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the traditional nineteen in which x, y, z are restricted as
in § 5 ; and four of these unrestricted syllogisms, namely,
(Darapti),,, (Felapton),,, (Bramantip),,, and (Fesapo),,, fail
in certain cases. (Darapti) w fails in the case y 7 '(".:)\
(Felapton),, and (Fesapo) w fail in the case y%ez / ) TI , 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
belongs to the traditional logic, with restricted values
of its constituents x, y, z; while the other is a more
general syllogism, of which the traditional syllogism is a
particular case. Now, as shown in § 45, when a general
law F w , with unrestricted values of its constituents, implies
a general law F,., with restricted values of its constituents,
if the former is true absolutely and never fails, the same
may be said of the latter. This is 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,. ; for though F, e , : F;. is a valid formula, the implication
F M e : F; e (which is equivalent to the converse implica-
tion F e r : F e ,) is not necessarily valid. For example, the
general and non-traditional syllogism (Darapti),, implies
the less general and traditional syllogism (Darapti),..
The former fails in the exceptional case y\xzj i ; but
in the traditional syllogism this case cannot occur
because of the restrictions which limit the statement
y to the class 6 (see § 50). Hence, though this case of
failure necessitates the conclusion (Darapti);;*, we cannot,
from this conclusion, infer the further, but incorrect,
conclusion (Darapti); 6 . Similar reasoning applies to
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\$,

52 SYMBOLIC LOGIC [§§ 56, 57

be shown to be only particular cases of the formula
(x'.y)(y:z):(x:z), which expresses Barbara. Two or
three examples will make this clear. Lut <j)(x, y, z)
denote this standard formula. Referring to the list in
§ 54, we get

Baroko = (z : y)(x : //)' : (x : z)' ; which, by transposition,
= (.« : z){z : y) : (x : y) = (f)(x, z, //).

Thus Baroko is obtained from the general standard
formula <p(x, y, z) by interchanging y and z.

Next, take the syllogism Darii. Transposing as before,
we get

Darii = (y : z)(x : yj : (x : z)' = (y : z)(x : z) : (x : if)

= (y : z){z : x) : (y : x') = (p(y, z, x).

Next, take (Darapti) r . We get (see § 54)
(Darapti),. = (// : z)(y : x) : (x : z')' = (y:zx): (xz : n)'

= (// : xz)(xz : >j) : n = (y : xz){xz : >/) : (y : »/) ;

for, in the traditional logic, (y:rf) = »/, since, by the con-
vention of S 5 0, y must always be a variable, and, there-
lore, always possible. Thus, finally (Darapti),. = (f)(y, xz, n).
Lastly, take (Bramantip),.. We get

(Bramantip), = (z : y)(y : x) : (x : z")' = (z : y)(y : x ){x : z) : i]
= (z : y){z : x')(y : x) : n = (z : yx)(y : x) : >i
= (z : yx')(yx' : >i) :i] = (z: yx')(yx : r,) : (z : //) ;

for, in the traditional logic, (z:r]) = r), since z must be a
variable and therefore possible. Hence, finally, we get

(Bramantip),. = <p(z, yx\ >/).

57. By similar reasoning the student can verify the
following list (see §§ 54-56):

(p(x, y, z) = Barbara ; (p(x, y, z') = Celarent = Cesare :
ip(y y z , x') = Darii = Datisi ; (p{x, z, y') = Ferio = Festino

= Ferison = Fresison ; (p(z, y, x) = Camestres

= Camenes ;

§§57-59] TESTS OF SYLLOGISTIC VALIDITY 53

<p(y, x, z) = Disamis = Dismaris ; <p(x, z, y) = Baroko :
(p(y } x , v) = Bokardo; <f>(y, xz, »/) = (Darapti),. ;
(p(y, xz, »/) = (Felapton) r = (Fesapo) r ;
<p(z, yx', n) = (Bramantip),..

58. It is evident (since x : y = y' : x) that (f)(x, y, z) =
cb(z,' y', x) in the preceding list ; so that all these
syllogisms remain valid when we change the order ot
their constituents, provided we, at the same time, change
their signs. 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. Text-books on logic usually give rather compli-
cated rules, or " canons," by which to test the validity
of a supposed syllogism. These we shall discuss further
on (see §§ 62, 63); meanwhile we will give the following
rules, which are simpler, more general, more reliable, and
more easily applicable.

Let an accented capital letter denote a non-implication
(or " particular "), that is to say, the denial of an impli-
cation : while a capital without an accent denotes a
simple implication (or " universal "). Thus, if A denote
x : y, then A' will denote (x : y)' . Now, let A. B, C denote
any syllogistic implications, while A', B', C denote their
respective denials. Every valid syllogism must have
one or other of these three forms :

(1) AB:C; (2) AB r : C ; (3) AB : C ;

that is to say, either the two premises and the conclusion
are all three implications (or " universals ") as in (1);
or one premise only and the conclusion are both non-
implications (or "particulars") as in (2); or, as in (3),
both premises are implications (or " universals "), while
the conclusion is a non-implication (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.

54 SYMBOLIC LOGIC [§ 59

The second form may be reduced to the first form by
transposing the premise B' and the conclusion C, and
changing their signs ; for AB' : C is equivalent to AC : B,
each being equivalent to AB'C : >?. When thus trans-
formed the validity of AB' : C, that is, of AC : B, may be
tested in the same way as the validity of AB : C. The
test is easy. Suppose the conclusion C to be x : z, in
which z may be affirmative or negative. If, for example,
z — He is a soldier; then z' = He is not a soldier. But it
z—He is not a soldier; then z' — He is a soldier. The
conclusion C being, by hypothesis, x:z, the syllogism
AB : C, if valid, becomes (see § 11) either

(x :y:z):(x: z), or else {x : y' : z) : (x : z),

in 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
of these two forms, it is not valid ; if it can be reduced
to either form, it is valid. To take a concrete example,
let it be required to test the validity of the following
implicational syllogism :

If no Liberal approves of Protection, though some Liberals approve
of fiscal Retaliation, it follows that some person or persons who approve
of fiscal Retaliation do not approve of Protection.

Speaking of a person taken at random, let L = He is
a Liberal; let P = He approves of Protection; and let
R = He approves of fiscal Retaliation. Also, let Q denote
the syllogism. We get

Q=(L:P')(L:R'/:(R:P)'.

To get rid of the non-implications, we transpose them
(see § 56) and change their signs from negative to
affirmative, thus transforming them into implications.
This transposition gives us

Q = (L:P , )(R:P):(L:R').

§§59, 00] TESTS OF SYLLOGISTIC VALIDITY 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

either to L : P : R' or else to L : P' : R' ;

in which P is the letter left out in the new consequent
or conclusion L : R'. Now, the factors L : P and P : R'
of L : P : R' are not equivalent to the premises L : P' and
R : P in the second or transposed form of the syllogism
Q ; 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.
Hence Q is valid.

As an instance of a non-valid syllogism of the form
AB : C, we may give

(x:y')(y:z'):(x:z');

for since the y's in the two premises have different signs,
the one being negative and the other affirmative, the
combined premises can neither take the form x:y:z nor
the form x : y' : z' , which are respective abbreviations for
(x>\y){y:z) and (x t y')(y' : /). The syllogism is there-
fore not valid.

00. The preceding process for testing the validity of
syllogisms of the forms AB : C and AB' : C apply to all
syllogisms without exception, whether the values of their
constituents x, y, z be restricted, as in the traditional
logic, or unrestricted, as in my general logic of state-
ments. But as regards syllogisms in general logic of the
form AB : C (a form which includes Darapti, Felapton,
Fesapo, and Bramantip in the traditional logic), with
two implicational premises and a non-implicational con-
clusion, they can only be true conditionally ; for in general
logic (as distinguished from the traditional logic) no
syllogism of this type is a formal certainty. It therefore
becomes an interesting and important problem to deter-

56 SYMBOLIC LOGIC [§§ GO, 61

mine the conditions on which syllogisms of this type can
be held valid. We have to determine two things, firstly,
the iveakest premise (see § 33, footnote) which, when
joined to the two premises given, would render the syllo-
gism a formal certainty ; and, secondly, the weakest con-
dition which, when assumed throughout, would render
the syllogism a formal impossibility. As will be seen, the
method we are going to explain is a general one, which may
be applied to other formulae besides those of the syllogism.

The given implication AB : C is equivalent to the
implication ABC : y, in which A, B, C are three impli-
cations (see § 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
variable independently. Let a denote the strongest con-
clusion deducible from ABC and containing no reference
to the eliminated x. Similarly, let /3 and y respectively
denote the strongest conclusions after the elimination of
y alone (x being left), and after the elimination of z alone
(x and y being left). Then, if we join the factor a or /3'
or y' to the premises (ix. the antecedent) of the given
implicational syllogism AB : C, the syllogism will become
a formal certainty, and therefore valid. That is to say,
ABa' : C will be a formal certainty ; and so will AB/3' : C
and AB?' : C. Consequently, AB (a +fi'+ y) : C is a
formal certainty ; so that, on the one hand, the weakest
premise needed to be joined to AB to render the given
syllogism AB : C valid {i.e. a formal certainty) is the
alternative a' + fi' + y', and, on the other, the weakest
datum needed to make the syllogism AB : C a formal
impossibility is the denied of a + /?' + y , that is, a(3y.

61. Take as an example the syllogism Darapti.
Here we have an implication AB : C in which A, B, C
respectively denote the implications (y : x), (y : z), (x : z).
By the method of § 34 we get

ABC = yx + yz' + xz : >; = M* + N./ + P : r,, say,

§61] CONDITIONS OF VALIDITY 57

in which M, N, P respectively denote the co-factor of x,
the co-factor of %', and the term not containing x. The
strongest consequent not involving x is MN + P : *), in which
hero M = z, N = y, and P = yz' ; so that we have

MN + P : n = zy + yz' : n = //( - + z') : 1
= ye : >/ = y : v\.

Thus we get a = y: >/, so that the premise required when
we eliminate x is (y : >;/ ; and therefore

( r .x)(y.z)(y.ri) f -(x:z , ) t

should be a formal certainty, which is a fact ; for, getting
rid of the non-implications by transposition, this complex
implication becomes

(y : x)(y : z){x : z) : (y : 17),
which = (y : xz)(xz : n) ■ (y ■ n) ;

and 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,
we find that (3 = xz : *i = x : z' ; so that the complex
implication

{y:x)(y:z)(x:zy:(x:z')'

should be a formal certainty. That it is so is evident by
inspection, on the principle that the implication PQ : Q,
for all values of P and Q, is a formal certainty. Finally,
we eliminate z, and find that y = y: n- This is the same
result as we obtained by the elimination of x, as might
have been foreseen, since x and z are evidently inter-
changeable.

Thus we obtain the information sought, namely, that
« / + /3 / + 7 / , the weakest premise to be joined to the
premises of Darapti to make this syllogism a formal
certainty in general logic is

(y : >/) / + (xz : >/)' + (// : •?)', which = y*> + (xz)- 1 " ;

58 SYMBOLIC LOGIC [§§ 61, 62

and that a/3y, the Aveakest presupposed condition that
would render the syllogism Darapti a logical impossi-
bility, is therefore

/ ,p + (,,.,) - ; j ' t w hich = y\ocz)\

Hence, the Darapti of general logic, with unrestricted
values of its constituents x, y, z, fails in the case y\xzy ;
but in the traditional logic, as shown in § 50, this case
cannot arise. The preceding reasoning may be applied
to the syllogisms Felapton and Fesapo by simply chang-
ing z into z! .

Next, take the syllogism Bramantip. Here we get

ABC = yx' + zy' + xz : >i,

and giving u, /3, y the same meanings as before, we
get a = z r >, /3 = z\ y = (x'y)\ Hence, a^y — z\xyf, and
a' + ft' + y' = z~ n + (£c'y)~ r '. Thus, in general logic, Bra-
mantip is a formal certainty when we assume z~ v + {x'yY*,
and a formal impossibility when we assume &{x'yf ; but
sible, since z v is inadmissible by § 50, while the former is
obligatory, since it is implied in the necessary assump-
tion 2f.

62. The validity tests of the traditional logic turn
mainly upon the question whether or not a syllogistic
' term ' or class is ' distributed ' or ' undistributed.' In
ordinary language these words rarely, if ever, lead to
any ambiguity or confusion of thought ; but logicians
have somehow managed to work them into a perplexing
tangle. In the proposition " All X is Y," the class X is
said to be ' distributed,' and the class Y ' undistributed.'
In the proposition " No X is Y," the class X and the
class Y are said to be both ' distributed.' In the pro-
position " Some X is Y," the class X and the class Y are
said to be both 'undistributed.' Finally, in the pro-
position " Some X is not Y," the class X is said to be
' undistributed,' and the class Y ' distributed.'

§ 6 2] < DISTRIBUTED — < UNDISTRIBUTED , 59

Let us examine the consequences of this tangle of
technicalities. Take the leading syllogism Barbara, the
validity of which no one will question, provided it bo
expressed in its conditional form, namely, " If all Y is Z,
and all X is Y, then all X is Z." Being, in this form
(see § 52), admittedly valid, this syllogism must hold
good whatever values (or meanings) we give to its con-
stituents X, Y, Z. It must therefore hold good when
X, Y, and Z are synonyms, and, therefore, all denote the
same class. In this case also the two premises and the
conclusion will be three truisms which no one would
dream of denying. Consider now one of these truisms,
say " All X is Y." Here, by the usual logical convention,
the class X is said to be ' distributed,' and the class Y
1 undistributed.' But when X and Y are synonyms they
denote the same class, so that the same class may, at the
same time and in the same proposition, be both ' dis-
tributed' and 'undistributed.' Does not this sound like
a contradiction ? Speaking of a certain concrete collec-
tion of apples in a certain concrete basket, can we con-
sistently and in the same breath assert that " All the
apples are already distributed " and that " All the apples
are 'still undistributed " ? Do we get out of the dilemma
and secure consistency if on every apple in the basket we
stick a ticket X and also a ticket Y ? Can we then con-
sistently assert that all the X apples are distributed, but
that all the Y apples are undistributed ? Clearly not ; for
every X apple is also a Y apple, and every Y apple an X
apple. In ordinary language the classes which we can
respectively qualify as distributed and undistributed are
mutually exclusive ; in the logic of our text-books this
is evidently not the case. Students of the traditional
logic should therefore disabuse their minds of the idea
that the words ' distributed ' and ' undistributed ' neces-
sarily refer to classes mutually exclusive, as they do in
everyday speech ; or that there is anything but a forced
and fanciful connexion between the ' distributed ' and

60 SYMBOLIC LOGIC [§ 02

' undistributed ' of current English and the technical
' distributed ' and ' undisturbed ' of logicians.

Now, how came the words ' distributed ' and ' undis-
tributed ' to be employed by logicians in a sense which
plainly does not coincide with that usually given them ?
Since the statement " No X is Y " is equivalent to the
statement "All X is "Y," in which (see §§ 46-50) the
class Y (or non-Y) contains all the individuals of the
Symbolic Universe excluded from the class Y, and since
" Some X is not Y " is equivalent to " Some X is *Y," the
definitions of ' distributed ' and ' undistributed ' in text-
books virtually amount to this : that a class X is dis-
tributed 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
this is not the case, then the class X is undistributed with
regard to the class Y (or'Y). Hence, when in the state-
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
the same class. In other words, the proposition that
" All X is Y " ought to imply that " Some Y is not X."
But as no logician would accept this implication, it is
clear that the technical use of the words ' distributed '
and ' undistributed ' to be found in logical treatises is
lacking in linguistic consistency. In answer to this
criticism, logicians introduce psychological considerations
and say that the proposition " All X is Y " gives us infor-
mation 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
'distributed' and the term Y 'undistributed.' To this
explanation it may be objected, firstly, that formal logic
should not be mixed up with psychology — that its for-
mulae are independent of the varying mental attitude of
individuals ; and, secondly, that if we accept this ' infor-
mation-giving ' or ' non-giving ' definition, then we should

§62] 'DISTRIBUTED'— < UNDISTRIBUTED 1 fil

say, not that X is distributed, and Y undistributed, but
that X is known or inferred to be distributed, while Y is
not known to be distributed — that the inference requires
further data.

To throw symbolic light upon the question we may
proceed as follows. With the conventions of 8 50 we
have

(1) All X is Y = x:y; (2) No X is Y = x : //
(3) Some X is Y = (x : //)'; (4) Some X is not Y = (x : //)'.

The positive class (or ' term ') X is usually spoken of by
logicians as the subject'; and the positive class Y as
the ' predicate.' It will be noticed that, in the above
examples, the non-implications in (3) and (4) are the
respective denials of the implications in (2) and (1). The
definitions of ' distributed ' and ' undistributed ' are as
follows.

(a) The class (or ' term ') referred to by the ante-
cedent of an implication is, in text-book language, said to
be ' distributed ' ; and the class referred to by the conse-
quent is said to be ' undistributed.'

(/\$) The class referred to by the antecedent of a non-
implication is said to be ' undistributed ' ; and the class
referred to by the consequent is said to be ' distributed.'

Definition (a) applies to (1) and (2); definition
(/3) applies to (3) and (4). Let the symbol X d assert
that X is ' distributed' and let X u assert that X is ' un-
distributed.' The class 'X being the complement of the
class X, and vice versa (see 8 46), we get (*X)* = X M ,
and (X)" = X d . From the definitions (a) and (/3), since
(Y) d = Y", and ( y Y) u = Y d , we therefore draw the following
four conclusions : —

In (1) X d Y u ; in (2) X d Y d ; in (3) X U Y U : in (4)
X u Y d . For in (2) the definition (a) gives us X d ( r Yf ,
and CY) u = Y d . Similarly, in (3) the definition (/3) gives
us X u CY) d , and ( , Y) d = Y M .

If we change y into x in proposition (1) above, we

62 SYMBOLIC LOGIC [§§ 62, 63

get " All X is X "=x:x. Here, by definition (a), we have
X d X" ; which shows that there is no necessary antagonism
between X rf and X" ; that, in the text-book sense, the
same class may be both ' distributed ' and ' undistri-
buted ' at the same time.

63. The six canons of syllogistic validity, as usually
given in text-books, 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
\//('', y, z) to denote any valid syllogism. The 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, \|/(.'", ,/', :>-,) is valid (§ 13, footnote). Yet this is
a case which Canon (1) appears arbitrarily and need-
lessly to exclude. Canon (2) is simply a definition, and
requires no comment. The second part of Canon (3)
applies to all arguments alike, whether syllogistic or not.

* Violation of Canon (4) is called "Illicit Process." When the term
illegitimately distributed in the conclusion is the major term, the fallacy
is called " Illicit Process of the Major " ; when the term illegitimately dis-
tributed in the conclusion is the minor term, the fallacy is called " Illicit
Process of the Minor " (see § 54).

§63] 'CANONS 1 OF TRADITIONAL LOGIC 63

It is evident that if we want to avoid fallacies, we must
also avoid ambiguities. The first part of Canon (3)
cannot be accepted without reservation. The rule about
the necessity of middle-term distribution does not apply
to the following perfectly valid syllogism, " If every X is
Y, and every Z is also Y, then something that is not X
is not Z." Symbolically., this syllogism may be expressed
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 ;
finding it impossible to contest the validity of this syllo-
gism, refuse to recognise it as a syllogism at all, on the
ground that it has four (instead of the regulation three)
terms, namely, X, Y, Z, % the last being the class con-
taining 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
above syllogism. For Darapti is

_ {y:x\y:z):{x:zy (3);

and by virtue of the formula a : (3 = /3' : a, the syllogism
(l) in question becomes

(/:*')(/ :*'):(*':*)' (4).

Thus, if \^(f;, y, z) denote Darapti, then y\s(x', //', ;')
will denote the contested syllogism (1) in its form (4);
and, vice versa, if ^(x, y, z) denote the contested syllo-
gism, namely, (1) or (4), then ^(a/, y ', z') will denote
Darapti. To assert that any individual is not in the
class X is equivalent to asserting that it is in the com-
plementary class 'X. Hence, if we call the class 'X the
non-X class, the syllogism in question, namely,

(/:./)(/:/) :(,/:*)' (4),

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