thereThus, finally (Darapti),. = (f)(y, xz, n). always possible. We get Lastly, take (Bramantip),..
vention of lore,
(Bramantip),
= (z = (z = (z
:
y)(y x) :
y){z
:
yx')(yx'
:
:
and therefore
By
y
,
:
:
:
:
:
:
:
:
r,)
:
i]
>i
:
:
//)
r),
Hence,
=
yx\
finally,
we
;
a
get
>/).
similar reasoning the student can verify the list
(see §§
5456):
= Barbara (p(x, y, = Celarent = Cesare y') = Ferio = Festino (p{x, x') = Darii = Datisi = Ferison = Fresison (p(z, y, x) = Camestres = Camenes
(p(x, y, z)
ip(y z
z")'
x)
possible.
(Bramantip),.
following
:
:
>i) :i]
the traditional logic,
for, in
variable
57.
= (z y)(y x ){x z) n = (z yx)(y x) = (z: yx')(yx (z (z:r]) = since z must be (x
:
x')(y
:
z')
;
:
z,
;
;
;
TESTS OF SYLLOGISTIC VALIDITY
§§5759]
z)
x
v)
(p(y
}
,
(p(y, xz,
58. It
is
53
= Disamis = Dismaris
;
= (Felapton) = (Fesapo) n) = (Bramantip),.. evident (since x y = y' x) »/)
:
»/)
r
r
;
that
:
(f)(x, y, z)
:
so
that
=
these
in the preceding list cb(z,' y', x) syllogisms remain valid when we change the order ot their constituents, provided we, at the same time, change ;
all
For example, Camestres and Camenes may each be expressed, not only in the form cp(z, y, x'), as in the list, but also in the form (p(x, y, z). 59. Textbooks on logic usually give rather complicated rules, or " canons," by which to test the validity These we shall discuss further of a supposed syllogism. on (see §§ 62, 63); meanwhile we will give the following rules, which are simpler, more general, more reliable, and their signs.
more
easily applicable.
Let an accented capital letter denote a nonimplication (or " particular "), that is to say, the denial of an impliwhile a capital without an accent denotes a cation Thus, if A denote simple implication (or " universal "). :
Now, let A. B, C denote x y, then A' will denote (x y)' denote their any syllogistic implications, while A', B', Every valid syllogism must have respective denials. one or other of these three forms .
:
:
C
(1)
that
is
AB:C;
AB C r
(2)
to say, either the
:
;
(3)
AB
:
C
;
two premises and the conclusion
three implications (or " universals ") as in (1); or one premise only and the conclusion are both nonimplications (or "particulars") as in (2); or, as in (3), both premises are implications (or " universals "), while are
the
all
conclusion
is
a nonimplication
(or
"
particular
").
If any supposed syllogism does not come under form (1) nor under form (2) nor under form (3), it is not valid that is to say, there will be cases in which it will fail.
;
SYMBOLIC LOGIC
54
The second form may be reduced
59
[§
the
to
first
form by
transposing the premise B' and the conclusion C, and is equivalent to AC B, changing their signs for AB' When thus transAB'C >?. to each being equivalent of AC B, may be is, that C, formed the validity of AB' The of AB C. validity the tested in the same way as in z, be x C to conclusion Suppose the test is easy. example, If, for negative. which z may be affirmative or :
;
C
:
:
:
:
:
:
— He
z
is
z—He
is
a soldier; then
AB
C,
:
not a soldier.
is
a
a soldier; then z' being, by hypothesis, x:z,
C
if valid,
(x
= He
— He
not
conclusion
z'
soldier.
But it The
the syllogism
(see § 11) either
becomes
:y:z):(x:
is
or else {x
z),
y'
:
:
z)
:
(x
:
z),
which the statement y refers to the middle class (or term ") Y, not mentioned in the conclusion x z. If any supposed syllogism AB C cannot be reduced to either if it can be reduced of these two forms, it is not valid a concrete example, take To valid. it is form, to either
in "
:
:
;
be required to test the validity of the following implicational syllogism let
it
:
If
no Liberal approves
of fiscal Retaliation, of fiscal Retaliation
it
of Protection,
do not approve of
Protection.
Speaking of a person taken a
Liberal;
R = He
let
P = He
approves of
the syllogism.
though some Liberals approve who approve
follows that some person or persons
We
at
approves
random,
let
L = He
of Protection;
fiscal Retaliation.
Also, let
is
and let Q denote
get
Q=(L:P')(L:R'/:(R:P)'. To get (see
§
rid of the nonimplications,
56)
affirmative,
change
and thus
their
signs
we transpose them from negative
transforming them into
This transposition gives us
Q = (L:P
,
)(R:P):(L:R').
to
implications.
TESTS OF SYLLOGISTIC VALIDITY
§§59, 00]
55
Since in this form of Q, the syllogistic propositions are all three implications (or " universale "), the combination of premises, (L P')(R:P), must (if Q be valid) be equi:
valent
L P R'
either to
which P
in
:
the letter
is
L
or conclusion
:
:
:
L
or else to
:
P'
:
R'
new consequent L P and P R' premises L P' and
out in the
left
Now, the
R'.
L P R' are not R P in the second
of
:
factors
equivalent to the
:
:
:
or transposed form of the syllogism but the factors L P' and P' R' (which is equivalent to R P) of L P' R' are equivalent to the premises in the second or transformed form of the syllogism Q. :
Q
:
:
;
:
:
:
Hence Q is valid. As an instance of AB C, we may give
a nonvalid syllogism of the form
:
(x:y')(y:z'):(x:z');
two premises have different signs, one being negative and the other affirmative, the combined premises can neither take the form x:y:z nor
for since the y's in the
the
the form x y' :
:
z'
,
which are respective abbreviations
(x>\y){y:z) and (x t y')(y' /). :
The syllogism
is
for
there
fore not valid.
The preceding process
00.
testing the validity of
for
C
apply to all syllogisms of the forms AB C and AB' syllogisms without exception, whether the values of their :
constituents
x,
y,
z
ments.
But
AB
traditional
be restricted, as in the
or unrestricted, as in
logic,
:
my
general logic
of state
as regards syllogisms in general logic of the
C
(a form which includes Darapti, Felapton, in the traditional logic), with Bramantip Fesapo, and and a nonimplicational conpremises two implicational
form
:
clusion, they can only be true conditionally logic
(as
distinguished from the
syllogism of this type
is
;
for in general
traditional
a formal certainty.
logic)
no
It therefore
becomes an interesting and important problem
to deter
SYMBOLIC LOGIC
56
mine the
on which syllogisms of this type can We have to determine two things, firstly,
conditions
be held valid. the
61
[§§ GO,
iveakest
premise
(see
when
which,
33, footnote)
§
joined to the two premises given, would render the syllogism a formal certainty ; and, secondly, the weakest condition which, when assumed throughout, would render
As will be seen, the the syllogism a formal impossibility. general one, which may method we are going to explain is a of the syllogism. be applied to other formulae besides those
AB
The given implication
ABC
implication
:
y,
in
:
C
equivalent to the
is
which A, B, C are three impli
59) involving three constituents x, y, z. Eliminate successively x, y, z as in § 34, not as in finding the successive limits of x, y, z, but taking each cations (see
§
variable independently.
Let a denote the strongest con
clusion deducible from ABC and containing no reference Similarly, let /3 and y respectively to the eliminated x.
denote the strongest conclusions after the elimination of y alone (x being left), and after the elimination of z alone Then, if we join the factor a or /3' (x and y being left). or y' to the premises (ix. the antecedent) of the given implicational syllogism AB C, the syllogism will become :
a formal certainty,
ABa'
:
and therefore
C will be a formal certainty
and AB?' C.
;
premise needed
to
AB
alternative a'
:
C
be joined to valid
+ fi' + y',
datum needed
to
(a
will
is
to say,
AB/3'
+fi'+ y)
C
:
C
:
is
a
so that, on the one hand, the weakest
formal certainty syllogism
and so
;
AB
Consequently,
:
That
valid.
{i.e.
AB
to render the given
a formal
certainty)
the
is
and, on the other, the weakest
make
+ /?' + y
an example 61. Take as Here we have an implication
the
AB
:
,
:
>;
C
:
that
syllogism
:
x),
= M* + N./ + P
(y
:
r,,
a formal a(3y.
is,
C in which
respectively denote the implications (y By the method of § 34 we get
ABC = yx + yz' + xz
AB
the syllogism
impossibility is the denied of a
:
Darapti.
A, B, z),
say,
(x
:
C z).
CONDITIONS OF VALIDITY
§61]
57
which M, N, P respectively denote the cofactor of x, The %', and the term not containing x. in which strongest consequent not involving x is MN + P hero M = z, N = y, and P = yz' so that we have in
the cofactor of
*),
:
;
MN + P
:
= zy + yz' = ye = y n
>/
:
Thus we get a = y: we eliminate x is (y
:
= //( + z') 
1
:
v\.
so that the premise required
>/,
(
n
:
>;/
:
when
and therefore
;
r.x)(y.z)(y.ri)
f
(x:z
should be a formal certainty, which rid of the nonimplications
by
,
t
)
a fact
is
;
getting
for,
complex
transposition, this
implication becomes (y
x)(y
:
which
and
= (y
z){x
:
:
:
z)
xz)(xz
:
:
(y
17),
:
(y
n)
n)
;
this is a formal certainty, being a particular case of
the standard formula
(f)(x, y, z),
which represents Barbara
both in general and in the traditional logic (see § 55). Eliminating y alone in the same manner from AB C, = x z' so that the complex we find that (3 = xz :
:
:
*i
;
implication
{y:x)(y:z)(x:zy:(x:z')'
That it is so is evident by should be a formal certainty. inspection, on the principle that the implication PQ Q, Finally, for all values of P and Q, is a formal certainty. we eliminate z, and find that y = y: n This is the same :
we obtained by the elimination of x, as might have been foreseen, since x and z are evidently inter
result as
changeable.
Thus we obtain the information sought, namely, that «
/
/
+ /3 + 7
/
premise
the weakest
,
premises of Darapti to
make
certainty in general logic /
(y
:
>/)
+ (xz
:
>/)'
+ (//
the formal
be joined
to
this
syllogism
to
a
is
:
•?)',
which
= y*> + (xz)
1
" ;
SYMBOLIC LOGIC
58
[§§ 61,
62
and that a/3y, the Aveakest presupposed condition that would render the syllogism Darapti a logical impossibility,
therefore
is
'
+
,p
/
(,,.,);
j
t
w hich = y\ocz)\
Hence, the Darapti of general values of
constituents
its
x, y,
with
logic,
unrestricted
in the case
fails
z,
y\xzy
;
but in the traditional logic, as shown in § 50, this case The preceding reasoning may be applied cannot arise. to the syllogisms Felapton and Fesapo by simply changing
z into z!
Here we get
Next, take the syllogism Bramantip.
ABC = yx' + zy' + xz and giving
u,
:
>i,
y the same meanings
/3,
we
before,
as
= z\ y = (x'y)\ Hence, a^y — z\xyf, and Thus, in general logic, Braa' + ft' + y' = z~ + (£c'y)~ a
get
=z
r
/3
>,
r
n
'.
mantip is a formal certainty when we assume z~ v + {x'yY*, and a formal impossibility when we assume &{x'yf but ;
assumption
in the traditional logic the latter sible,
z v is
since
inadmissible by
obligatory, since
inadmis
50, while the former
§
is
implied in the necessary assump
is
it
is
tion 2f.
The
62.
validity
traditional
the
of
tests
logic
turn
mainly upon the question whether or not a syllogistic In undistributed.' or distributed term or class is to ever, lead rarely, if words these language ordinary logicians thought but of confusion or any ambiguity have somehow managed to work them into a perplexing '
'
'
'
'
;
tangle.
In the proposition
said to be
'
distributed,'
class
Y
position said
to
position '
Some
be "
X
All
X
is
X
'
undistributed,'
X
X
Y," the class
is
and
Y
the class
Y
'
is
X
and the
X
and the
In the proclass
Finally, in
not Y," the class
X
undistributed.'
'
distributed.'
both 'undistributed.'
Some
Y," the class
is
class
Y," the class
is
be both
are said to "
No
"
In the proposition
"
and the
is
distributed.'
Y
are
the pro
said to be
§ 6
2]
<
— UNDISTRIBUTED
DISTRIBUTED
,
59
<
Let us examine the consequences of this tangle of Take the leading syllogism Barbara, the technicalities. validity of which no one will question, provided it bo expressed in
conditional form, namely, "
its
If
Y
all
is
Z,
Y, then all X is Z." admittedly valid, this syllogism must hold good whatever values (or meanings) we give to its conIt must therefore hold good when stituents X, Y, Z. X, Y, and Z are synonyms, and, therefore, all denote the In this case also the two premises and the same class.
and
X
all
(see §
Being, in this form
is
52),
three truisms which no one would Consider now one of these truisms,
conclusion will be
dream
of denying.
X is Y." Here, by the usual logical convention, X is said to be distributed,' and the class Y But when X and Y are synonyms they undistributed.'
say
"
All
the class 1
'
denote the same class, so that the same class may, at the same time and in the same proposition, be both disDoes not this sound like tributed' and 'undistributed.' '
a contradiction
Speaking of a certain concrete
?
collec
tion of apples in a certain concrete basket, can we consistently and in the same breath assert that " All the
apples are already distributed are
'still
undistributed "
"
and that
Do we
?
"
All the apples
get out of the
dilemma
and secure consistency if on every apple in the basket we Can we then constick a ticket X and also a ticket Y ? sistently assert that all the
that
all
every apple.
X
the
Y
apple
X
apples are distributed, but Clearly not for ?
apples are undistributed is
Y
also a
apple,
Y
apple an
X
In ordinary language the classes which we can
and
respectively qualify
as
mutually exclusive
in the logic of
is
;
and every
;
evidently not the
distributed
undistributed
are
our textbooks this Students of the traditional
case.
minds of the idea necesundistributed and that the words distributed do in they as exclusive, mutually sarily refer to classes forced but a anything is there everyday speech or that and fanciful connexion between the distributed and
logic
should
therefore disabuse their '
'
'
'
;
'
'
SYMBOLIC LOGIC
60 '
undistributed
'
distributed
'
current English and the undisturbed of logicians.
of
and
'
[§
technical
'
'
Now, how came the words tributed to be employed by '
'
distributed
and
'
'
logicians in a sense
plainly does not coincide with that usually given "
Since the statement
statement "All
X
is
No X
"Y," in
is
Y"
which
them
?
4650) the
(see §§
Y (or nonY) contains all Symbolic Universe excluded from the
undis
equivalent to the
is
which
the individuals of the
class
"
02
Some
X
is
not
definitions of
Y
" is
equivalent
distributed
'
'
and
'
to "
and since
class Y,
Some
X
undistributed
is '
*Y," the
in text
books virtually amount to this that a class X is distributed with regard to a class Y (or *Y) when every individual of the former is synonymous or identical with :
some individual
or other of the latter
;
and that when
then the class X is undistributed with Hence, when in the stateregard to the class Y (or'Y). ment " All X is Y " we are told that X is distributed with regard to Y, but that Y is undistribided with regard to X, this ought to imply that X and Y cannot denote exactly this is not the case,
In other words, the proposition that to imply that " Some Y is not X." But as no logician would accept this implication, it is distributed clear that the technical use of the words
the "
same
X
All
is
class.
Y"
ought
'
and
'
undistributed
lacking
'
to
linguistic
in
be found in logical treatises is In answer to this
consistency.
criticism, logicians introduce psychological considerations
and say that the proposition " All X is Y " gives us information about every individual, X 1; X 2 &c, of the class X, but not about every individual, Y v Y 2 &c, of the class Y and that this is the reason why the term X is said to be To this 'distributed' and the term Y 'undistributed.' ,
,
explanation it may be objected, firstly, that formal logic that its forshould not be mixed up with psychology mulae are independent of the varying mental attitude of individuals and, secondly, that if we accept this informationgiving or nongiving definition, then we should
—
'
;
'
'
'
'DISTRIBUTED'— UNDISTRIBUTED
§62]
X
say, not that
X
that
distributed,
is
known or
is
known
not
1
fil
<
Y
and
undistributed, but
Y
inferred to be distributed, while
—
to be distributed
is
that the inference requires
further data.
To throw symbolic light upon the question we may With the conventions of 8 50 we
proceed as follows.
have (1) All
Some X
(3)
The
Y = (x
:
'
'
No X is Y = x // Some X is not Y = (x
(2) (4)
//)';
positive class (or
logicians
the
X is Y = x:y;
is
term
')
:
X
is
predicate.'
It
//)'.
usually spoken of by
the subject'; and the positive class
as
:
Y
as
be noticed that, in the above
will
examples, the nonimplications in (3) and (4) are the respective denials of the implications in (2) and (1). The definitions of
'
distributed
and
'
'
undistributed
are
'
as
follows.
The
term ') referred to by the antean implication is, in textbook language, said to distributed and the class referred to by the conse
(a)
(or
class
'
cedent of
be
'
'
;
quent
is
(/$)
said to be
The
'
undistributed.'
class referred
implication
is
to
said to be
'
by the
antecedent of a non
undistributed
referred to by the consequent
is
and the
;
said to be
'
class
distributed.'
to (1) and (2); definition and (4). Let the symbol X d assert that X is distributed' and let X u assert that X is undistributed.' The class 'X being the complement of the class X, and vice versa (see 8 46), we get (*X)* = XM and (X)" = X d From the definitions (a) and (/3), since (Y) d = Y", and ( Y) u = Y d we therefore draw the following
Definition
(/3)
applies
applies
(a)
to
(3)
'
'
,
.
y
,
four conclusions
In
XY u
X d Yu
(1)
d .
For
in
:
in
;
Xd Y d
(2)
;
in (3)
XUY U
(2) the definition (a) gives us Similarly, in (3) the definition
:
in
(4)
Xd Yf r
(
and CY) u = Y d (/3) gives us X u CY) d and ( Y)d = YM If we change y into x in proposition (1) above, we .
,
,
.
SYMBOLIC LOGIC
62
[§§ 62,
63
X is X "=x:x. Here, by definition (a), we have which shows that there is no necessary antagonism between X and X" that, in the textbook sense, the same class may be both distributed and undistributed at the same time. get " All
X dX"
;
rf
;
'
'
'
'
63. The six canons of syllogistic validity, as usually given in textbooks, are (1) Every syllogism has three and only three terms, namely, the major term,' the minor term,' and the :
'
'
middle term (see § 5 4). (2) Every syllogism consists of three and only three propositions, namely, the major premise,' the minor premise,' and the 'conclusion' (see § 54). (3) The middle term must be distributed at least once in the premises and it must not be ambiguous. (4) No term must be distributed in the conclusion, unless it is also distributed in one of the premises.* (5) We can infer nothing from two negative pre'
'
'
'
;
mises. (6) If one premise be negative, the conclusion must be so also and, vice versa, a negative conclusion requires one negative premise. Let us examine these traditional canons. Suppose The syllogism \//('', y, z) to denote any valid syllogism. being valid, it must hold good whatever be the classes to which the statements x, y, z refer. It is therefore valid when we change y into x, and also z into x that is to ;
;
say,
\/(.'",
a case
,/',
:>,)
valid
is
(§
13,
Yet this is and needsimply a definition, and
footnote).
which Canon (1) appears
arbitrarily
Canon (2) is comment. The second part of Canon (3) all arguments alike, whether syllogistic or not.
lessly to exclude.
requires no applies to
*
Violation of
Canon
(4) is called
"Illicit Process."
is
called " Illicit Process of the Major "
tributed in the conclusion
Process of the Minor " (see
is
;
the term
the major term, the fallacy
when
the term illegitimately dis
the minor term, the fallacy
§ 54).
When
is
illegitimately distributed in the conclusion
is
called " Illicit
'CANONS
§63] It
is
evident that
1
if
OF TRADITIONAL LOGIC we want
ambiguities.
63
we must Canon (3)
avoid fallacies,
to
The
part of
also avoid The rule about cannot be accepted without reservation. distribution does not apply middleterm the necessity of " If every X is syllogism, perfectly valid to the following that is not X something then Y, and every Z is also Y, expressed may be syllogism Symbolically., this is not Z." first
in either of the two forms
(x.y){z:y):{x :z)'
(1)
{xy'nzyj'.ix'z'r
(2)
Conservative logicians who still cling to the old logic it impossible to contest the validity of this syllogism, refuse to recognise it as a syllogism at all, on the ;
finding
ground that
has four (instead of the regulation three) the last being the class containing all the individuals excluded from the class X. Yet a mere change of the three constituents, x, y, z, of the syllogism Darapti (which they count as valid) into their denials x', //, z' makes Darapti equivalent to the it
terms, namely, X, Y, Z,
above syllogism.
%
For Darapti
is
{y:x\y:z):{x:zy
(3);
_
and by virtue of the formula a (l) in question becomes
:
(3
= /3'
a, the syllogism
:
(/:*')(/ :*'):(*':*)' Thus,
if
\^(f;, y, z)
denote
(4).
Darapti,
then
y\s(x', //', ;')
denote the contested syllogism (1) in its form (4); and, vice versa, if ^(x, y, z) denote the contested syllogism, namely, (1) or (4), then ^(a/, y z') will denote will
',
To
Darapti. class
X
is
that
class 'X.
class,
be read,
Hence,
if
we
is
it is
call
not
in the
in the
com
the class 'X the
the syllogism in question, namely,
(/:./)(/:/)
may
any individual
equivalent to asserting that
plementary
nonX
assert
"
:(,/:*)'
If every
nonY
(4), is
a nonX, and every non
SYMBOLIC LOGIC
64
[§
03
For then some nonX is a nonZ." , z )' which asserts that it is possible for an individual to belong at the same time In both to the class nonX and to the class nonZ. Thus other words, it asserts that some nonX is nonZ.
Y
also a nonZ,
is
(x':z)'
is
r>
equivalent to (./
,
becomes a case of Darapti,
read, the contested syllogism
Z
being replaced by their respective It is evident that complementary classes 'X, 'Y, 'Z. when we change any constituent x into x in any syllothe classes X, Y,
gism, the words
change
'
distributed
and
'
'
undistributed
inter
'
places.
Canon
(4)
of the traditional logic asserts that "
No
term' must be distributed in the conclusion, unless it is This is another also distributed in one of the premises."
Take the
canon that cannot be accepted unreservedly. syllogism Bramantip, namely, (z
and denote the
within
by
it
:
y)(y x) :
:
z')'
Since the syllogism
\f/(V).
restrictions
(x
:
of
the
traditional
is
logic
valid (see
should be valid when we change z into /, and We should then get consequently z into z. § 50),
it
>},{/)
Here
(see § 02)
= (*' :y)(y:x):(x:z)'.
we get Z w
in the first premise,
and Z
rf
the conclusion, which is a flat contradiction to the Upholders of the traditional logic, unable to deny the validity of this syllogism, seek to bring it
in
canon.
within the application of Bramantip by having recourse to distortion of language, thus " If every nonZ is Y, and every Y is X, then some X :
is
nonZ."
Z" in d premise and Z in the conclusion, which would contradict the canon, would have ( Z)'' in the first premise and ( Z) u in the conclusion, which, though it means exactly the same thing, serves to "save the face" of the canon
Thus
the
treated, the syllogism, instead of having
first
V
y
and
to hide its real failure
and
inutility.
§
TESTS OF SYLLOGISTIC VALIDITY
G3]
Canon
(5) asserts that "
A
two negative premises."
Avhich into
The example
is
:0(^*') :(*':*)',
obtained from Darapti by simply changing
is
and x into x
z',
can infer nothing from show the
single instance will
unreliability of the canon. (2,
We
65
It
.
may
"
be read,
If
Y
no
is
z
X,
and no Y is Z, then something that is not X is not Z." Of course, logicians may " save the face " of this canon " If also by throwing it into the Daraptic form, thus all Y is nonX, and all Y is also nonZ, then some nonX is nonZ." But in this way we might rid logic of all negatives, and the canon about negative premises would then have no raison d'etre. Lastly, comes Canon (6), which asserts, firstly, that " if one premise be negative, the conclusion must be :
negative
and,
;
secondly,
that
requires one negative premise."
negative
a
The
conclusion
objections to the
preceding canons apply to this canon also. In order to give an appearance of validity to these venerable syllogistic tests, logicians are obliged to have recourse to distortion of language, and by this device they manage to
make
their negatives look like affirmatives.
But when
logic has thus converted all real negatives into
affirmatives the canons about negatives
through refer.
want of negative matter to which they can The following three simple formulae are more
easily
applicable and will supersede
canons
:
(1) (a: (2)
(z
:
first
y x)
:
(x
:
the
traditional
Barbara.
Bramantip.
z)'
....
Darapti.
of these is valid both in general logic
the traditional logic
;
and
in
the second and third are only valid
in the traditional logic. all
all
:z):(x:z) :
(3) (y:x)(y:z):(x:z')'
The
seeming
must disappear
Apart from
this limitation, they
three hold good whether any constituent be affirmaE
SYMBOLIC LOGIC
66 tive
or negative,
and
64
[§§ 03,
whatever order we take the
in
Any
syllogism that cannot, directly or by the /3' a and a/3' y' ay fi, formulae of transposition, a /3 letters.
=
:
=
:
:
be brought to one or other of these forms
:
is invalid.
CHAPTER IX Given one Premise and the Conclusion, to find the missing Complementary Premise.* 64. When in a valid syllogism we are given one premise and the conclusion, we can always find the complementary premise which, with the one imply the conclusion. AVhen the given conclusion is an implication (or " universal ") such as x z or x z\ the complementary premise required is found For example, suppose we readily by mere inspection. f have the conclusion x:z and the given major premise The syllogism required must be z y (see § 5 4).
weakest
given,
will
:
:
:
either {x:y :z'): (x
:
z')
or (x y :
r :
z')
:
(x
:
z'),
The major prethe middle term being either y or y'. is which is not equivalent mise of the first syllogism y z' ',
:
Hence, the first syllomajor premise z y. The major premise of the gism is not the one wanted. y' z', and this, by transposition and second syllogism is change of signs, is equivalent to z y, which is the given major premise. Hence, the second syllogism is the one wanted, and the required minor premise is x y' to the given
:
:
:
:
When
the conclusion, but not the given premise, is a nonimplication (or " particular "), we proceed as follows. Let P be the given implicational (or " universal ") premise, and
C the given nonimplicational (or "particular")
conclusion. *
A
Let
W be the required weakest premise which,
syllogism with one premise thus left understood
enthymeme.
is
called an
§§ G4,
TO FIND A MISSING PREMISE
05]
joined to P, will imply
We
C.
have
shall then
which, by transposition, becomes
PC W. :
67
PW
C,
:
Let S be the We shall then
strongest conclusion dcducible from PC. have both PC S and PC W'. These two implications having the same antecedent PC, we suppose their consequents S and W' to be equivalent. We thus get S = = S'. The weakest 'premise required W', and therefore :
:
W
therefore
is
PC
from
denial of the strongest conclusion dedueible
the
and
{the given premise
the
of the given
denial
conclusion).
For example,
the given premise be y
let
given conclusion (x
We
r
z )'
:
.
:
x,
and the
are to have
(y:x)W:(x:z'y. Transposing and changing signs, this becomes \{y:x){x:z')'.W. But, by our
fundamental
syllogistic
formula,
we have
also (see § 5G)
(y:x)(x:z'):(y:z').
We
therefore assume f
(y (y
:
:
z
f
)
W=
y:z' and, consequently, )
The weakest premise required
.
//, and the required syllogism (//
:
%)(y
*')'
(«
The only formulae needed complementary premise are 65.
The
*
is
W=
therefore
is
«')' :
in finding the weakest
= (3':a'.
(1)
a:(3
(2)
(a:/3)(/3:
7 ):(a: 7 ).
(3) (a:/3)(a:
7 ):(/3 7 r\
two are true universally, whatever be the statethe third is true on the condition a*, (3, y that a is possible a condition which exists in the first
ments
a,
;
—
* The implication y «, since would also answer as a premise footnote, and § 73).
in the traditional logic
:
;
but
it
it
implies (y
would not be the weakest
:
s')',
(see § 33,
SYMBOLIC LOGIC
68
[§§ 65,
any of the statements
traditional logic, as here
66
a, (3,
y
represent any of the three statements x, y, z, or any every one of which six stateof their denials x y', z ments is possible, since they respectively refer to the six
may
,
,
%Y
Z, every one of which classes X, Y, Z, stood to exist in our Universe of Discourse.
is
under
Suppose we have the major premise z:y with the z')' and that we want to find the weakest complementary minor premise W. We are to have
conclusion (x
:
',
(z:y)W:(x:z'y, which, by transposition and change of signs, becomes
(z:y)(x:z'):W. This,
by the formula a
:
/3
= ft'
:
a
,
becomes
(z:y)(z:x'):W.
But by Formula
(3)
we have
also
(z:y)(z:x'):(yx'y.
We therefore assume W' = (yz')' and consequently W = (yx'y = y:x. The weakest minor premise required 71
,
is
therefore y x :
and the required syllogism
;
:
y)(V
.')
')'
('• :
is
:
As the weakest which is the syllogism Bramantip. premise required turns out in this case to be an implication, and not a nonimplication, it is not only the weakest complementary premise required, but no other complementary premise is possible. (See § 64, second footnote.) 66. When the conclusion and given premise are both nonimplications (or " particulars "), we proceed as follows. Let P' be the given nonimplicational premise, and
C
W
denotes the the nonimplicational conclusion, while shall required weakest complementary premise.
We
C
or then have P'W transposition. obtain by :
its
equivalent
WC
The consequent P
:
P,
which we
of the second
§§66, 66
THE STRONGEST CONCLUSION
(a)]
69
being an implication (or " universal ") we have only to proceed as in § 64 to find W. For example, let the given nonimplioational premise be (// z)'\ and implication
:
the given nonimplicational conclusion {x
:
z)'.
:
z
We
are
have
to
(yri/W :(*:«)'. By
becomes
transposition this
W(x:z):(y:z).
The
missing in the consequent y P must therefore be
letter
syllogism
WC
is
The
x.
:
either (y x z) :
:
(y
:
z)
:
or else (y:x':z):(y:z);
one or other of which must contain the implication C, which the given nonimplicational conclusion C, re
of
presenting (x
:
and not the second contains
that
W=y
Hence
to
position,
for it is the first
;
:
WC
Now,
x.
:
:
P
of these two syllogisms,
first
the implication
WC
The syllogism
the denial.
is
z)',
must therefore denote the
P
and not the second
or
C, is
its
synonym x
:
z.
equivalent, b}r trans
WP' C, which is the syllogism required. W, P', C, we find the syllogism sought :
Substituting for to be
(//
:
*)'
'<)(>/
(?
:
*)',
and the required missing minor premise to be y x. 66 (a). By a similar process we find the strongest conclusion derivable from two given premises. One Suppose we have the combination example will suffice. Let S denote the strongest of premises (z y)(x y)' :
:
:
conclusion required. (z
:
y){x
The
:
//)'
letter
:
S,
'.
We
get /
which, by transposition,
is
(z
:
//)S
:
(x
:
y).
missing in the implicational consequent of the
second syllogism must be
is
z,
so
that
either x z y or else x :
:
antecedent
its
:
z'
:
>/.
(z
:
y)S
/
SYMBOLIC LOGIC
70 first
so that its other factor x
y,
by
antecedent
is
:
Hence, we get S'=x:z, and S
S'.
G7
(a),
the one that contains the factor z must be the one denoted
The z
:
6G
[§§
strongest * conclusion required
= (#:«)'.
therefore (x
is
The
z)''.
:
CHAPTER X
We
will now introduce three new symbols, Wcp, which we define as follows. Let A v A 2 A 3 A m be m statements which are all possible, but of which Out of these m statements let it be one only is true. A r imply (each sepaunderstood that A r A 2 A 3 A s imply that A r+1 Ar+2 A.,. +3 rately) a conclusion cp cp' and that the remaining statements, A s+1 As+2 A m neither imply cp nor cp'. On this understanding we 6 7.
Yep, Sep,
,
,
.
,
,
,
.
the following definitions
(5) (6)
W'cp means
W(/)
.
2
1
.
.
.
:
=A +A +A + +A W^) = Ar+1 + Ar+2 + ... +A V4> = V<£' = A s+1 + Ag+2 + ... +A m S^ = W^ + V^ = W
(1)
.
,
,
down
.
.
.
;
lay
.
.
.
;
.
,
,
.
.
r
3
,
(2) (3)
S
.
.
,
(4)
(7)
.
means
S'
(W(f>)',
The symbol Wcp denotes the cp
;
while
Sep
than
A+
A + BfC,
denotes
weakest statement that implies
the
33, footnote). B, while A + B
implies (see
the denial of W.
(S<£)', the denial of Sep.
strongest
As
§
and
so
on,
we
is
is
statement
formally
that (p implies.
cp,
and
that
stronger formally
stronger
are justified in
the weakest statement that implies strongest statement
A
calling
than
Wcp
in calling S(p the
Generally
Wcp and
Sep
* Since here the strongest conclusion is a non implication, there is no other and weaker conclusion. An implicationcU conclusion x z would also admit of the weaker conclusion (x z')'. :
:
EXPLANATIONS OF SYMBOLS
68]
§§ 67,
present themselves as logical sums or alternatives
may
exceptional cases, they
From
terms. formulae, r
(\\ (^
= S<£ =
the
(£).
preceding
The
but, in
present themselves as single
W^S'0;
(1)
;
71
we
definitions
= W'<£;
S4>'
(2)
the
get
(3)
V°<£
=
last of these three formulas asserts
that to deny the existence of Y(p in our arbitrary uni
A A2
verse of admissible statements,
and
to affirming that W<^>, Sep,
,
(p
are
&c,
,
all
is
equivalent
three equivalent,
The statement Y°
,
.
.
.
,
;
that
an
is
it
impossibility,
in
or,
contradicts our data or definitions.
may be
true
;
Y
the statement
v
other words,
that
it
The statement Y°cp The cannot be true.
statement Y°
,
.
.
.
>/
every statement of the class
implies both
tj
;
and
',
since
a is always true, proved in § 18) the implication The statewhatever be the statement represented by a. ment Y^cp also contradicts the convention laid down that
(as
all
>/
the statements
A A2
we may have W°<£ 68.
,
or
,
W
.
.
A w are
.
:
possible.
Similarly,
will illustrate
the mean
^/.
The following examples
Suppose our ings of the three symbols Wcp, Y(p, Sc£. " hypotheses to consist universe ") of possible total (or multiplication of of the nine terms resulting from the 9 The the two certainties A' + A^ + A and B« + B" + B product is fl
.
A B + A^ + A B" + A^B' + A"B" + A"B* + A*B + A B" + A B e
e
e
e
e
9
9
.
SYMBOLIC LOGIC
72 Let
(p
e denote (AB)
We
.
68
get
W(AB)* = A B« + A B S(AB) e = A*B 9 + A*B + A e B* = A""B e + A e B"". W(AB) e = S , (AB) 9 = A" + B + A B (See § 9 9 e S( AB) = W'(AB) = A" + B" + A'B' + A B €
(1)
[§
e
fl
.
f
(2)
e
T
(3)
f
'
.
fl
(4)
.
§
The
first
A
of the above formulae asserts that the weakest
we can conclude
AB +A B
certain
B
and
9
e
the alternative is
(See
69.)
data from which is
69.)
e
AB
that
is
a variable
€
which affirms that either else A variable and B certain. ,
variable, or
The second formula asserts that the strongest conclusion we can draw from the statement that AB is a variable is
the alternative
A
is possible
A B + A^^, 9
_T?
B
which
A
asserts that either
and B possible. Other formulae which can easily be proved, when not evident by inspection, are the following and
variable, or else
variable
:
(5
(6
W<£
:
(p
:
S(f>.
= Sep) = (Wdj =
<j>).
e
e
£
e
(8 (9
(10 (11
(12 (13
(14 (15
(16 (1<
(18
e
.
6
e
e
e
e
.
e
e
I)
e
9
£
9
e
e
!
e
.
:
e
6
:
£
:
e
:
f
:
:
.
The formulae (15) and (16) may evidently be deduced from (13) and (14) by changing B into B'. Formula (17) asserts that the weakest data from which we can
APPLICATIONS OF SYMBOLS
G9]
§§ 68,
A
conclude that
A
either
B
:
B)'
the alternative that
is
and B uncertain, or else A possible and The formula may be proved as follows
certain
is
impossible.
W(A
B
does not imply
73
= S'(A B) = (A" + B + A BV = (A") (B') = A^B^A" + B' = A*B + A"B" e
,
fl
/
:
e
9
(A e B e )
/
e
)
for,
A^A^M and B
evidently,
B = B". may be proved e
e
from first § 68 though some may be deduced more readily
69. All the formulae of principles,
Take, for example,
from others.
We
(1), (2), (3).
are
We first S(AB) W(AB)" 9 the product constitute which terms nine the write down required
W(AB)
find
to
fl
fl
.
,
,
A + A" + A
and B' + B" + B as term that implies every underdot we done, This in § 68. 9 variable we underline is a AB that asserts (AB) which 5 that AB is asserts which (AB)" implies every term that term every brackets in enclose we not a variable; and 9 thus We get (AB)nor (AB) that neither implies e
of the two certainties
fl
fl
,
;
,
,
.
A B + A'B' + A B + A"B + A»B*» + A"B + A B + A B" + (A B e
e
€
1
9
e
9
9
9
9
e
9
).
By our
definitions in § 67
we thus have
W(AB) 9 = A B + A B By
9
9
£
definition also Ave
e
(1)
9 have V(AB) 9 = A B 9 and therefore ,
+ V(AB) = A B + A°B + A B = A B + A B + A B + A B for a = a + a = + A )B + A*(B' + B = A"B (2). + A B"
S(AB) 9 = W(AB) 9
9
e
e
9
e
9
9
f
9
9
9
e
9
9
,
e
9
9
(
9
9
)
fl
We may similarly deduce
(3) and (4) from first principles, more easily from the two deduced but they may be
formulae
+ ^) = W(£ + Wxfr S(
....
(a) (£)>
SYMBOLIC LOGIC
74
[§§ 69,
70
as follows
+ AB)" = W(AB)« + W(AB/> = A B + A" + B", from § 08, Formulae 7, 13. = S { AB) + AB)" } = S( AB) + S(AB)» = A B« + A" + B" + A B from § 08, Formulae
W(AB)" = W{(AB) C
S( AB)
9
f
(
(
€
£
e
(
(
e
e
9
,
14.
7,
The
70.
following
is
an example of inductive, or rather
inverse, implicational reasoning (see §§ 11, 112). The formula (A x) + (B x) (AB x) is always :
when (if ever) is the (B
:
x), false
while
We
denotes
converse, implication
Let
?
true
:
:
:
(AB
:
x)
:
(A
:
x)
;
+
denote the first and valid formula, converse formula to be examined.
its
get
=(ABxy:(Ax'y + (Bx'y
= (Ax' = (a(3) Hence
(see e
r,
a
Bx'f :
§
11),
7,
fir
{Ax'y
we
i
+ (Baj')" Ax, and
putting a for
!
(a/3)"a"/3~"
!
it
for Bx'.
(a/SjV/S*
Bx')\kxy(Bxy (ABxy(Ax') (Bxy !
implication
(p c
the
in
fails
case
which represents the statement
(ABa/yCAa/r^V and
(3
get
+ /3")'
converse
the r,
.
:
+ ffr,
oP
(Ax'
!
(a{$)
<
(a/3)Xa"
I
(f>'
Thus,
.
r
•
•
the case (afiy>a
therefore also fails in
•
•
9
fi
,
(
1 );
which
represents the statement (ABa/)"(A#')"(Ba/)6 for the
....
(2)
The failure second statement implies the first. may be illustrated by a diagram as
of
(A) implies that the individual represented by A has a real Firstly,
on opposite page.
Out
of the total ten points
take a point A, B, x
P
assert
at
marked
random, and
respectively
(as
in this diagram,
the three symbols propositions) that the let
§§ 70,
CERTAIN DISPUTED PROBLEMS
71]
75
be in the circle A, that P will be in the It is evident circle B, that P will be in the ellipse x. propositions A, B, four the of chances that the respective 2 variables. all are that they so x, AB are T T%, £>> T o It is also clear that the respective chances of the three
point
P
will
%
;
statements AB./,
9
have (ABx'y(Axy(Bx') we found to be insr, ,
We may
failure.
by
direct
as
follows.
asserts
appeal
in both the circles
A
diagram,
a
is
B
and
being also in the ellipse
ment which
this
AB
x
:
P cannot be
point
;
of
show
the
The implication
that the
,
case
a
also to
^
2 so that we also iG reasonsymbolic pure by which,
Bx', are 0,
Axe',
W
without a state
x,
material
certainty,
from the The implication diagram (see § 109). A x asserts that P cannot be in A without being in x, a statement which is a material impossibility, as it is and B x is inconsistent with the data of our diagram Thus we have AB x = e, impossible for the same reason.
as
it
necessarily
follows
special data of our :
:
;
:
A
:
x = v\,
B
ip cf) c
:
x
=
»/,
so that
= (A x) + (B = AB x) (A :
(
x)
:
:
:
we
:
:
x)
get
(AB
+ (B
:
:
= + v *= e x) = e n + n = h
x)
:
>i
>
:
and (p c equivalent, because they draw no distinction between the true (t) and the certain (e), nor between the false (i) and the Every proposition is with them either impossible (>/). propositions which I call or impossible, the certain
The Boolian
variables (6)
logicians
consider
being treated as nonexistent.
ing illustration
makes
it
clear that this
is
The preceda serious
and
fundamental error. 71. The diagram above will also illustrate two other propositions which by most logicians are considered equivalent, but which, according to
the word
if,
arc not equivalent.
my
They
interpretation of are the
complex
SYMBOLIC LOGIC
76 conditional, " If
A
is true, "
the simple conditional, is true!'
Expressed in
pretation
of the
70
point P),
the
that
it
is
is
x
true
is
true" and
are both true, then
and with
my
x
inter
10), these con
§
A (B x) and AB x. Giving x, AB the same meanings as in :
:
:
the
to
evident that
same
B
subject,
which
x,
:
random point P cannot be
the
asserts
B
the circle
in
ellipse x, contradicts our data,
without being also in the
and
is
B
notation,
having reference
(all
B
and
conjunction if (see
to the propositions A, B,
random
A
my
ditionals are respectively
§
then if
If
72
[§§ 71,
The statement A, on the
therefore impossible.
other hand, does not contradict our data
neither does
;
its
denial A', for both, in the given conditions, are possible
though uncertain. Hence, A is a variable, and B x being impossible, the complex conditional A (B x) becomes which is equivalent to 0", and therefore an im6 But the simple conditional AB x, instead of possibility. :
:
:
:
1},
:
being impossible,
in the given conditions, a certainty,
it is clear from the figure that P cannot be in both Hence, though and B without being also in x.
for
A A
is,
(B x) always implies AB x, the latter does not always imply the former, so that the two are not, in all cases, :
:
:
equivalent.
A
72.
the
"
question
Existential
make an
much
affirmation
same time,
AB
"
No A
is
or a denial A"
,
B ,
B
Do
?
do we, at the
"
Some A
is
B,"
"
A
?
Do we
the four technical propo
the traditional logic, namely, B,"
is
When we
Propositions."
of
implicitly affirm the existence of
affirm the existence of sitions of
discussed amongst logicians
Import
Some A
is
"
All
A
is
B,"
not B," taking
each separately, necessarily imply the existence of the Do they necessarily imply the existence of the class A ? My own views upon this question are fully class B ? explained in Mind (see vol. xiv., N.S., Nos. 5355); here The convention a brief exposition of them will suffice. of a
"Symbolic Universe"
(see
leads to the following conclusions
§§ :
4650)
necessarily
EXISTENTIAL IMPORT
§§72,73]
77
when any symbol A denotes an individvM any intelligible statement
;
then,
;
existence depends
Secondly,
upon the
context.
when any symbol A denotes
a
class,
then,
statement <£(A) containing the symbol A implies that the whole class A has a symbolic existence but whether the statement (p(A) implies that the class
any
intelligible
;
A
is
wholly
unreal,
wholly unreal, or partly real and partly
real, or
depends upon the context.
As regards
this question of
one important point in which other symbolists
the
is
"
Existential Import," the I
appear to
The
following.
differ
from
null class
0,
which they define as containing no members, and which I,
for convenience of symbolic operations, define as con
the null or unreal
sisting of is
members
V
2,
3,
&c,
be contained in every class, real consider it to be excluded from every
understood by them
to
whereas I Their convention of universal inclusion leads real class. " Every to awkward paradoxes, as, for example, that form squares round because triangle," round square is a conbe to understood is them) (by a null class, which
or unreal
;
tained in every
class.
My
convention leads, in this case,
to the directly opposite conclusion, namely, that "No round square is a triangle," because I hold that every
purely unreal class, such as the class of round squares, is necessarily excluded from every purely real class, such as the class of figures called triangles.
73. Another paradox which results from this convention of universal inclusion as regards the null class 0, "
paradox that the two universals " All X is Y " and No X is Y " are mutually compatible that it is possible for both to be true at the same time, and that
is
their
;
this is necessarily the case
nonexistent.
My
when
convention of a
the class "
X
is
null or
Symbolic Universe
"
SYMBOLIC LOGIC
78 leads,
on the contrary,
to
the commonsense conclusion "
of the traditional logic that the two propositions
X
74
[§§ 73,
All
and " No X is Y " are incompatible. This may be proved formally as follows. Let (p denote the proposition to be proved. We have is
Y"
= (x:y)(x:y ):v = (xy )\xyy:f = (V + xy = {,/•(/ + y) = (x = (xe — (6 /
/
]
(t>
:
tj)
:
>])
:
t]
:
>/
tj)
:
:
tj
:
assumed
In this proof the statement x
is
by the convention
See also
noticed that lent to {x y) :
implies
"
(x
Some
46.
§
:
>/}
y')' :
which
',
X
is
n
:
>/
:
to
be a variable It will
5 0.
§
the proposition just proved,
(p, :
of
tj)
asserts that
"
All
be
equiva
is
X
Y"
is
Y."
Most symbolic logicians use the symbol A~< B, or some other equivalent (such as Schroeder's A=£ B), to 74.
A
assert that the class
is
wholly included in the class
B
and they imagine that this is virtually equivalent to my symbol A B, which asserts that the statement A implies That this is an error may be proved the statement B. :
easily
as
equivalent to the statement
A
hold good when the statement
>/
:
statement
the
If
follows.
A
denotes
by
e,
A B
be always
:
< B, the equivalence
>;,
and
definition,
B
denotes
e.
must Now,
synonymous with
is
which only asserts the truism that the impossibility (For the compound statement yja, an impossibility. whatever a may be, is clearly an impossibility because But by their definition it has an impossible factor tj.) (ye'y, r\e
is
the statement
n < e
included in the class
asserts that the class e;
that
to say,
is
>?
wholly
is
asserts
it
that
every individual impossibility. v 2 3 &c, of the class e or e &c.) of the is also an individual (either e 3 r or 2 e is a Thus, which is absurd. class of certainties e tj
>/
,
>;
,
,
>j
;
formal certainty, whereas (See 8 18.)
>;
,
y < e is a
:
formal impossibility.
CLASS INCLUSION
75]
§
75. to
Some
my
drag
AND IMPLICATION
logicians (see § 74)
have
also
79
endeavoured
formula
(A:B)(B:C):(A:C) into their systems
(1)
under some disguise, such as
(A < B)(B < C) < (A < C)
The meaning
of (1)
is
clear
....
(2).
and unambiguous; but how
can we, without having recourse to some distortion of The symbol language, extract any sense out of (2) ? < A B (by virtue of their definition) asserts that every individual of the class A is also an individual of the Consistency, therefore, requires that the complex statement (2) shall assert that every individual of the class (A < B)(B < C) is also an individual of the
class B.
class
statement class
But how can the doublefactor compound C). (A < B)(B < C) be intelligibly spoken of as a
(A <
contained in
It is true that the
the singlefactor statement (A
the single statement (A
implies
implication
(A
(3);
The two formulae (1) is quite another matter. and (3) are both valid, though not synonymous; whereas their formula (2) cannot, without some arbitrary departure from the accepted conventions of language, be made to convey any meaning whatever. but that
The inability of other systems to express the new ideas xy kxyz &c, may be shown represented by my symbols A This Take the statement A 80 by a single example. T (unlike formal certainties, such as e and AB A, and ,
,
.
:
e
such as 6 and 6 >/) may, in impossibility, or a variable, an certainty, be a system, my our problem or investiof special data the to according
unlike formal
impossibilities,
:
But how could the proposition gation (see §§ 22, 109). 09 In these it could systems ? other in expressed A be
SYMBOLIC LOGIC
80
not be expressed at
hypothesis
with
certain,
consider
for its recognition
all,
the abandonment of
76
[§§ 75,
would involve unworkable is synonymous
erroneous and
their
(assumed always) that true and false with impossible. If they ceased
their
A
equivalent to their
(when
(A=
denotes
it
and
1),
their
a proposition)
A'
(or
to
as
their corre
sponding symbol for a denial) as equivalent to their (A = 0), and if they employed their symbol (A=l) in the sense of my symbol A and their symbol (A=0) in the sense of my symbol A v they might then express my but the expression statement A ee in their notation long extremely and intricate. would be Using (in accordance with usage) as the denial of (A = B), my statement A e would then be expressed by (A=/=0)(A=/r l), e
,
,
;
A^B
and
my A
80
by
{(A^0)(A^l)^0}{(A^0)(A=/=l)=£l}. This example of the difference of notations speaks for itself.
CHAPTER XI Let
76.
A
denote the premises, and
Then A .\ B (" A is or its synonym B v A (" B is each of which synonyms
B
the conclusion,
of any argument.
true, therefore
true
"),
true because
true
"),
A(AB
/
r ',
)
argument
denotes the asserts,
argument.
firstly,
A
is
That
B
A
equivalent is
to
say,
is is
to
the
that the statement (or collec
true, and, secondly, that the coupled with the denial of B constitutes an impossibility^ that is to say, a statement that is incomWhen the person patible with our data or definitions. to whom the argument is addressed believes in the truth / of the statements A and (AB )' he considers the argument
tion
of statements)
affirmation of
is
A
)
,
valid
;
if
invalid.
he disbelieves
either,
he considers the argument
This does not necessarily imply that he
dis
'BECAUSE
§§76,77]
1
AND
THEREFORE'
<
81
A or the conclusion B he be firmly convinced of the truth of both without accepting the validity of the argument. For the truth of believes either the premises
;
may
A
coupled with the truth of B does not necessarily imply the truth of the proposition (AB') though it does that 17
,
The statement (AB')
of (AB')'. (see § 23)
and therefore
A(AB')
A
But
AB T
T .
..
B, like
A(AB')\
A'
+
equivalent to (AB')'
is
Hence we have
B.
= A(A' + B) = AB = A B\ T
synonym A(AB / )
its
A(AB
Like
1
to
1
/
1
)
,
asserts that
it
>
',
A
asserts is
more than
true, but, unlike
AB' is false, but that it incompatible with our data or definitions. For example, let k He turned yah, and let B Ife is guilty. Both statements may happen to be true, and then we have A T B T which, as just shown, is equivalent to A(AB') yet the argument A B (" He is
it
asserts not only that
impossible
—
that
is
it
=
=
,
1
..
;
turned
therefore he is guilty ") is not valid, though the weaker statement A(AB')' happens on
pale
:
for
this
occasion to be true, the stronger statement A(AB')'' is not true, because of its false second factor (KB'f. I call this factor false,
because
merely (AB') that it is false that he turned pale without being guilty, an assertion which may be true, but also (AB')'', that it is impossible he should turn pale without being guilty, an assertion which is not true. 77.
it
The convention that to A(A:B), and
equivalent
1
asserts not
A
..
to
B its
,
shall
be considered
synonym A(AB'y,
us however to accept the argument A ,\ B as even when the only bond connecting A and B is the fact that they are both certainties. For example, let A denote the statement 13 + 5 = 18, and let B denote the statement 4 + = 10. It follows from our symbolic obliges valid,
conventions that in this case A .\ B and B A are both valid. Yet here it is not easy to discover any bond of connexion between the two statements A and B we know the truth of each statement independently of ..
;
F
SYMBOLIC LOGIC
82 all
We might, it is true, deduction somewhat as
consideration of the other.
follows
logical
appearance of
give the
78
[§§ 77,
:
By our data, 13 + 5 = 18. From each of these equals take away 9. This gives us (subtracting the 9 from the 13) 4 + 5 = 9. To each of We then, finally, get these equals add 1 (adding the 1 to the 5). 4 + G = 10 quod end demonstrandum. ;
the unreality (from a psychoyet much logical point of view) of the above argument demonstrations rigorous mathematical of our socalled
Every one must
feel
;
'
'
A
are on lines not very dissimilar.
striking instance is
Euclid's demonstration of the proposition that any two sides of a triangle are together greater than the third
—
proposition which the Epicureans derided as patent even to asses, who always took the shortest cut to any place
As marking the
they wished to reach.
A
tween
B and
..
noticed that though
first
A
false
e
..
;
for
though factor
78.
i]
its
7]
is
:
e
>/
difference beB,
it
be
to
is
:
»/ .*.
fails
.: x,
:
The A, can be accepted as valid. and the second is always when A = like its synonym >?(>/ x), is false, because,
and
evidently
A
A are formal certainties and the two other and stronger state
A
(see § 18), neither of
ments,
implied factor
its
j/,
:
second factor
>j
:
x
is
necessarily true,
its first
necessarily false by definition.
Though
in purely formal or symbolic logic
generally best to avoid,
when
it is
possible, all psychological
considerations, yet these cannot be wholly thrust aside
when we come
of first principles,
to the close discussion
and of the exact meanings of the terms we use. The In ordinary speech, words if and therefore are examples.
when we true,
say, " If
therefore
B
A
is
is
true,"
true,
then
we
B
is
suggest,
true," if
or "
A
is
we do not
knowledge of B depends in upon previous knowledge of A. But
positively affirm, that the
some way
or other
in formal logic, as in mathematics,
absolutely necessary, to
it is
convenient,
if
not
work with symbolic statements
§§ 78,
CAUSE AND EFFECT
79]
83
whose truth or falsehood in no way depends upon the mental condition of the person supposed to make them. Let us take the extreme case of crediting him with absolute omniscience. On this hypothesis, the word therefore, or its symbolic equivalent would, from the ..
,
subjective or 'psychological standpoint,
be as meaningless, in no matter what argument, as we feel it to be in the argument (7x9 = G3) therefore (2 + 1 = 3); for, to an omniscient mind all true theorems would be equally selfevident or axiomatic, and proofs, arguments, and logic generally would lay
word
have no raison
psychological
aside
'therefore,' or its
d'etre.
considerations,
synonym
.*.
But when we and define the
as in
,
7G,
§
it
ceases
and the seemingly meaningless argu63)/. (2 + 1 = 3), becomes at once clear,
to be meaningless,
ment, (7 x definite,
9
=
and a formal
79. In
order to
certainty.
make our symbolic
formula?
and
operations as far as possible independent of our changing individual
opinions,
we
will
lay
arbitrarily
following definitions of the word
'
cause
'
and
down '
the
explana
Let A, as a statement, be understood to assert the existence of the circumstance A, or the occurrence of the event A, while asserts the posterior or simultion.'
V
taneous occurrence of the event V and let both the statement A and the implication A V be true. In these circumstances A is called a cause of V V is called ;
:
;
the
effect
A.*. V,
is
of
A
;
and the symbol A(A V), or :
its
synonym
called an explanation of the event or circum
V. To possess an explanation of any event or phenomenon V, we must therefore be in possession of two pieces of knowledge we must know the existence or occurrence of some cause A, and we must know the law or implication A V. The product or combination of these two factors constitute the argument A/. V, stance
:
:
which call
A
.•.
A
V
an explanation of the event V. We do not the cause of V, nor do we call the argument the explanation of V, because we may have also is
SYMBOLIC LOGIC
84
B
.•.
V,
B would B
which case
in
cause of V, and the argument
be ..
V
[§§ 79,
another
80
sufficient
another sufficient
explanation of V.
we want
80. Suppose
event or phenomenon or otherwise) that x certain
number
discover the
to
We
x.
cause of an
notice (by experiment
first
each of a
invariably found in
is
circumstances, say A, B,
of
therefore provisionally
(till
We
C.
an exception turns up) regard
each of the circumstances A, B, C as a sufficient cause of that we write (A x)(B x)(C x), or its equivalent A + B + C x. We must examine the different circum
x, so
:
:
:
:
cumstance
or
account for
C
whether they possess some circommon which might alone Let us suppose that they the phenomena.
stances A, B,
to see
factor
common
do have a
in
We
factor /.
thus get (see
§
28)
•
(A:/)(B:/)(C:/),wmch=A + B + C:/.
We
before possessed the knowledge
A+B+C
:
x,
so that
we have now
A + B + C:/,'. be not posterior to x, we may suspect it to be Our next step should be to alone the real cause of x. seek out some circumstance a which is consistent with that is to say, some circum/, but not with A or B or C stance a which is sometimes found associated with /, but If
/
;
If we find not with the cofactors of / in A or B or C. that is to say, if we that fa is invariably followed by x
—
—
then our suspicion is condiscover the implication fa x firmed that the reason why A, B, C are each a sufficient :
cause of x is to be found in the fact that each contains the factor /, which may therefore be provisionally considered as alone, and independently of its cofactors, a moreover, we discover that If, sufficient cause of x. while on the one hand fa implies x, on the other f'a that is to say, if we discover (fa %){fa x' our suspicion that / alone is the cause of x is confirmed implies x'
;
:
:
:
§
CAUSE AND EFFECT
80]
85
more strongly. To obtain still stronger confirmation we vary the circumstances, and try other factors, (3, y, S, consistent with /, but inconsistent with A, B, C and with If we similarly find the same result for each other. still
these as for a
so that
;
which =/a x :/+ a (//3 x)(fp x'), which = /]8 x :f + /3' (/? x )(f'y x ')> which =fy x :/+ y' (/<M(/'<S: •<•'), which =fS:x:f+S' (fa
x)(f'a
:
:
:
x'),
:
:
:
'•
:
:
our conviction that / alone is a sufficient cause of x reBut by no ceives stronger and stronger confirmation.
we reach absolute certainty that / when (as in the investigation of natural laws and causes) the number of hypotheses or inductive process can
is a sufficient cause of x,
possibilities logically consistent
with
/ are
unlimited
;
for,
eventually, some circumstance q may turn up such that fq does not imply x, as would be proved by the actual Should this comoccurrence of the combination fqx'. and in natural phenomena it is bination ever occur always formally possible, however antecedently improbable the supposed law f:x would be at once disproved.
—
—
For,
since,
by hypothesis, the unexpected combination
fqx' has actually occurred, we may add this fact to our data e e e &c. so that we get ,
2
,
3
;
,
e:fqx' :(fqx'r
This
may
be read,
" It is
:(fx'r
'•(/'*)'.
certain that fqx' has occurred.
The occurrence fqx' implies that fqx is possible. The possibility of fqx' implies the possibility of fx' and the possibility oifx' implies the denial of the implication /: x." ;
The inductive method here described will be found, upon examination, to include all the essential principles of the methods to which Mill and other logicians have given the names of Method of Agreement and Method '
'
'
of Difference
'
(see § 112).
SYMBOLIC LOGIC
86
CHAPTER
We
will
now
[SS 81,
82
XII
give symbolic solutions of a few miscel
laneous questions mostly taken from recent examination papers. 81. Test the validity of the reasoning, "All fairies are
mermaids, for neither fairies nor mermaids exist." Speaking of anything S taken at random out of our a symbolic universe, let/= It is a fairy" let m = "it is a " mermaid," and let e = it exists." The implication of the argument, in symbolic form, is
(f:e){m :/):(/: m) which = (/: e')(e m') (/: :
:
ra).
Since the conclusion /: m is a "universal" (or implication), the premises of the syllogism, if valid, must (see § 59) be either f:e:m or /: e m. This is not the case, so that :
the syllogism
is
not valid.
Most symbolic
Of
course,
logicians, however,
may
replace
e
.
would consider
this
By
our
syllogism valid, as they would reason thus
:
"
therefore /= m. Hence, all fairies and m = data, /= and mermaids are fairies" (see § 72). mermaids, all are argument " It is not validity the Examine the of 82. and it is incompounds, the case that any metals are it may therecorrect to say that every metal is heavy heavy, and are not fore be inferred that some elements also that some heavy substances are not metals." Lete "it is an element" = " it is not a compound"; " it is a metal " and let h = " it is heavy." let m The above argument, or rather implication (always supposing the word " If " understood before the pre;
:
;
=
=
mises)
;
is
(m e)(m :
Let
A=m
:
e,
let
B=m
K)'
:
:
:
(e
h, let
:
h)\h
C=e
:
:
m)'.
h, let
D — h: m
,
and
§§ 82,
MISCELLANEOUS EXAMPLES
83]
denote the implication of the given argument.
let
We
then get
= AB' CD' = (AB' C')(AB' since x yz = (x y) (x :
:
:
In
order
AB'
:
87
that
may
C and AB'
:
:
D'),
be valid, the two implications
D' must both be
:
:
z).
Now, we have
valid.
(see § 59)
AB
C = AC
7 :
:
B = (m :e)(e: h) (m .
:
h),
Hence, C, which asserts (e:h)', is valid by § 56. some elements are not heavy " is a legitimate conWe next examine clusion from the premises A and B'.
which
that "
the validity of the implication
AB' D' = (m :
AB
7
We
D'.
:
have
e)(m K)' (h m)'.
:
:
:
:
Now, this is not a syllogism at all, for the middle term m, which appears in the two premises, appears also in Nor is it a valid the conclusion. implication, will
as
figure
Let the eight points in constitute the class m
show.
m
the circle
subjoined
the
;
the twelve points in the circle e and let the constitute the class e let
;
five
points in the circle h constitute
the class
m is
is
h.
not h
"
Here, the premises " Every m is e, and some are both true yet the conclusion, " Some h ;
not m" is false. Hence, though the conclusion
conclusion 83. for
D
r
is
C
r
is
legitimate, the
not.
Examine the argument,
only experience can
"
No young man
give wisdom,
is
wise;
and experience
comes only with age." Lety = "he is young") letw = "he is wise" and let Also, let (p denote the has had experience." e = "he We have implication factor of the given argument. ;
cj>
= (/
:
w'){y
:
e')
:
(y
:
w')
= (y
f :
e
:
w')
:
(y
:
w').
SYMBOLIC LOGIC
88
The given implication
[§§
8385
therefore valid (see §§ 11, 56,
is
59).
Examine the argument, " His reasoning was but as I knew his conclusion to be false, I was led to see that his premises must be false also." 84.
Let clusion
P=" was
were true," and
his premises
Then P C =
true."
:
implication) was valid."
Let
(p
at once
C = "his
con
his reasoning (or rather
"
denote the implication of
We
the argument to be examined.
let
correct,
get (see
105)

= (P:C)C':P'
<£
= the
Modus
valid form of the
tollendo tollens.
Thus interpreted (p is valid. But suppose the word premises " means P and Q, and not a single compound
"
We
statement P.
then get
<£=(PQ:C)C:P'Q'
;
an interpretation which fails in the case CP'Q1 and also in To prove its failure in the latter case, the case C^P^Q ,
6
.
we
substitute for C, P,
values
e,
t},
r\,
=
Q
exponential
their respective
and thus get (>/e
:
rfirf
:
i/e'
=
(rj
:
?])e
:
et]
= ee
:
>/
=
rj.
in the argument: mistakes are culpable for mistakes are sometimes quite unavoidable." "it is culpable," let Let "it is a mistake," let c u " it is unavoidable," and let
Supply the missing premise
85.
"
Not
all
;
m=
=
we get
=
(see §§ 59, cp
= (m
:
64)
m')'Q
:
(m
x :
c)
= (m
:
c)Q
(m
:
:
u').
For this last implication to be valid (see § 64), we must have its premises (or antecedent) either in the form
m The
first
:
c
:
vf or else in the form ,
m
:
c
:
u
form contains the antecedent premise
.
m
:
c;
the
MISCELLANEOUS EXAMPLES
8587]
§§
89
The first form is therefore the second form does not. one to be taken, and the complete syllogism is
(m
:
c
:
u) (m :
n),
:
Q being c vf which asserts that The original reasoning nothing culpable is unavoidable." in its complete form should therefore be, " Since mistakes are sometimes unavoidable, and nothing culpable is un
the missing premise
:
,
"
avoidable,
some mistakes
are not culpable."
Supply the missing promise in the argument, " Comets must consist of heavy matter for otherwise they would not obey the law of gravitation." 86.
;
= "it
=
"it consists of heavy let A obeys the law of gravitation." Putting
Let
c
matter" and
is
let
a comet"
# = "it
= (h':g')Q:(c:h)=:(c:g:h):(c:h), application of §64; for g:h = h':g', so (P
by
the
that
missing minor premise Q understood is c g, which asserts The full that " all comets obey the law of gravitation." :
reasoning
is
therefore (see
§11)
(c:h)\(c:g)(g:h),
§11)
or its equivalent (see (c
the
In
form
first
:
it
g){g :h):(c: h).
may
be
read,
"
Comets consist of
obey the law of gravitation, heavy matter law of gravitation consists the obeys that and everything ;
for
all
comets
of heavy matter."
87.
the "
following
Some
is
Supply the missing proposition which
enthymeme
professional
men
*
into
a
valid
will
make
syllogism:
are not voters, for every voter
a householder."
Let
P = "he *
is
a professional man," let
An enthymeme
is
V = "he
a syllogism incompletely stated.
is
a
SYMBOLIC LOGIC
90
and
voter"
H = " he
let
[§§
a householder."
is
the implication of the argument, and additional premise required to justify
We
have <£
W
8789
Let
the
(see § 11)
= (P = (P
V)' !(V
:
:
H)W = (V H)W :
V)(V H) :
:
:
(P V)' :
W' = (P V H) W.
:
:
:
:
deducible from P V H is assume P H = W', and conse= (P H)', which is therefore the weakest quently The complete argument is therefore premise required.
The
We
:
this
conclusion
strongest
P H.
W
"
:
voter
Some
:
therefore :
men
professional
a householder, and
is
:
:
are not voters, for every
some professional men are
not householders." 88. Put the following argument into syllogistic form, and examine its validity " The absence of all trace of :
paraffin and matches, the constant accompaniments of arson, proves that the fire under consideration was not
due
to that crime." F " it was the fire
=
Let " it
was due to arson and matches " and
"
We
given argument.
= (¥
:
T')(A
:
T)
let
;
let
;
:
under consideration
T="
" ;
A=
let
it left a trace of paraffin
denote the implication of the
get
(F
= (F T')(T' A (F A = (F:T' :A'):(F: A 7
:
A')
:
:
)
:
:
7
)
7
).
The implication of the given argument is therefore valid. The argument might also be expressed unsyllogistically (in the
Let T = " the let A = " the let
(p
(see §
modus fire
fire
tollendo tollens) as follows (see § 105). left
89. •'
How
;
denote the implication of the argument.
We
get
105) (j)
which
a trace of paraffin and matches " and to the crime of arson "
was due
= T'(A
:
T)
:
A'
the valid form of the Modus tollendo tollens. Put the following argument into syllogistic form can any one maintain that pain is always an evil,
is
:
§§ 89,
TECHNICAL WORDS EXPLAINED
90]
sometimes be a Let
R = " It
E = " it
an the argument. is
is
good
real
remorse "
evil "
and
?
let
;
P = " it
causes pain
is
" ;
let
denote the implication of (f) get (as in Figure 3, Bokardo)
;
let
We <£ = (R:P)(R:E) :(P:E) = (R:P)(P:E):(R:E), ,
which
may
that remorse involves pain, and yet
who admits
91
/
But to reduce we have been obliged to
a syllogism of the Barbara type.
the reasoning to syllogistic form
Remorse may sometimes be a real weaker premise (R E)', which only asserts that " Remorse is not necessarily an evil." As, however, the reasoning is valid when we take the weaker premise, it must remain valid when we substitute consider the premise,
"
good," as equivalent to the
the stronger premise
;
:
only in that case
it
will not
be
strictly syllogistic.
CHAPTER
XIII
be given definitions and explanations of some technical terms often used in treatises on
In this chapter
logic.
90. Sorites.
Barbara.
will
— This
an extension of the
is
syllogism
Thus, we have
Barbara =
(A:B:C):( A: C) C D) (A D) C D E) (A
= (A B (Sorites), = (A B (Sorites)!
:
:
:
:
:
:
:
:
:
:
:
E)
&c, &c.
Taken
we get what may
in the reverse order (see § 11)
be called Inverse
Sorites,
thus
:
Barbara=(A!C)!(A!B!C) (Sorites^ = (A D) (A B C !
&c.
!
!
!
!
D).
SYMBOLIC LOGIC
92 91. Mediate
and Immediate
[§§9194
When
Inferences.
from a
we infer another proposition \j/(a?, z) in which one or more constituents of the first proposition are left out (or " eliminated "), we call it Mediate Inference. proposition
(p(x, y, z)
If all the constituents
of the first proposition are also
found in the second, none being eliminated, we have For example, in what is called Immediate Inference. Barbara we have mediate inference, since from x y z we infer x z the middle term y being eliminated. On the other hand, when from x y we infer y' x', or ax y, we have immediate inference, since there is no elimination of any constituent. :
:
Law
92.
This is the name given of Excluded Middle. B + A~ B or its equivalent a a. The
the certainty
individual
:
:
:
to
:
;
A
A
+
,
B
either belongs to the class
B
belong to the class a formal certainty. 93. Intension
and
—an
alternative
or
which
Extension, or Connotation
does not
it
is
evidently
and Denota
Let the symbols (AB), (ABC), &c, with brackets, tion. in as § 100, denote the collection of individuals, (AB)^ (AB) 2 &c, or (ABC) r (ABC) 2 &c, common to the classes (AB) so that S will not be synonymous inside the brackets ABC (ABC) AB With this interwith nor S S (see § 9). with S be any individual pretation of the symbols employed, let S taken at random out of our universe of discourse, and X let S S (AB) be our definition of the term or class X. The term X is said to connote the properties A and B, and to denote the individuals X 1> X 2 &c, or (AB) r (AB) 2 &c, possessing the properties A and B. As a rule the greater the number of properties, A, B, C, &c, ascribed to X, the fewer the individuals possessing them or, in other words, the greater the connotation (or intenIn A a sion), the smaller the denotation (or extension). ,
,
;
,
=
,
,
Aa
the symbol a connotes as predicate, and in
it
denotes
as adjective.
All
The two
Contrary and Contradictory.
94. "
X
is
Y"
(or x
:
y)
and
"
No
X
is
Y"
propositions
(or
x
f :
y
)
are
TECHNICAL WORDS EXPLAINED
9498]
§§
called
93
each being the contrary of the other.
contraries,
The propositions
X
Y"
X
Some is not Y," respectively represented by the implication x y and its denial (x y)' are called Contradictories, each being the contradictory or denial of the other (see § 50). Similarly "
All
is
"
and
:
:
"No X
Y"
is
and "Some
sented by the implication x
:
X
is
y'
and
Y," respectively repredenial (x
its
y')
:
f
are
,
called Contradictories.
Some
X
X
The propositions "Some
95. Subcoutraries.
and
Y"
is
not Y," respectively represented by the r nonimplications (x y') and (x y)' are called Subcontraries. It is easily seen that both may be true, but that both cannot be false (see § 73). "
is
:
The
96. Subalterns.
Y," or x (x
:
y')
f ;
y,
:
',
and the universal
X
Some
"
"
No
"
is
X
X
Some
Y," or x
is
not Y," or (x
y' :
'
f :
y)
is
Y," or
is
implies
,
In each
.
the implication, or universal,
cases
X
universal proposition "All
implies the particular
the particular of these
:
called
is
the Subalternant, and the nonimplication, or particular, is called the Subalternate or Subaltern. That x y implies {x:y')' is proved in § 73; and by changing y into y' :
and
proves that x
vice versa, this also
r :
implies (x
y
:
y)'
This is the name given by some logicians to the formula x y ?/ x, which, with the conventions of §§ 46, 50, asserts that the proposition 97.
Contraposition.
=
:
"
All
X
nonY
is
is
Y"
:
proposition
" All
logicians define the
word
equivalent to
is
But other
nonX."
the
differently.
98.
let
Let
Conversion.
A, E,
I,
\j/(y,
x) denote
implies, the
plication
denote any proposition,
letters
y)
:
§ 50); and any other proposition which the first x and y being interchanged. The im
implying
x)
is
called
y)
and
y]/(y,
the two implications
each
(p(x, y)
or O, of the traditional logic (see
the
other,
as
(x:y'y = (y:x'y, the conversion version.
When
the proposition
x\y
in
is
(p(x,
When
Conversion.
\/(y,
x) are equivalent, r
— y:x,
called
and in
Simple
Con
y) implies but is not
SYMBOLIC LOGIC
94 implied by
\^(v/, x),
conversion
is
In
accidens.
called
the
as in the case of (x
Conversion
called all
these
Convertend
:
y)
:
(y
:
the Per
.«')',
Limitation
by
the antecedent
cases, ;
98100
[§§
or
y) is
and the consequent ^{y,
x)
is
called the Converse. 99. Modality.
In the traditional logic any proposition
AB
of the first degree is called a pure proposition, while any of my propositions A BC or A BCU &c, of a Mr/her degree ,
would generally be considered a modal proposition ; but upon this point we cannot speak with certainty, as logicians are not agreed as to the meaning of the word For example, let the pure proposition A B modal.' " then A Be might assert that " Alfred will go to Belgium be read " Alfred will certainly go to Belgium" which would Again, the proposition be called a modal proposition. A" B which asserts that " Alfred will not go to Belgium" would be called a pure proposition whereas A B or its synonym (A B )\ which asserts that A B is false, would, by most logicians, be considered a modal proposition (see §§ 21, 22, and note 2, p. 105). 7 100. Dichotomy. Let the symbols (AB), (AB ), (ABC), '
;
,
',
;
1
with brackets, be understood to denote classes (as in 7 Boolian systems) and not the statements AB, AB ABC, &c.
&c.,
,
We
get*
A = A(B + B ) = A(B + B
)(C + C) = &c. = (AB) + (AB = (ABC) + (ABC + (AB = &c. 7
7
7
7
)
A
)
7
C)
7
+ (AB C
7
)
may be mutually exclusive divisions then, by similar subdivision of each of these, into four This process mutually exclusive divisions and so on. of division into two, four, eight, &c, mutually exclusive Thus any divided,
class
first,
in our universe of discourse
into
two
;
*
B
;
The symbol (AB) denotes the
total of individuals
the symbol (AB') denotes the total
so on.
number
in
A
common
A and
to
but not in
B
;
and
§§
TECHNICAL WORDS EXPLAINED
10010:.]
divisions
The
called Dichotomy.
is
Bamean
Porphyry, or
"
enthusiastically of
of
Tree, affords a picture illustration
Jeremy Bentham wrote
by Dichotomy.
of this division
Tree
celebrated
95
the matchless beauty of the
Ramean
Tree."
101. Simple symbolically,
Dilemma.
Constructive
:
may
either
or
B
:
A
" If
be read,
A
expressed
the implication
is
(A aO(B x)(A It
This,
implies
true, then x
is
+ B)
:
x.
and B implies
x,
x,
and
true."
is
This
102. Complex Constructive Dilemma.
is
the im
plication
(A:aOCB:yXA + B):s + y. 103. Destructive Dilemma.
(A:;r)(B:
y)(
t
It
may
A
" If
be read,
t
This
is
/ + //):A' + B'.
implies
B
and
x,
implies
y,
and
then either A or B is false." 104. Modus ponendo ponens (see Dr. Keynes's "Formal There are two forms of this, the one valid, the Logic "). other not, namely, either x or y
is false,
(A B)A :
:
B
and (A B)B A. :
:
the second form fails in e e for, denoting the the case A^B"' and in the case A~ B
The
first
form
is
selfevident
;
1
;
second form by
we
Wc£
get (see /
;
the
= A 'B + A B r
e
>
e
'
105. Modus tollendo
forms
67—69)
§§
tollens.
.
Of this
also there are
(A B)B' A' and (A B)A' :
:
The case
first
is
A^B"*,
evident
and
in
;
two
the second not, namely,
first valid,
:
the second
the case A~
e
fails,
B
e .
:
B'.
as before, in the
For, denoting the
SYMBOLIC LOGIC
96
[§§
105108
=
A^B" + A" 6 B £ (See second form by (p, Ave get Wc// 6769.) §§ This also has two forms; 106. Modus tollendo ponens. They are the first valid, the other not. (A
The
first
may
+ B)A
/
:B and (AB)'B':A.
be proved formally as follows
+ B)A' B = A'B'( A + B)
(A
:
:
= The second
is
.
:>]
>j
=
r,
:
= + (,,
:
A = A'B'(AB)'
= (A + B)
,/
n
:
= A'B'
;
We
denote the given implication. (p
=
(
AB)'B'
:
in the case case, let
(p
get
A = (A + B)
e ,
A + B = 0,
Therefore, putting
as already proved.
n
:
e
which fails both in the case (A + By and (A + B)". To prove its failure in the last
= e* = n
:
not valid, for
(AB)'B'
(p
>,)
e.
we
get
.
107. Modus poncndo
tollens.
This also has a valid and
an invalid form, namely,
(AB)'A B' and (A :
The
+ B)B
:
The second
is
(A
which
n
=1 1= :
e.
not valid, for
+ B)B:A' = AB(A + B):>/ = AB:>,, e
the
=
:
€ both in the case (AB) and in the case (AB) the given implication becomes e first case which and in the second case it becomes 6
fails
which = also
A'.
first is valid, for
(AB)'A B' = AB(AB)'
In
:
.
:
t]
:
;
>;,
>/,
>].
Let x 108. Essential (or Explicative) and Ampliative. be any word or symbol, and let
§§
108110]
TECHNICAL WORDS EXPLAINED
containing x (see
word
(p(x)
is,
or follows neces
which explains the meaning of the
of words) x
collection
(or
When
§ 13).
sarily from, a definition
97
;
then the proposition
called an essential, or an explicative, proposition.
is
Formal
certainties
are essential propositions (see §
When we
109).
have a proposition, such as xa or x~ a or a x + vf, which gives information about x not contained in any definition of x such a proposition is called ,
,
;
ampliative.
109. Formal and Material A proposition is called a formal certainty when it follows necessarily from our definitions, or our understood linguistic conventions, without further data and it is called a formal impossi;
when
bility,
it
is
inconsistent
linguistic conventions.
when
it
It is
our
with
follows necessarily from
some
necessarily contained in our definitions. called a material impossibility
definitions or
called a material certainty
when
it
special data not
Similarly,
contradicts
it is
some
datum or data not contained in our definitions. In this book the symbols e and n respectively denote certainties and impossibilities without any necessary implication as to whether formal or material. When no special data are given beyond our definitions, the special
and impossibilities spoken of are understood be formal when special data are given then e and n respectively denote material certainties and impossibilities. 110. Meaningless Symbols. In logical as in mathematical researches, expressions sometimes turn up to certainties to
;
which we cannot,
for
a time, or in
considered, attach any meaning.
the circumstances
Such expressions
are
not on that account to be thrown aside as useless. The meaning and the utility may come later; the symbol
^/
—
1
in
mathematics
is
the fact that a certain
a wellknown instance.
simple
or
From
complex symbol x
happens to be meaningless, it does not follow that every statement or expression containing it is also meaningless. For example, the logical statement A^ + A'*, which
G
SYMBOLIC LOGIC
98 asserts that
belong to
A
A
either belongs to the class x or does not
whether A be meanwhether x be meaningless or not. meaningless and x a certainty. We get
it,
a formal certainty
is
ingless or not,
Suppose
[§110
Next, suppose
and
also
+ (P = + e =
A*
+ A" x =
A
a certainty
e
>/
e.
and x meaningless.
A x + A = e° + t° = + e = f
We
get
r
>;
Lastly, suppose
A
.
We
and x both meaningless.
A x + A"* = 0° +
0°
=e+ = >/
get
e.
Let A x denote any function of x, that is, any expression containing the symbol x and let
;
Ax
though intelligible for most happens to be meaningless when x has a particular value a, and also when x has a Suppose also that the statement particular value /3.
now
that
the symbol
values (or meanings) of
,
x,
true (and therefore intelligible) for
all
values of
x except the values a and /3, but that for these two values of x the statement
becomes true (and therefore intelligible) also for = a and x = ft provided we lay
the exceptional cases x
down the convention meaningless symbol
or
Aa
definition
shall
that
the
have a certain
hitherto
intelligible
meaning m., and that, similarly, the hitherto meaningsymbol A^ shall have a certain intelligible meanThen, the hitherto meaningless symbols A a and ing m 2 Ap will henceforth be synonyms of the intelligible symbols m1 and m2 and the general statement or formula
.
,
of x
without exception.
It
is
on
this
principle
that
MEANINGLESS SYMBOLS
§§110,111]
99
mathematicians have been led to give meanings to the meaningless symbols a° and a n the first of which is now synonymous with 1, and the second
originally
,
with
—
an
Suppose we have a formula,
In this case, since
intelligible.
thesis,
meaningless,
we
are
at
by hypoit any with any
is,
(p(?)
liberty
give
to
meaning that does not conflict previous definition or established formula. In order, therefore, that the formula
=
the meaningless value 9), we may legitimately lay down the convention or definition that the hitherto meaningless expression (£(?) shall henceforth be synonymous with the always intelligible expression yf(s). With this convention,
had only a
the formula,
(j)(x)
restricted validity, will
=
y(s(x),
which before
now become
true in
all cases.
=
111. Take, for example, the formula, s/x >/x x in mathematics. This is understood to be true for all positive
values of x; but
the
symbol
^/x,
and conse
quently also the symbol Jxjx, become meaningless when x is negative, for (unless we lay down further conventions) fractions
the are
square
roots
nonexistent.
of negative numbers or Mathematicians, therefore,
have arrived tacitly, and, as it were, unconsciously, at the understanding that when x is negative, then, Avhatever
meaning may be given combination y/x^x, like synonymous with x and, ;
it
may
in future be
that meaning
to its
the symbol
Jx
itself,
synonym {^/xf,
further, that whatever
shall
the
be
meaning
found convenient to give to */— 1, conflict with any previous formula
must not
SYMBOLIC LOGIC
100
Those remarks bear solely on the
or definition.
symbol
*J —
illustration
previously.
by in
1,
which we have given merely
it is
wider
the
of
general
—
algebraic
as a concrete
principles
In geometry the symbol *J
itself a clear
no way
111, 112
[§§
1
discussed
now conveys
and
conflicts
intelligible meaning, and one which with any algebraic formula of which
a constituent.
112. Induction.
— The reasoning by which
we
infer, or
rather suspect, the existence of a general law by observation of particular cases or instances
is
called Induction.
Let us imagine a little boy, who has but little experience of ordinary natural phenomena, to be sitting close to a clear lake, picking up pebbles one after another, throwing them into the lake, and watching them sink. He might is a stone" (a); "I "It sinks" (7). These
reason inductively as follows: "This
throw
it
into
the water"
(/3)
;
three propositions he repeats, or rather tacitly and as
it
were mechanically thinks, over and over again, until finally he discovers (as he imagines) the universal law a/3 y, that a/3 implies y, that all stones thrown into ivatcr sink. He :
continues the process, and presently, to his astonishment, discovers that the inductive law a/3 true.
An
exception has occurred.
:
y is One
not universally
of the pebbles
which he throws in happens to be a pumicestone and Should the lake happen to be in the crater of an extinct volcano, the pebbles might be all pumicestones, and the little boy might then have does not sink.
arrived inductively at the general law, not that all stones sink,
but that
all stones float.
called " law of nature."
So
The whole
it
is
with every so
collective experience
mankind, even if it embraced millions of ages and extended all round in space beyond the farthest stars that can ever be discovered by the most powerful telescope, must necessarily occupy but an infinitesimal portion of infinite time, and must ever be restricted to a mere Laws founded upon infinitesimal portion of infinite space. data thus confined, as it were, within the limits of an of
§ 1
1
"
2]
LAWS OF NATURE "
infinitesimal can never be regarded (like
and
101
most formulae
in
mathematics) as absolutely certain they should not therefore be extended to the infinite universe of time and space beyond a universe which must necessarily remain for ever beyond our ken. This is a logic
in
;
—
truth which philosophers too often forget (see
Many theorems
in mathematics, like
most
§
80).
of the laws
of nature, were discovered inductively before their validity
could be rigorously deduced from unquestionable premises. In some theorems thus discovered further researches have
shown that
their validity
limits than
was
in
the
is
restricted
at first supposed.
Differential
Calculus
is
within narrower Taylor's
Theorem
a wellknown example.
Mathematicians used to speak of the " failure cases " of Taylor's Theorem, until Mr. Homersham Cox at last investigated and accurately determined the exact conditions of its validity. The following example of a theorem discovered inductively by successive experiments may not be very important but as it occurred in the course of my own researches rather more than thirty years ago, I venture to give it by way of illustration. Let C be the centre of a square. From C draw in a ;
random
direction a straight line CP, meeting a side of
the square at P.
What
whose variable radius
is
is
the average area of the circle
CP ?
The question is very easy for any one with an elementary knowledge of the integral calculus and its applications, and I found at once that the average area required is equal to that of the given square. I next took a rectangle instead of a square, and found that the average area required (i.e. that of the random circle) was equal to that of the rectangle. This led me to suspect that the same law would be found to hold good in regard
symmetrical areas, and I tried the ellipse. The was what I had expected taking C as the centre of the ellipse, and CP in a random direction meeting the curve at P, I found that the average area of the variable to all
result
:
SYMBOLIC LOGIC
102
[§112
is CP must be equal to that of the Further trials with other symmetrical figures confirmed my opinion as to the universality of the law. Next came the questions Need the given figure be symmetrical ? and might not the law hold good for any point C in any area, regular or irregular ? Further trials again confirmed my suspicions, and led me to the discovery of the general theorem, that if there be any given areas in the same plane, and we take any point C anywhere in the plane (whether in one of the given areas or not), and draw any random radius CP meeting the
circle
whose radius
ellipse.
:
boundary
of any given area at a variable point P, the average area of the circle whose radius is CP is always
equal to the
sum
of the given areas, provided
when P when P is
sider the variable circle as positive exit
from
any
area,
entrance, and zero
negative
when P
random radius meets none Next came the question theorem be extended
to
is
is
nonexistent,
a
we con
a point of
point
of the given boundaries.
Might not the same general any number of given volumes :
instead of areas, with an average sphere instead of
Experiment again led
of
because the
—
circle
?
an affirmative answer that is to say, to the discovery of the following theorem which (as No. 3486) I proposed in the Educational Times as follows to
:
Some
lie about matter where they be Within such solid, or without,
shapeless solids
No
Let's take a centre C.
From
centre C, in countless hosts,
Let random radii run, And meet a surface each at P, Or,
may
be,
meet with none.
Those shapeless solids, far or near, Their total prove to be The average volume of the sphere Whose radius is CP.
FINITE, INFINITE, ETC.
§§112, 113]
The
sphere, beware,
When
out at
P
positive
they
But, changing sign,
When
is
103
fly
'tis
negative
you spy. One caution more, and I have done entrance there
The sphere
is
naught
when P
there's none.
In proposing the question in verse instead of in plain prose, I merely imitated the example of more dis
Mathematicians,
contributors.
tinguished folk,
moments
have their
burst forth into song just to relieve their theorem thus discovered inductively was
ductively by Mr. G. clearer
S.
A
Carr.
proof was afterwards
who succeeded Mr.
Miller
like
other
when they The feelings.
exuberance,
of
proved
de
and therefore
fuller
given by Mr. D. Biddle,
as
mathematical
editor
of
the Educational Times.
Much 113. Infinite and Infinitesimal. is caused by the fact that each of
confusion of
those words used in different senses, especially by mathematicians. Hence arise most of the strange and inadmissible paraTo doxes of the various non Euclidean geometries. avoid all ambiguities, I will define the words as follows. The symbol a denotes any positive quantity or ratio too large to he expressible in any recognised notation, and any
ideas is
such ratio
is
called a positive infinity.
course of an investigation,
have
to
As we may,
in the
speak of several such
the symbol a denotes a class of ratios called infinities, the respective individuals of which may be designated by a a 2 a g &c. An immensely large number is not
ratios,
,
,
For example, let M denote a million. which denotes the millionth power of a
necessarily infinite. M
The symbol
million,
is
a
M
,
number
so inconceivably large that the ratio
which a million miles has to the millionth part of an inch would be negligible in comparison yet this ratio M M is too small to be reckoned among the infinities a a a y &c, of the class a, because, though inconceivably ;
,
SYMBOLIC LOGIC
104
[§113
in our decimal nota
large, its exact value is still expressible
tion ; for we have only to substitute 10° or 1,000,000 The for M, and we get the exact expression at once. in— negative any a, denotes synonym its symbol /3, or negative different denote &c, finity ; so that fi v j3 2 /33 ratios, each of which is numerically too large to be Mathematicians expressible in any recognised notation. ,
,
and — co pretty much in the and /3 but unfortunately they
often use the symbols oo to a
sense here given
employ
also
oo
and
1 3 as , ,
sions such
—
consider oo and
meeting
it
is
They speak a point
at
ratios at
but mire
all,
Mathematicians equivalent when they are employed (see §
class
oo
in this sense; but
equivalent.
indifferently to denote expres
&c. which are not
nonexistences of the
—
;
oo
at
clear
6).
—a
that a and
infinity
but this
;
are not
straight
parallel
of all
lines
only an
is
which &c, or fi v or /33 &c, can never be distinguished by any or /8 and possible instrument from parallel straight lines may, therefore, for all practical purposes, be considered
abbreviated
meet
way
any
at
of saying that all straight lines
av or a 2 or
infinite distance
,
a,,
,
;
parallel.
The symbol any
positive
expressible 7c,
called
h,
called a positive
quantity or ratio
in a
any
negative
quantity or ratio in
any
recognised
any positive
recognised
finite
a ratio neither
to
be
and the symbol denotes any negative
small numerically
Let
notation.
number
denotes
notation;
infinitesimal,
too
infinitesimal,
small numerically
too
or
too large nor
c
ratio
to
be
expressible
temporarily
—
that
too small
is
denote to
say,
be expressymbols of the to
our ordinary notation; and let forms xy, x + y, x — y, &c„ have their customary mathematical meanings. From these conventions we get various selfevident formula?, such as sible in
§
FINITE, INFINITE, ETC.
113]
(2) (ch)\ (ckf
(1) (cay, (c(3f;
105
 c)\
(3) («
;
;
fl
(4) (,±/0
(7)
(10)
The
Q\ of
first
finite
c
:
(5)
;
(£)*;
afar*
;
( 1 1 )
+ cf;
«
a
+ s^
:
2
(/S )";
ar°
;
()
;
(9) (aflP;
(12) (M)*.
positive infinite is a positive infinite
neither
if
any
ratio
x
is
;
and a
difference
between
positive
the tenth
a positive
a positive nor a negative infinite.
formula asserts that the infinite
(«Y,
(8)
(f)",
(6)
formula asserts that the product of a
and a
formula asserts that is
((3
finite, it
The
third
a positive
positive finite is a positive infinite.
and the infinieighth article on " Symbolic Reasoning" in Mind. The article will probably appear next April. Note 2.— The four " Modals " of the traditional logic are the four terms f T This proin the product of the two certainties A + A' and A + A' + A".
Note 1.— A
fuller discussion of the finite, the infinite,
tesimal will be found in
my
A^ + A^ + A^A^ + A'A"; it asserts that every statement A is either (A € ), or necessarily false (A''), or true in the case considered but not always (A T A"), or false in the case considered but not always (A'A"). See § 99. duct
is
necessarily true
CALCULUS OF LIMITS CHAPTER XIV
We will begin by applying this calculus to problems in elementary algebra. Let A denote simple ratio, or fraction. A x asserts number, symbol any The the belongs to class that A x, the symbol x denoting as positive, or negative, or zero* or some such word The symbols A*B», A^ + B A* B y A~ x imaginary, &c. &c, are to be understood in the same sense as in §§ 4For example, let Y= positive, let N = negative, and 10. 114.
2
= zero*
let
;
while
numbers
all
',
:
,
,
or ratios not included
in one or other of these three classes are excluded from
our Universe of Discourse out (3

3)°,
x
(f),
(3
p
+ N 2)N
— that
(3PJi*
0)°,
,3,
(P 1
+ P2
),
(N 1
,
is
to
Thus we get
consideration.
of
(P^/,
say, left entirely
(6
—
4)
p ,
(W,
(4
—
6)
N ,
(N^f
and many other selfevident
for
mulas, such as
(AB) P = A P B P
+ AN B N N P P N N (2) (AB) = A B + A B (3)(AB)° = A° + B°.
(1)
(4)
{Ax
 B) p =
.
.
B B Ux  A/J )Y = k{x  ?Y + A*(x A A) \ I
\
V
* In this chapter and after, the symbol 0, representing zero, denotes not simple general nonexistence, as in § G, but that particular nonexistence through which a variable passes when it changes from a (See § 113.) positive infinitesimal to a negative infinitesimal, or vice verm.
106
CALCULUS OF LIMITS
§§114, 115]
(5)(A
107
,B,={4B)^4By + 4_By.
(7) (ax
= ah) = (ax  ab)° = { a(x b)}° = a" + (x  b)°.
greater and less have a wider meaning In algebra, when than in ordinary speech. we have (.« — a) p we say that " x is greater than a," whether a is positive or negative, and whether x is Also, without any regard to the positive or negative. sign of x or a, when we have (x — ctf, we say that " x Thus, in algebra, whether x be positive is less than a!' or negative, and whether a be positive or negative, we have
The words
115.
in algebra
,
(x
From
this (x
— of = (x > a), follows,
it
(x
— «) N = (x < a).
by changing the sign of
+ af = (x >  a), > and <
the symbols
and
and
(x
a,
that
+ af = (x <  a)
;
being used in their customary
algebraic sense.
For example,
a 
let
3.
We
(,rsy = (x>3), and In other
words,
that
equivalent to asserting that x
is
—
is
assert
that x
that x
is less
Next,
3
than
let a
is
negative
 3f = (x < 3).
(x
assert
to
get
x
—
greater
3
is
than 3
positive ;
equivalent to asserting
3.
=  3. We get x  a y = (x + 3) = (x >  3 N = (x <  3 (x  af = (x + 3 p
(
)
Let x (x
Let
= 6,
> 
x= 0,
(x
> 
we 3)
p
)
(a certainty).
get
= {x + 3 p = (0 + 3 = e p
3)
).
get
= (x + 3) = (6 + 3 p = e
we
)
is
while to
)
(a certainty).
SYMBOLIC LOGIC
108
[§§
x= — 1, we get p (x >  3) = (x + 3) =  1 + 3) = e
115117
Let
p
(a certainty).
(
= — 4, we get p = (x >  3) = (x + 3) =  4 + 3 Let
a?
)''
(
It
is
evident that
between
> —
3)
is
(an impossibility).
a certainty
for all
(e)
and for all negative values of x and — 3 but that x> — 3 is an impossibility negative values of x not comprised between
positive values
(>?)
(,/;
>/
of
x, ;
for all
and —3. With (x< —3) the case is reversed. The statement (x< — 3) is an impossibility (>?) for all positive values of x and for all negative values between and — 3 while (x < — 3 ) is a certainty (e) for all negative values of x not comprised between Suppose, and — 3. for example, that x= — 8 we get ;
;
(x<  3) = (x + 3) N =  8 + 3) N = e (a certainty). Next, suppose x= — 1 we get (x<  3) =  1 + 3) N = (an impossibility). (
;
(
116.
From
>?
the conventions explained in
§
115,
we
get
the formulas
(A>B) = (A)<(B), and (A( B);
= {(A)(B)} N = (A + Bf = (AB) = (A>B), and{(A)>(B)} = {(A)(B)} p = (A + B) p = (AB) N = (A
P
number
117. Let x be a variable
a
is
a constant of fixed value.
fraction, while
or
When we
have
(x
— «) p
,
synonym (x > a), we say that a is an inferior limit oix\ and when we have (x — cif, or its synonym (x
+ a) p (x + cif (x
asserts that
asserts that
— a is — a is a
an
inferior
limit of x,
superior limit of
x.
and
CALCULUS OF LIMITS
§§118,119] 118. For
example,
let
it
109
be required
to
find
the
superior or inferior limit of x from the given inequality
Sx
—x—
3
x >x+ + 6
2
A
Let
3
denote this given statement of inequality.
We
get
2
\
= i6 3
Hence, —
2«
— —— _
)\

3
= (tx—3y=(x —
.
is
an inferior limit of
In other words, the
x.
7
A
given statement of x lower
values of
than
3 ,
impossible for any positive value
is
and
also impossible for all negative
x.
119. Given the statements
A
— — —<—
denotes Sx
and
B, in
which
We
of x.
—
and B denotes
4
2
Find the limits
,
A
3
3x
4'
have
A = Ux°^ j = (12xl() + 2xlf 1
=(^n>»4liy=(*
—
= (6
3xj = (244x36x3)»
= (214tor = (4te2ir' = Hence we 8get
AB =
(
—>x>— 40/
\14
1.
= (.,^J
a;
(
>l).
SYMBOLIC LOGIC
110
119, 120
[§§
Thus x may have any value between the superior 11 limit
....
and the

limit
inferior
21 
but any J value of
;
40
14
x not comprised within these limits would be incompatible with our data. For example, suppose x = 1
We
get
*=i  1Y Ya  2  1 Y /3 V '\
a (s
4/
2
\
B
6 :
(
^
3
4/
'U/
 ij Y  30
;
(an •
im ~
» >?
'
possibility).
(a certainty).
e
with
A
Thus, the supposition (x=0) is incompatible with though not with A. 120. Next, suppose our data to be AB, in which
B
Thus the supposition (#=1) though not with B.
A B
^
/ :
:
]

—
A
incompatible
get
N
1
(
(
We
x=0.
Next, suppose
is
)
e
:
:
n
denotes ox
(a certainty).
(an impossibility).
—  > 4a; + . 4
B
denotes Qx
3
—  < 4« + . 4
2
We
get 3 ^4
.
4;/:
IV = / 13\ p / 13 = (^12 (^12 3J )
/
CALCULUS OF LIMITS
120, 121]
§§
Hence we
get
AB = >£> — = 13
5
/5
,
t01
In this
denotes
what what
for
and
for
2xl — x— 6
=
1
T2J
>
13\ :
'
/
l2J
data AB are mutually A or B, is possible taken combination AB is impossible.
but the
121. Find positive,
/5
(8
our
Each datum,
incompatible. itself;
(an impossibility)
:
therefore
case
>i
13\
>aJ>
\8
by
111
positions
of
F
x the ratio
is
F
when
negative,
positions
28 — x
+ 84
2x2 29a;
2(x
4)(x

10£)
x(x  3)
x(x3)
in § 113, let a denote positive infinity, and let /3 Also let the symbol (to, n) denote 'negative infinity. assert as a statement that x lies between the superior limit m and the inferior limit n, so that the three
As
symbols (to, synonyms.
(m>x>ri), and
n),
We
have
consider
to
(m six
— x)\x — nf limits,
are
namely,
in descending order, and the five to the five statements corresponding intervening spaces 10i), Since x must lie (a, (10J, 4), (4, 3), (3, 0), (0, (3). a,
in
10i, 4,
3,
0,
(3,
one or other of these e
= (a,
10£)
five spaces,
+ (10l,
4)
+ (4,
3)
Taking these statements separately,
1
Oh
4)
(4, 3) (3,
+ (3,
0)
+ (0,
(3).
Ave get
 1 0)>  4)>  3) V F p p  3) FK (z  1 Offix  4) (x 1 0)> 4) N N ¥  3)V F p (x  10i) (fl  4) (x  ±)"(x  S) N 0) (x  3)V (x  10)>  ±f(x  3)V F N N Fp /3) x" x\x  3 f{x  4) (sc  1 0i) ( 0+)
(a, 1 (
we have
:

(x
p
p
1
0)
:
(x
:
:
:
Thus, these
(.v
:
five
:
:
:
:
:
:
,
V
(;v
:
:
.
:
statements respectively imply
F
p ,
FN Fp ,
,
SYMBOLIC LOGIC
112
F N Fp
[§§
121, 122
the ratio or fraction F changing its sign four times as x passes downwards through the limits 1 Oi, 4, ,
,
Hence we get
3, 0.
F p = («, 10*)+(4, 3) + (O,0); F N = (10i 4) + (3, 0). That
is
and
or between 4
ment that F that x 3
and
3,
is
either between
is
equiva
is 'positive is
and 10 \,
either between a
is
or between
negative
is
F
statement that
to say, the
lent to the statement that x
and
ft
;
and the
state
equivalent to the statement
10i and 4 or
between
else
0.
2«l_28 122. Given that
values of It is
—
—
x
the value or
find
to
,
x
3
x.
evident by inspection that there are two values of
x which do not satisfy this equation
m When x=0, n
.
we get 6
2a;
1 = 1
x3
;
...
while
and
they are
—x = — 28
28
3'
.
and
;
3. .
evi
dently a real ratio  cannot be equal to a meaningless o
— 28
ratio or unreality
2rel
.
get 6
— = —5 xS 
be equal to
28 —
...
while
,
— = 28 —
28
x
.
Excluding
denote our data, and
let
5
.,
,
;
3
(x=0) and (x=o) from our
A
Again when x=3, we
(see § 113).
fl and evidently J 
therefore
cannot
the suppositions
universe of possibilities, let
F=
—x ——  —x
.
We
get
3
A Fo .
.
/ 2a
_
\x3 :
28\°.
f
2(x
xj'l
 4)(^10i)} {(X
:(x
=
4:)
+ (x=10i).
x(x'S) :
J
(x 4f + (x10if
§§
CALCULUS OF LIMITS
122124]
From our
we conclude
data, therefore,
113
must be
that x
either 4 or 10i.
„ 123. Suppose
„^ n
.
,
we nave given
13j;
3
8
4
>
3«
6
4
— 7% 8
to find the limits of x.
Let
A= .
A
/13a;
3x
3

G
4
\ 8
4
=
'
we
,13a;
3
ment that
,
=
N
than
— 7x
6
4
—
the statement that
is
—
Q
7x
TT
Hence
.
4
whatever value we give
is
—
,
13x
3
8
4'
sign
=
,
which,
for
all
values
given If in the b for the sign
> we ,
,
so that, in this case,
the value of
124. Let the limits of
A
is
G
—
7a? ,
4 8 evident from the fact
is
to its
simplest form, r
of
is
x,
equivalent
is
to
shall get
G7,y =
8/
4
4
8
ox
J
statement we substitute the
/13a_3_3. \
than
equal to *
8
6
less
'
This
to x.
when reduced
.
4
\2x
7x
6
2,x
,
must be
4
8
8
that 
3
13a;
.
nnposl
4
8
3%
.
is
8 Q
'tQ,,
and so
sible,
for
Thus, the state
>/.
3x
,i
,
greater
<
substitute the sign
4
8
_ XPp 7«)
v.
A=
.
is
=
we
shall get
.
1Q  B6  6x B = (13# + ,
l
If in the given statement
the sign >,
have
1'
7.A
8/

,,
We
denote the given statement.
()0
=
a formal certainty, whatever be
x.
A x.
denote the statement
We
A = (x2 
x}
+ 3>2>x\
have
= { (x  2x + = {{x l) + 2}" =
2x + 3) p
2
1
)
+ 2 }p
2
e.
H
to find
SYMBOLIC LOGIC
114
Here
A
is
124128
a formal certainty whatever be the value of
no
so that there are If
[§§
we put the
sign
=
x,
limits of x (see § 113). for the sign > we shall get
real
finite
A={(,el)°
+ 2}° =
>
h
Here A is a formal impossibility, so that no real value of 2 It will be remem2x. x satisfies the equation x + 3 bered that, by § 114, imaginary ratios are excluded from our universe of discourse. 125. Let it be required to find the value or values of
=
We get (x Jx=2) = (x  Jx  2)° = (x + x* + x°) _ J x _ 2 )° = x\x  Jx  2)° = x {(x  2)(xi + 1)}° = A (^  2)° = (x = 4) N for (x = 4) implies x and x° and « are incompatible the datum (x  Jx  2)°.
x from the datum x
— s/x= 2.
v
;>J
(
p
h
'
P
v
,
126. Let
with
be required to find the limits of x from
it
datum (x— Jx>2).
the
(xJx>2) = (xJx2y = (c '+x"+x°)(xJx2y i
= x (xJx2y 2)(x + 1)}^ = ,^ 2) = p
=
p cc
{(x
i
p
i
F
for
(v>4) implies x and datum (x — Jx — 2) ,
the
1
127. Let the
(x
it
x°
and
N re
(.> ;
>4)
;
are incompatible with
'.
be required to find the limits of x from
datum (x— Jx<2).
Jx<2) = (x Jx 2) N = (.^+^M')<> Jx= (x + x°)(x  Jx  2) N = of(x  Jx  2) N + x° = x {^  2)(x$ + 1)} N + x°=x*(xi  2f+x° = x\x* < 2) + x° = x\x < 4) + x° = (4>^>0) + O=0).
2)
N
v
¥
Here, therefore, x
may have any
value between 4 and
zero, including zero, but not including 4.
128. The symbol
gm
denotes any
number
or
ratio
§§
CALCULUS OF LIMITS
128, 129]
115
than m, while Im denotes any number or ratio less The symbols gx m g2m, g3m, &c., than m (see § 115). denote a series of different numbers or ratios, each greater Similarly, than vi, and collectivelyforming the class gm. the symbols l{m, l2 m, l3m, &c, denote a series of different numbers or ratios, each less than m, and collectively forming the class Im. The symbol xgm asserts that the number or ratio x belongs to the class gm, while x '" asserts that x belongs to the class Im (see § 4). The symbol xgm gn is short for xgm z gn the symbol xP mln is short iorxgm x ln ; and so on (see § 9, footnote). These symbolic conventions give us the formulae
greater
}
1
'
;
m = (x>m) = (xmy.
(1) x^
x
(2) (
129. Let
We
lm
3) x
m
= (x<m) = (x mf. = x° mx = (x > m)(x < n) = (x — mY(x — iif = (n> x > m).
gm
ln
ln
and n be two
different
numbers
or ratios.
get the formula (
To prove numbers) af
1
this
m.gn
)
,,:<""
•
9*
V
= X^V + Xa n = (x > m > n) + (x > n > m). 71
we have
(since
m
and n are different
_ ^m.gn^jn + ^m^ for ^jn + ngm _ g = xgmx*nm9n + xgmx n9m = {x9mm9W)x + nnam)x°m = xgmmgn + x9 ngm = (x > m > n) + (x > u > m ffn
<)n
{,:»P
for
term the outside
in each
factor
may
),
be omitted,
compound statement in the bracket, since x>m>n implies x>n, and x>n>m implies x>m. Similarly, we get and prove the formula because
it
lm ln
(2) x
implied in the
is
= J mm + aV = (x < m < n) + (x < n < m).
This formula
ln
may
be obtained from (1) by simply sub
SYMBOLIC LOGIC
116 stituting
for g
I
and the proof
;
is
129131
[§§
obtained by the same
substitution.
130. Let m,
We
ratios.
aT m rf
fi 
1)
(
(
2)
be the three different numbers or
r
n,
get the formulae
= m n Jr + — Jm ni m + x
gn gr ln
lr
,
7>iP
in
ln
lr
,/;?
W'" + aWV*. lm
n nlr + J r r lm r ln
ln
.
These two formulae are almost selfevident but they may be formally proved in the same way as the two for since m, n, r are, by hypothesis, formulas of § 129 or ratios, we have numbers different ;
;
mgn or + ngm gr + ^m gn _ ^ m + nlm.lr + rlm.ln = €^ jf n 9n.gr = x gm.gn.gr e ^ by fas formula .
.
ln.lr
while
x
im.in.i r=.
x
multiplied
o:°
m
e^ 9n 'jr
implied factors, as in
§

ln

lr
omitted implied factors, as in r, s,
and
131.
principle evidently applies to four ratios, m, n,
so If,
on
to
any number.
we suppose m, n, r to be inferior terms of the alternative ev namely,
in § 130,
limits of x, the three
mgnir
i
When get Formula (1). by the alternative e 2) and § 129, we get Formula (2).
we
129,
we have multiplied xlm The same
= ae,
and ^q same formula. When we have by the alternative ev and omitted a
^
im.in.ir
ngm ° r
gm an
r
,

,
respectively assert that <m
nearest inferior limit, that n
is
is
the
the nearest inferior limit,
And if we suppose be superior limits of x, the three terms of the ln lr n lmAr r lmAn respectively alternative e2 namely, m assert that m is the nearest superior limit, that n is the that r
m,
is
the nearest inferior limit.
n, r to

,
,
nearest superior limit, that r
For of any number of
is
the nearest superior limit.
inferior limits
nearest to x is the greatest; whereas,
superior limits, the nearest to x
,
,
is
the
of a variable
of least.
x,
the
any number of
And
since in
each case one or other of the limits m, n, r must be the nearest, we have the certain alternative e1 in the former case, and the certain alternative e2 in the latter.
CALCULUS OF LIMITS
131133]
§§
It is evident that
m may ln
that
mPn may be replaced by (m—n) v
be replaced by (m
replaced by (m
117
— n)"(m — r)
N ;
— ?i) N
and
,
that
mlnAr may
be
so on.
CHAPTER XV When
we have to speak often of several limits, &c, of a variable x, it greatly simplifies and shortens our reasoning to register them, one after another, as they present themselves, in a tabic of reference. The * symbol ®m>, n asserts that xm is a si^erior limit, and x n an inferior limit, of x. The* symbol xm n rs asserts that xm and xn are superior limits of x, while x r and xs 132.
x x x2 x3 ,
,
,
,
are inferior limits of
aW.n means xm'.n'.r.
and
Thus
x.
(x
means
,
 m f{x  n J or (xm >x>xn  mf(x  xn f{x  r) p(x 
),
(x
s
f,
so on.
The symbol
(with an acute accent on the osm m) always denotes a proposition, and is synonymous with (x — xm y, which is synonymous with
133.
numerical
.
suffix
It affirms that the mth limit of x our table of reference is a superior limit. xm (with no accent on the numerical suffix), a proposition, asserts that the mth limit of x
(x<xm ).
our table of reference xm means (xxm ) p
is
an
registered in
The symbol when used
as
registered in
Thus
inferior limit of x.
.
my memoir
* In
on La Logique Symbulique et ses applications in the du Congres International de Philosophic, I adopted the symbol x™ (suggested by Monsieur L. Couturat) instead of iy„, and .vm " instead ofxm n rs The student may employ whichever he finds the more conBibliotheque
r
>
t
.
From long habit I find the notation of the text easier but the other occupies rather less space, and has certain other advantages in the process of finding the limits. When, however, the limits have been venient.
;
found and the multiple integrals have to be evaluated, the notation of the text is preferable, as the other might occasionally lead to ambiguity (see §§ 151, 150).
SYMBOLIC LOGIC
118
134, 135
[§§
134. The employment of the symbol xm sometimes to denote the proposition (x — x,m) v and sometimes to denote the simple number or ratio xm never leads to any ,
,
ambiguity
for the context always
;
—I
X
it
is
which
fraction ,
of
reference
outside the
z)
supposed
is
3,
bracket denotes the
the
be marked in the table
to
the third limit of x; whereas the x3
as
bracket,
is
— x3 Y,
statement
(x
Similarly,
when we
A=
— \x — x — x
the xs inside
that
clear
makes the meaning
For example, when we write
perfectly evident.
,
affirmed to be equivalent to the
and
is
therefore
statement
a
also.
write
+ 8 4 > 2 9x) = (x  1 0) + (x  4)N = {x — x^f + (x — x2 Y = x + Xg, p
2
( 2,,;
l
we
assert that the statement
+ x^,
native statement x l (as a statement)
A
equivalent to the alter
is
of which the
first
term x1
that the limit x1 (denoting
and the second term
inferior limit of x,
10)
asserts
an
is
asserts that
»_,
Thus, the limit x 2 (denoting 4) is a superior limit of x. the alternative statement x \x2 asserts that "either xl >
is
an
inferior limit
of
x,
or else x 2
is
a superior limit
x.
135. The
operations of
calculus
this
of
limits
are
mainly founded on the following three formula? (see
§§
129131): (
In the /„,./„,
/
xm
.
n
= xm ~ x n) xn\xn ~ xm) = xm \xm xn + xn \xn — xm
xm n
\° )
x m'.n
first
and
1
(Z)
,
""
''m\
>
)
''
m' .n\'
vi
''
n)
)
.
'
symbol xm n means and xn are both inferior limits
of the above formulae, the
asserts that xm
•
CALCULUS OF LIMITS
135, 136]
§§
119
The statement (xm  xn f asserts that Xm is greater x. than xn and therefore a nearer inferior limit of x while the statement (xn xmY asserts, on the contrary, that xn and not xm is the nearer inferior limit (see §§ 129, In the second formula, the symbol xm n asserts 131). The statethat xm and xn are both superior limits of x. ment (xm  xj" asserts that xm is less than xn and therewhile the statement fore a nearer superior limit of x K — that xn and not xm is contrary, on the asserts, x (x m) of
;
>
.
;
n
The
the nearer superior limit.
third formula
is
equiva
lent to '
and
of
§
.n
•
\
xm
xn)
>
a superior limit, and xn an inferior
xm then xm must be greater than xw
asserts that
if
is
When we
have
limit, of x,
13G.
m
three inferior limits,
Formula
(1)
135 becomes %m .n.r = xm «
in which a asserts that xm inferior limits,
ft
xr
is
asserts that
+ Xnfi + X y, r
the nearest of the three
is
asserts that xn is the nearest, In other words, the nearest.
and y
— xm ~ xn) xm ~ X p = {xn xm (xn — xr y=(xr xmf(xr xny. a
\
r)
\
)
)
When we
have
Formula
three superior limits,
(2) of §
135
becomes xm'. W. ?
= xm' a + x
n'ft
+ xr'7>
in which, this time, a asserts that xm
the nearest of the
is
ft asserts that x n is the nearest, In other words, is the nearest.
three superior limits,
y
asserts that
xr
= (xm
and
xr ) xn \Xm ft=(xn Xmf(xn xrY a
)
y = (xr — xm f(xr — xn ) Evidently the same principle
number
may
of inferior or superior limits.
s .
be extended to any
SYMBOLIC LOGIC
120
[§§
137, 138
137. There are certain limits which present themselves so often that (to save the trouble of consulting the Table of Limits) it is convenient to represent them by special
These are positive infinity, negative infinity, and Thus, when we have zero (or rather an infinitesimal). any variable x, in addition to the limits x v x 2 x 3 &c, registered in the table, we may have always understood symbols.
,
,
the superior limit xa which will denote positive infinity, the limit xQ which will denote zero (or rather, in strict logic, a positive or negative infinitesimal), and the always ,
,
understood inferior limit xp infinity (see § 113).
variable
y,
,
which
will
denote negative
Similarly with regard to any other
we may have the
three understood limits ya
,
&c. y yp in addition to the registered limits yv y 2 y 3 Thus, when we are speaking of the limits of x and y, we ,
,
,
,
= =
 a. x (or dx or dy) x yp y the other hand, the statement xa m asserts that x lies between positive infinity xa and the limit xm registered in the table of reference; whereas xm p asserts that x lies
have xa — y a = a
;
= =
;
fi
On
,_
,
,
Simibetween the limit xm and the negative infinity xp larly, xm tQ asserts that x lies between the superior limit while ;% n asserts that x lies xm and the inferior limit and the inferior limit xn limit superior the between that x is positive, and implies statement « Thus, the m the statement xQ is Also, negative. is that x xff n implies s the statement x is and X statement synonymous with the p shown in § 134, As x statement synonymous with the to denote a sometimes symbol x the employment of the Q not lead need statement, a limit, and sometimes to denote .
,
;
.
,.
,
;
.
any ambiguity.
to
138. Just as in finding the limits of statements in pure logic (see §§ 3340) we may supply the superior limit n when no other superior limit is given, and the inferior limit
e
when no other
inferior limit is given, so in find
ing the limits of variable ratios in mathematics, we may supply the positive infinity a (represented by xa or y a or z &c, according to the variable in question) when no ,
§§
CALCULUS OF LIMITS
138, 139]
other superior limit
and the negative
given,
is
121 infinity
(3
(represented by .^ or yp or zp &c.) when no other inferior Thus, when xm denotes a statement, limit is given. ,
namely, the statement (x — x^f, it may be written xa m and, in like manner, for the statement xn which denotes (x — xn y, we may write xn tP (see § 137). 139. Though the formulae of § 135 may generally be dispensed with in easy problems with only one or two variables, we will nevertheless apply them first to such ,
;
>,
,
make
problems, in order to
meaning and object
their
apply them afterwards to more complicated problems which cannot dispense with their aid. Given that 7a?— 53 is positive, and 67 — 9a; negative;
clearer
when we come
required the limits of
Let
A
denote the
to
x.
datum, and
first
B
the second.
We
get TABLE
A = (7x5Sy = (x~X=x1 =xa
,,
1
B = (679*) N = (9,:G7) p = (^y
)
Hence, we get
AB = av. By Formula
(1) of § 135,
xa
x
,,
2
=x
a ._
j
we get
= Xjlfa — x + x (x — x^f p
a5j _
2
2
2)
2
53_67
Y
,67
53V
9~
= r (477469) + r (469477) p ,forQ = (63Q) = x1e + aw = x (see § 11, Formula? 22, 23). From tne aata AB thereThus we get AB = a 12 = p
p
p
t
t
2
1
1
,i'
fore
we
infer
that
between positive greater than
53
—

.r

x
infinity ,_4
or 7.
.i
between xa and ,i\ that 53 In other words, x and
lies
;
—
is.
is
SYMBOLIC LOGIC
122
[§§
139, 140
Now, here evidently the formula of § 135 was not for it is evident by mere inspection that u\ is
wanted
;
greater than
,r 2
,
that a\ being therefore the nearest
so
inferior limit, the limit
out of account.
AB = A =
,r aU 140. Given
positive
Let
;
A
,r
2
is
In fact
A
—
53
superseded and
may
implies B, so that
be
left
we get
.
7x
that
is
required the limits of
denote the
first
negative
datum, and
B
get—
A = (7£53) N = (x
53
x x,
and
07
—
9*
x.
—
the second.
We
CALCULUS OF LIMITS
141]
123
CHAPTER XVI 141. We will now consider the limits of two variables, and first with only numerical constants (see § 156). Suppose we have given that the variables x and y are both positive, while the expressions 2y — 3# — 2 and 3^ + 2^ — 6 are both negative; and that from these data we are required to find the limits of y and x in the order
Table op Limits.
y, x.
A denote We have
Let data.
our whole o 2/i
A=y
r
x p (2y

3x

6) Beginning bracket
with
factor,
(2y

3x
we
N
2) (3?/
+ 2x
=^+ l
N .
the
first
get*
N 2Y = (y^xlJ = (y A ) = Vv
Then, taking the second bracket
?
factor,
we get
SYMBOLIC LOGIC
124
Substituting this alternative for y v A, we get
2
.
[§141
in the expression for
A = (y rx v + y^\)y^. o = (yv. 0% + feiftK.o
= VV. VC
V
a.
+
.
y
==
C a'. 1.
2/l'.
0^1'.
2/2'.
'
'1
O^a'.
1
first term the superior limit xa because it the nearer superior limit x x and omitby is superseded term the limit x because it is supersecond ting in the The next step is to apply limit x v nearer seded by the We ^factors the Formula (3) of § 135 to yvo and yz
omitting in the
;
,
.
get
= Vv.
yv.
0(2/1
 y<>Y = yv. 0(2/1)* = yv. d
= yv.o(3x + 2) = yvJx + ^ =
®
+
l
1
(*
'
yi'.
 x^f
J
—
2/1'.
0^2 P
2  x = = = J = y (6  2^ = ^,0(3 xf = y^ Q(x 3) N = 2/2'.0<%P
y%.
2/2'.
2
0(2/2
2/2'.
?7o)
o(2/ 2 )
for
}Jx. cftv.
evidently
o(
,
Substituting these equivalents of
A=
2/2'.
x
2.0
is
"J~ 2/2'.
O^a'.
3'.
1
?y
=
r
2/l'.
and 0^1'.
?/ 2
2/2'.
1
O'^V. 1
a nearer inferior limit than
therefore supersedes
,v
2
;
x3
while
is
we get
in A,
.
a nearer
>
,r
2
,
and
superior
xa (which denotes positive infinity), and theresupersedes fore xa We have now done with the ?/stateonly remains to apply Formula (3) of § 135 ments, and it It is evident, however, to the ^'statements xvo and xsi this is needless, as table, that of the by mere inspection nor any inconfactor, discover it would introduce no new than is, than greater that x sistency, since x x is evidently process The zero, and x3 is evidently greater than xx limit than
.
.
,
.
therefore here terminates, and the limits are fully deter
CALCULUS OF LIMITS
§141]
1
25
We
have found that either x varies between xx and zero, and y between y1 and zero or else x varies between xB and xv and y between y2 and zero. The figure below will illustrate the preceding process The symbol x denotes the and table of reference. mined.
;
distance of any point
P
(taken at random out of those in line x and the symbol y
the shaded figure) from the
,
denotes the distance of the point The first equivalent of the data
P from
A
is
the line y
.
the statement
xz xo x
r
ox o>
asserts that y1 and y 2 are superior limits (or zero) is an inferior limit of y, and that
which
llv
2
of
y,
x
(or zero) is
this
that y
an
inferior limit of x.
compound statement
A
is
It is evident that
true for every point
the shaded portion of the figure, and that it for any point outside the shaded portion. equivalent
+ Vv. o
of
the
A
data
is
the
is
P
in
not true
The
final
y v% x r true for every point alternative
_
term of which is P in the quadrilateral contained by the lines yv y x v xQ and the second term of which is true for the triangle contained by the lines y 2 y0) xv tl
V.
i>
the
first
,
,
;
SYMBOLIC LOGIC
126
[§142
2 2x — 4 y — 4./.' is negative and y required the limits of y and x. positive get Let A denote our data. Table of Limits. p r)\y 2,,;4) (v/4 A +
142. Given
+
tliat
;
We = = 4 N (yyi)"; tf  ±xy = {(y2 JxXy + 2 Jx)Y = {y2 s/xr(y+2 sfxy t
2
tt)
(7/
—
for (y
2
s/^YiV
+ 2 x/^) N
We
impossible.
*s
therefore
get
a= By Formula 2/ 3 i .
2/2'.
(1) of
3(2/
§
 2/i) p =
2/2'. s2/i
=
y2.3. i
135 we get
 Vif + y/yi  7hY =  4) + Vl{2x 2jx 4)* = y (2tf  2 = y (#  x/«  2 + y^a?  s/x  2 N (see §§
^
2/3(2/3
3
p
p
126,
)
)
3
127)
^X ~~l)
slxl
~i*
Y
"((•"DlM^i)!}' = (.j4) +2/i(*4) n  N = 2/3^i + 2/r*r= y (# ~ «i)P + p
^
?/3
3
^'i)
Therefore
A = 2/2'.3^1+//2'.l^l'
We !h.
now apply Formula
3
2/2'. 1
= =
2/2'.
2/2'.
(3) of § 135, thus
 VsY = Ik. s( 2 */« + 2 xA')'' = y*. 3 e P r 1(2/2  2/i) = i(2# + 2 V'/'  4) 3(2/2
2/2'.
= yr. i(* + «/*  )" = V*. i{( V* + 2J (2) } x ~ = = i(« 1 = h: M' zY i#2r
2/2'.
)
2/2'.
CALCULUS OF LIMITS
142, 143]
§§
127
Thus the application of Formula (3) of § 135 to y2 3 introduces no new factor, but its application to the other compound statement y2 introduces the new statement Hence we x2 and at the same time the new limit x 2 finally get (since Form 3 of § 135 applied to xa and Xy 2 makes no change) ,
,
1
.
,
.
a
A^y.,.3^+^1%.2
(see
§§137, 138).
" either x lies between x a and x., and y between the superior
This result informs us that (positive
infinity)
oc
jcz
and the inferior limit y 3 or else x lies beand x2 and y between y2 and y v The above figure will show the position of the limits. With this geometrical interpretation of the symbols x, y, &c., all limit y2
tween
;
£&,
,
the points marked will satisfy the conditions expressed by the statement A, and so will all other points
bounded by the upper and lower branches of the parablank area cut off by the
bole, with the exception of the
line
yv 2 143. Given that y — ±x is negative, and y also negative required the limits of y and x.
+ 2x — 4
;
Here the required
limits (though they
may
be found
SYMBOLIC LOGIC
128
independently as before) the diagram this
in
may
problem and that of
case y
+ 2x — 4
143145
be obtained at once from difference between
The only
142.
§
[§§
142
§
is
that in the present
negative, instead of being, as before,
is
Since y 2 — 4a; is, as before, negative, y.2 will be, as before, a superior limit, and y 3 an inferior limit of y so that, as before, all the points will be restricted within But since y + 2x — 4 the two branches of the parabola. positive.
;
has now changed sign, all the admissible points, while still keeping between the two branches of the parabola, The result will be that the only will cross the line y v now be restricted to the blank will points admissible portion of the parabola cut off by the line y v instead of being, as before, restricted to the shaded portion
and extending indefinitely A positive infinity. towards in the positive direction that the show will glance at the diagram of § 142 within
the
two
required result
branches
now
is
1J2'.
with, of course, the
3'%.
same
'
V\'. 3^1'. 2>
table of limits.
CHAPTER XVII 144.
The symbol
A —
,
when
the
numerator and denomi
nator denote statements, expresses the chance that A is true on the assumption that B is true; B being some state
ment compatible with the data
of our problem, but not
necessarily implied by the data.
145.
The symbol
A
denotes the chance that
A
is
true
e
when nothing is assumed but the data of our 'problem. This is what is usually meant when we simply speak of the "
chance of A."
§
CALCULUS OF LIMITS
146, 147]
The symbol^—,
146.
A ——A —
A
or
B
and
;
upon
S(A, B), denotes
this is called the dependence* of the statement
the statement
(when negative) the
— when
chance
synonym
its
129
B.
It
decrease,
indicates
the
increase,
or
undergone by the absolute
B
the supposition
is
added
to our data.
€
The symbol
<5°
D B
or
,
A
dependence of
synonym
its
upon B
Fig.
1.
A
said
is
which implies, is
to
be independent
,
€
symbols
a!',
— — — ,
,
e
,
I/,
c
',
the
on
statement
B
(see S 149), that
€
,
&c. (see
S
145); and the
€
&c, respectively denote the chances
&c, so that we get
e
1
*
oj
3.
&c. (small italics) respectively
b, c,
ABC— chances—,
represent the
Fig.
2.
as will be seen further
independent of A. The symbols a,
147.
e
E
E
Fig.
B
In this case the state
is zero.
E
ment
S°(A, B), asserts that the
Obscure
= n + a' = b + b' = c + c' = &c.
dependence and independence in prosome writers (including Boole) into serious errors. The
ideas about
bability have led
'
definitions here proposed are,
'
'
I
believe, original.
'
SYMBOLIC LOGIC
130
[§148
148. The diagrams on p. 129 will illustrate the preceding conventions and definitions.
Let the symbols A,
B
that a point P, taken at
assert respectively as propositions
random out
of the total
number
of points in the circle E, will be in the circle A, that
it
be in the circle B. Then AB will assert that P will be in both circles A and B AB' will assert that P will be in the circle A, but not in the circle B and similarly will
;
;
for the statements
In Fig.
1
A'B and
we have
A'B'.
§§
CALCULUS OF LIMITS
149, 150]
The following formulae
149.
133
are easily verified
:—
*}&
<•>£*?£(•>
The second of the above eight formulae shows that if any statement A is independent of another statement B,
B
independent of A for, by Formula (2), it is B) implies S°(B, A). To the preceding eight formulae may be added the following then
is
;
clear that <S°(A,
:
AB = A B B A = e
"
e
e"B
*A
B A B (11)^± = + _ AB
AB_A B _B A Q^~Q'AQ~QBQ
(10)
;
.
(12)
A+B
150. Let A be any statement, and proper fraction; then A x is short
A—
=%), which
\ €
that
asserts
= A + B _^?
x be any positive the statement
let
for
chance
the
of
A
is
x.
/
(AB)* means
Similarly,
convention gives us a and
the
—=
AB
\
x);
following
A b (as
before) are short for
e
(1)
A^:^=^ A V
(3)
(AB)x(A + By>:(x + y = a + b);
(5) (AB)"
= (A + B)
(2)
a+& ;
and so
on.
This
in
which
formulae,
— and
;
B . e
AB^AB/^A + B)*; (4) S°(A, B)
= (AB)
f
'»;
SYMBOLIC LOGIC
132
<6
>(s4)=(s=f)=*( A /A B\ /A = [B = A) \B
„ (7)
\
[§§150,151
B >;

+ {a = b):(AB)V + (a = h)
:
!

It is easy to
may
prove all these formulae, of which the last be proved as follows
A_B\
/A_Z> A\ /K_b A\° (A/ 6\)° B~Ay' \B~a'B/ \B a'B/ \ B\ X ~ a/ ;
:
:
J
\
:
A
V
/A
jjj(a&)
:(
The following chapter
B
\
=
+ («^)°:(ABr+(a = &).
0J
some knowledge
requires
of the
integral calculus.
CHAPTER XVIII 151. In applying the Calculus of Limits to multiple integrals,
will
it
be
convenient
to
the
use
following
which I employed for the first time rather more than twenty years ago in a paper on the " Limits of notation,
Multiple Integrals " in the Proc. of the Math.
The symbols
^>{x)xm!n
and x m

meaning. tion
The symbol
(p(x)dx,
xn
commonly expressed '
'
™
vm\n to the
>.
n
differ
differ
,
in
also in
>.
taken between the superior limit xm and
the inferior limit
fX
xm
n is short for the integra
m
<J>(%)%
which
n (p(x),
the relative positions of
Society.
\
an integration which would be
The symbol xm
.
left, is
short for
n
m)—
(j>(x
For example, suppose we have
j
I
so
dx(p(x) or
the
symbol
= ^(x).
Then,
l
m<
^
') (
=
p{x)x m n
(
xn p{ that we can thus ,
J\
v
'
with
by substitution of notation, we get
= #m.»' K#) = ^GO  ^G''»)
ex m
the form
either in
l
'
,
entirely
CALCULUS OF LIMITS
151153]
§§
133
with the symbol of integration,
dispense
the
as in
/,
following concrete example.
Let
be required to evaluate the integral
it
Table op Limits.
'
Cz
C' 1
C'¥
dy
"'I
I
JVi
J«a
dx,
I
J'o
=
za
=X V2 = h
«!
=«
h'o
=0
Vl
c
the limits being as in the given table. The full process is as follows, the order of variation being z, y, x.
= (z  z )yv x r =  c)y v xv = (k  ^/>% = { (hA cyd (hvl ~ cy } xv = { (I.*  «b)  (W  cb) x v = (h^  ex  \tf + bc)x r = #i\ o(^^ — ikr — 2# + &«') = £a — lea — \b + bca.
Integral zr
.
2
yv
&.
.
2
1
.
2
.
(//
.
.
9
,
2
?/r 2
.
.
o
2)
2
,
2
}
.
2
3
3
2
The
152.
evident (
ci
\
'
'.
n
.
;
.
J
.
^'
(5)
6
self
=  %n' m (2) #«>*V „ = ~ <£OX'. m *W. „<£(») = Xn'.rn
(3)
(
following formulae of integration are
:
*W
1)
.
//?»'
''
/
m'
n'
.
71
.
r'
'
~r"
(xm „
(9)
.
.
.
+ xr
153.
.
, .
.
"m'
s
.
s
rnfir'
.
)(p(x)
mt
.
r
J
I
.
s
2/m'
s
.
"
J >
•
r
.
n
.
s)
.
.
2/«'
>•
.
,
,
.
stated, the
.
mP^s' r .
5
'
= (xm + ^ n )(p(x) = 0(#)(#m< +
+ *V
As already
,
<.
.
i/n'
"V
r
.
a?
r
s
>
r)
.
.
.
/
.
%
iH>
;
„)•
.
symbol
,
when
A
and
B
A is true on the x and y be any
are propositions, denotes the chance that
assumption that
numbers
B
or ratios.
is
Now,
true.
The symbol
let

means
3/B
when
either of these two
suppose the number PU Ihus,
r
1
numbers

y is
x
— B
missing, Ave
understood.
x A xA  means  x — B
IB
;
and
—A means x x —AB 1

xB
;
and
may
V34>
SYMBOLIC LOGIC
xx
=l
[§§
154, 155
§
CALCULUS OF LIMITS
155]
135
Substituting these results in our expression for Q, shall
we
have
Multiplying by the given certainty xv 0 (see table), we get
XV. oH == lV i\ Applying Formulae 137)
+ 'V.
0^2'
1'. 3.
r. 2.02/2
and (2) of
(1)
we
135,
§
get (see
§
#3. _
X2.
(3C
'<0 )"
f
?,
•
o(''
,
expression for x v
results in our
Q,,
get #r. oQ
= %(%2/3 + AW3')y2' + == X2'
We
s
3)
^
3
Substituting these
we
^ V
= ^3 + •% = XlX + *oK = xs  xj* + ar^  x n = X^e + xv n = x + = X2* + ^ = «* = ff«te Z
t% r
.
31/2'
.
"r ^2' 0^3'
3
.
now apply Formula
"*2\ 3' ^2'.o'
''3'. 2'
.
•'3'
2^/2
.
#3
2' "•"
135
(3) of §
2^2'
'.
to the
statements
**nUS '2' .
3
*2' .0 "^3'
.
2
=V
l' .
== ^2'
=%
= r $%' X ^0) = ''V e — = XS 2V^3 Xi)
3(^2
'
.
Q\
<
2'
.
2
.
,?,
.
2)
.
iVl
This shows that the application of § 135, Form 3, introduces no new statement in y so that we have finished with the limits of x, and must now apply the formulas of §135 to find the limits of y. Multiplying the expres;
sion found for #i'.o 7/i'
By applying
tr ro
==, ''2 .
= y20
by the datum yv
s/Ar. r. 3.0
+
the formulae of
tion of the table, y.2
Q
we get y%
'''2 .
§
v=
0//3
.
2'.r.
<'V.2yi\oQ
we *
get
x3. 2?/i'.2.
o
135, or by simple inspec
y3 and substituting these results
side of the last equivalence,
Q,
y.y
;
.
=y
z
;
=
Vz' y$ 2 r righthand
in the
we get
= ''V.3y2'.3 + <>2\oy3\o + <'V.2/'r.2
<
.
.
'>
SYMBOLIC LOGIC
136
The
application of
Form
135,
§
155, 156
[§§
to the ystatements
3,
no fresh statements in z, nor destroy any that it contains an impossible factor showing term by found the nearest limits of y and it therefore We have Multiplying the last find the limits of z. to only remains get z we datum the expression by ro
will introduce
tj.
;
QA = Q,?v The
_
Q
yv
.
oZ r
application of
effect
= Ov
.
§
.
easy,
3
+ dfe
is
Int
.
o2/ 3
3, to
+%
o

.
.
s#r 2>r .
the factor
therefore over
We
;
zr
and
.
o
will _
The
p is a certainty. )
to evaluate the integrals.
it
pro
only
get
^
A
= Int(/e.r. $2 + x* ys + %? =l. The A = Int xv ,&v .
Int
.
{z — z x
no change, since
A
for
.
Form
135,
cess of finding the limits
remains
3y 2
3
.
.
o
.
Hl\
.
2K'
.
integrations
fs l..
are
. f
and the
result
log 2 (Naperian base),
is
which
is
±
5
a little above . 9
156. Given that a is positive, that n is a positive whole number, and that the variables x and y are each taken at random between a and — a, what is the chance that n+1
 «} positive ? {(x + y'T  a} is negative and {{x + y) let Q deTable); (see x Let A denote our data y Y 2 r#2 s n the denote R let and a} note the proposition {(x+y) p n+1 N exponent the in which a} proposition {(.?• + y) positive. P denotes negative, and the exponent _
,
,
We
have
„ , , to find the
.
chance
QR —^,
which = ,
.
,
Int
QRA .
In this problem we have only to find the limits of integration (or variation) for the numerator from the
compound statement QRA,
the limits of integration for ?/ 1 .2^1.2. the denominator being already known, since A
=
CALCULUS OF LIMITS
156]
Table of Limits
=
y1 a ~a y2 = i
y3 = a
n
—X 1
yi
= — an — x i
=a n + —x 1
7i •J
5
1
y,
=  « n+l  x
Vtf
SYMBOLIC LOGIC
138 #5
3 .
=
«'
s
when a> a"+i (
—a
We
=1
<'V. 5
;
(
an impossibili ty)
we have
1
]
x6
so that
Tl ;
must now apply
We
— an
2a
(
—
a;
§
135,
.
5
Form
y.
.
Q%1'. 2 = VS. 2
=
For
.r 3 ,
we have
1
the statements
3, to
=
z get ys 2 y&. &#\ yz>. 5 =yv.5 ai> Substituting these results, we get
in

always positive.
is
3
and Xg
;
and when ct<
,
]
[§156
+ y&. 5%. 3«1 + y V
— yv.^n
Vv.^
5%. 7
.
Having found the limits of the variable y, we must apply the three formulae of § 135 to the statements in x. Multiplying by the datum xVti we get ,
Q%V. g»l'. = 3fa. 2
for
We
xx
i
5
= =
2''
2/3'.
»/
V.
6'
ajj/ 5
;
.
+
=

+
5. 2
5^1'. 3 ft l
2/ 3 '.
B®1'.
ajji
;
Z'.
3
5'.
X V. 7 xZm 2 =
llv. b
.
+
2«1
?/l'.
5'?V.
1'.
7
.
2
I
'''3
=
#7.
!
^7*
,:
these results immediately by simple in
obtain
spection of the Table of Limits, without having recourse Applying the formulae of to the formulae of § 13 5. §
x which remain, we get
to the statements in
135
xVm v = ./v",, •%.
7
=
+ xr a
iV 3'. n
Substituting these values,
Q%r
2 Xy. 2
.
= QRA = ==
// 3

5
\
.
;
iV i'
I
l'.3
.
= «i = a2.i3
This limits,
QRA
is
limit of
Vi 3 a.2
'''v.
;
a 3
al
"I"
yv.sfls'. 7/
> ?
.
1
"1
#1'.
?"*. 3>
VM.ffiv.'flr\
— (Vb'.S^V. + an an( %.3 = i
7«2
//r. 5 (#3'.
(
(6
l
J
impossibility).
the final step in the process of finding the
and the
is
7
+
2
3
for ai.2
.
3
get
3 «. !
?,
=x =
#r
,
2
we
XV
?/3'.5'
.
a
result informs us that,
a.
greater than
when n
is
even,
when a x which =1) is an inferior In other words, when n is even and a is not
only possible
1,
chance when n
(
the chance of is
even and a
QR
is
is zero.
To
greater than
1,
find the
we have
§
CALCULUS OF LIMITS
L56]
139
only to evaluate the integrals, employing the abbreviated notation of
Thus
151.
§
yv r = (y — y )^v. — = x v {2ax) = 2cuc — 2ax = 4a QRA = yw # + y # =  y^ v + (Vi  y )'%.7
A = Int
Integral
2
,?;
2
2
x
2
,
2
Integral
=
= ®V. =
an
—a
a Tl
(
 a**
1
=
1l
+ l W,_ 3
5
+
~ aH + F + 1
a?l 3
 ^3 ) +
(^j
(
'
a
a
(
— ««+i+
r3'7
*"
 aw +*
an — a n + l Y 2a
'(
3,. 7
v. 3
(ys
2
2
1
Vm 6
r< 3
5
2a)#r> 2
(
]
,i
W
7
~ an+1 V + i^ )(a? 8
 x7 ) + £(#?,  ^)
 £a"  a™+
1
QR = Int QRA _ Int QRA A 7w« A 4a 2
=— We
have now
_ a^+i Y 4a  a  a"* 1
a"
(
Tl
QR A = (y
,/'
,
3
5
when %
the chance
to find
the same process as before r 3
we
+ yv
.
is
odd.
By
get
%. 7
5(
K +y
& &&. 2«3.
namely, a x and a3
Here we have £wo
inferior limits of a,
so that the
To separate the not yet over. cases, we must multiply the result
process
different possible
obtained by the certainty
reduces to a x
+a v + %, 3
For shortness sake
+a r )(a + a$),
(a 1
3
since a x let
M
x
is
which here
greater than ay
denote
the
bracket co
a x in the result already obtained denote yc 2 <% 2. tne coefficient of
efficient (or cofactor) of
for
QRA;
a3
We
.
and
let
,
is
M
3
«.
.
get
QRA = (M A + M 3a3 )(a1 + av + %) = (M + M 3 )a + M 3ar>3 .
1
1
3
;
SYMBOLIC LOGIC
140
156, 157
[§§
Hence, (an impossibility). and a9tl = is an odd when n cases there are only two possible which here a>a to say, number, the case a 1 (that is v For the latter, a r 3 means a>l) and the case a r 3 a 13
for
= ar
rj
.
we
,
get Lit M, 1 /_ OR ?= — jL_ = J 2a — ira* A 8a' \ A
N
4, n+1
2
For the
a>
namely, the case
first case,
QR_ 7^(M + M [
1
A ~
7w*
1,
we
get
8)
A
When will
the integrals in this case are worked out, the result be found to be
—
9? = _L( o»  a«+i Y 2a  a" + a^  a^+i A 8a\ / A 4a \ )
2
\2
1
+ _ The expression
for the
the expression for give the
same
it
(
2a
— a»+
easily seen to be the
1
chance——in the
in the case a
result
1
(
<1
a>l
case
and
evidently ought to
when we suppose a=l. This is fact; for when we put a=l, each
expression gives  as the value of the chance 8
157. The great advantage of this
"
——
A
Calculus of Limits
"
that it is independent of all diagrams, and can therefore be applied not only to expressions of two or three variables, but also to expressions of four or several variables. Graphic methods are often more expeditious when they
is
only require
known curves
straight ;
lines or easily traced
of integration are, in general, difficult
three
variables,
representation
and
well
but graphic methods of finding the limits
of
because
this
involves
when
there are
the perspective
the intersections of curved
surfaces.
CALCULUS OF LIMITS
§157]
When
there are four or
cannot be employed at
141
variables, graphic methods For other examples in pro
more all.
my sixth paper in Society (June Mathematical London the Proceedings of the Mathematical volumes of and to recent 10th, 1897), bability I
Questions
may
may
and
interest
refer the student to
Solutions
some
from
Educational
the
Times.
It
readers to learn that as regards the
155, 150, I submitted my results to the test of actual experiment, making 100 trials 1 and in each case, and in the latter case taking a The theoretical chances (to two figures) are re3. 7i spectively 56 and 43, while the experiments gave the
problems worked in
§§
=
=
close approximations of *53
and 41 
respectively.
THE END
Printed by Ballantyne,
Hanson
Edinburgh &* London
&
Co.
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