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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 so-called
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 so-called "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
:
alternative
e
or
cp,
:
<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
] = (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
(/),
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
,,
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
; b d' + c)(d c e)(e d e)(r) :e:e). :
:
:
:
:
:
:
»/
:
the final result with every limit expressed.
Omit-
UNRESTRICTED FUNCTIONS
42-44]
§§
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
+ /)',
{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 non-traditional 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
non-existent, 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
text-books, 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.
text-book 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 two-factor statement P(P:Q); the second or if-form asserts only the second factor P Q. The therefore-form vouches for the truth of P and Q, which are both false the if-form 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 over-imbued 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 text-books, 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
54-5 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 non-traditional 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 non-traditional 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 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
§§57-59] ?. 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 non-implications,
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 non-valid 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 non-implicational 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 co-factor 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 co-factor 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 non-implications
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]
/.
(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)',
The symbol Wcp denotes the cp
;
while
Sep
than
A+
A + B-f-C,
denotes
weakest statement that implies
the
33, footnote). B, while A + B
implies (see
the denial of W.
(S = 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
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
> 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
:
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 double-factor compound C). (A -< B)(B < C) be intelligibly spoken of as a
(A -
/
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 co-factors 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 co-factors, 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: •/
=
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
/
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 »4-liy=(* 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
2x-l — 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(x-3)
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
x-3
;
...
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
2re-l
.
get 6
— = —5 x-S -
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-
_
\x-3 :
28\°.
f
2(x-
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
G-7,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
,
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
124-128
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={(,e-l)°
+ 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
(x-Jx>2) = (x-Jx-2y = (c '+x"+x°)(x-Jx-2y i
= x (x-Jx-2y -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
