Hugh MacColl.

Symbolic logic and its applications online

. (page 6 of 11)
Online LibraryHugh MacCollSymbolic logic and its applications → online text (page 6 of 11)
Font size
QR-code for this ebook

may be read, " If every non-Y is a non-X, and every non-


Y is also a non-Z, then some non-X is a non-Z." For
(x':z)' is equivalent to (./ z , )' r> , which asserts that it is
possible for an individual to belong at the same time
both to the class non-X and to the class non-Z. In
other words, it asserts that some non-X is non-Z. Thus
read, the contested syllogism becomes a case of Darapti,
the classes X, Y, Z being replaced by their respective
complementary classes 'X, 'Y, 'Z. It is evident that
when we change any constituent x into x in any syllo-
gism, the words ' distributed ' and ' undistributed ' inter-
change places.

Canon (4) of the traditional logic asserts that " No
term' must be distributed in the conclusion, unless it is
also distributed in one of the premises." This is another
canon that cannot be accepted unreservedly. Take the
syllogism Bramantip, namely,

(z : y)(y : x) : (x : z')' ,

and denote it by \f/(V). Since the syllogism is valid
within the restrictions of the traditional logic (see
§ 50), it should be valid when we change z into /, and
consequently z into z. We should then get

>},{/) = (*' :y)(y:x):(x:z)'.

Here (see § 02) we get Z w in the first premise, and Z rf
in the conclusion, which is a flat contradiction to the
canon. Upholders of the traditional logic, unable to
deny the validity of this syllogism, seek to bring it
within the application of Bramantip by having recourse
to distortion of language, thus : —

" If every non-Z is Y, and every Y is X, then some X
is non-Z."

Thus treated, the syllogism, instead of having Z" in
the first premise and Z d in the conclusion, which would
contradict the canon, would have ( V Z)'' in the first premise
and ( y Z) u in the conclusion, which, though it means exactly
the same thing, serves to "save the face" of the canon
and to hide its real failure and inutility.


Canon (5) asserts that " We can infer nothing from
two negative premises." A single instance will show the
unreliability of the canon. The example is

(2, :0(^*') :(*':*)',

Avhich is obtained from Darapti by simply changing z
into z', and x into x . It may be read, " If no Y is 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
also by throwing it into the Daraptic form, thus : " If
all Y is non-X, and all Y is also non-Z, then some non-X
is non-Z." 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 a negative conclusion
requires one negative premise." The 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 seeming
affirmatives the canons about negatives must disappear
through want of negative matter to which they can
refer. The following three simple formulae are more
easily applicable and will supersede all the traditional
canons : —

(1) (a: y :z):(x:z) Barbara.

(2) (z : y : x) : (x : z)' Bramantip.

(3) (y:x)(y:z):(x:z')' .... Darapti.

The first of these is valid both in general logic and in
the traditional logic ; the second and third are only valid
in the traditional logic. Apart from this limitation, they
all three hold good whether any constituent be affirma-


66 SYMBOLIC LOGIC [§§ 03, 64

tive or negative, and in whatever order we take the
letters. Any syllogism that cannot, directly or by the
formulae of transposition, a : /3 = /3' : a and a/3' : y' = ay : fi,
be brought to one or other of these forms is invalid.


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
weakest complementary premise which, with the one
given, will 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
readily by mere inspection. For example, suppose we
have the conclusion x:z f and the given major premise
z : y (see § 5 4). The syllogism required must be

either {x:y :z'): (x : z') or (x : y r : z') : (x : z'),

the middle term being either y or y'. The major pre-
mise of the first syllogism is y : z' ', which is not equivalent
to the given major premise z : y. Hence, the first syllo-
gism is not the one wanted. The major premise of the
second syllogism is y' : z', and this, by transposition and
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' .

When the conclusion, but not the given premise, is
a non-implication (or " particular "), we proceed as follows.
Let P be the given implicational (or " universal ") pre-
mise, and C the given non-implicational (or "particular")
conclusion. Let W be the required weakest premise which,

* A syllogism with one premise thus left understood is called an


joined to P, will imply C. We shall then have PW : C,
which, by transposition, becomes PC : W. Let S be the
strongest conclusion dcducible from PC. We shall then
have both PC : S and PC : W'. These two implications
having the same antecedent PC, we suppose their con-
sequents S and W' to be equivalent. We thus get S =
W', and therefore W = S'. The weakest 'premise required
is therefore the denial of the strongest conclusion dedueible
from PC {the given premise and the denial of the given

For example, let the given premise be y : x, and the
given conclusion (x : z r )' . We are to have


Transposing and changing signs, this becomes


But, by our fundamental syllogistic formula, we have
also (see § 5G)


We therefore assume W = y:z' ) and, consequently, W =
(y : z f ) f . The weakest premise required * is therefore
(y : //, and the required syllogism is

(// : %)(y ■ *')' ■ (« : «')'■

65. The only formulae needed in finding the weakest
complementary premise are

(1) a:(3 = (3':a'.

(2) (a:/3)(/3: 7 ):(a: 7 ).

(3) (a:/3)(a: 7 ):(/3 7 r\

The first two are true universally, whatever be the state-
ments a, (3, y ; the third is true on the condition a*,
that a is possible — a condition which exists in the

* The implication y : «, since in the traditional logic it implies (y : s')',
would also answer as a premise ; but it would not be the weakest (see § 33,
footnote, and § 73).

68 SYMBOLIC LOGIC [§§ 65, 66

traditional logic, as here any of the statements a, (3, y
may represent any of the three statements x, y, z, or any
of their denials x , y', z , every one of which six state-
ments is possible, since they respectively refer to the six
classes X, Y, Z, %Y Z, every one of which is under-
stood to exist in our Universe of Discourse.

Suppose we have the major premise z:y with the
conclusion (x : z')' ', and that we want to find the weakest
complementary minor premise W. We are to have


which, by transposition and change of signs, becomes

This, by the formula a : /3 = ft' : a , becomes


But by Formula (3) we have also


We therefore assume W' = (yz')' 71 , and consequently
W = (yx'y = y:x. The weakest minor premise required
is therefore y : x ; and the required syllogism is

: y)(V ■ .') : (■'• : -')'-

which is the syllogism Bramantip. As the weakest
premise required turns out in this case to be an implica-
tion, and not a non-implication, it is not only the weakest
complementary premise required, but no other comple-
mentary premise is possible. (See § 64, second footnote.)
66. When the conclusion and given premise are both
non-implications (or " particulars "), we proceed as follows.
Let P' be the given non-implicational premise, and C
the non-implicational conclusion, while W denotes the
required weakest complementary premise. We shall
then have P'W : C or its equivalent WC : P, which we
obtain by transposition. The consequent P of the second


implication being an implication (or " universal ") we
have only to proceed as in § 64 to find W. For example,
let the given non-implioational premise be (// : z)' \ and
the given non-implicational conclusion {x : z)'. We are
to have

(yri/W :(*:«)'.

By transposition this becomes


The letter missing in the consequent y : z is x. The
syllogism WC : P must therefore be

either (y : x : z) : (y : z) or else (y:x':z):(y:z);

one or other of which must contain the implication C,
of which the given non-implicational conclusion C, re-
presenting (x : z)', is the denial. The syllogism WC : P
must therefore denote the first of these two syllogisms,
and not the second ; for it is the first and not the second
that contains the implication C, or its synonym x : z.
Hence W = y : x. Now, WC : P is equivalent, b} r trans-
position, to WP' : C, which is the syllogism required.
Substituting for W, P', C, we find the syllogism sought
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
example will suffice. Suppose we have the combination
of premises (z : y)(x : y)' '. Let S denote the strongest
conclusion required. We get

(z : y){x : //)' : S, which, by transposition, is (z : //)S / : (x : y).

The letter missing in the implicational consequent of the
second syllogism is z, so that its antecedent (z : y)S /
must be

either x : z : y or else x : z' : >/.

70 SYMBOLIC LOGIC [§§ 6G (a), G7

The first antecedent is the one that contains the factor
z : y, so that its other factor x : z must be the one denoted
by S'. Hence, we get S'=x:z, and S = (#:«)'. The
strongest * conclusion required is therefore (x : z)' '.


6 7. We will now introduce three new symbols, Wcp,
Yep, Sep, which we define as follows. Let A v A 2 , A 3 , . . .
A m be m statements which are all possible, but of which
one only is true. Out of these m statements let it be
understood that A r A 2 , A 3 , . . . A r imply (each sepa-
rately) a conclusion cp ; that A r+1 , A r+2 , A.,. +3 , . . . A s imply
cp' ; and that the remaining statements, A s+1 , A s+2 , . . .
A m neither imply cp nor cp'. On this understanding we
lay down the following definitions : —

(1) W(/) = A 1 + A 2 + A 3 + . . . +A r .

(2) W^) , = A r+1 + A r+2 + ... +A S .

(3) V4> = V<£' = A s+1 + A g+2 + ... +A m .

(4) S^ = W^ + V^ = W</) + V</) , .

(5) Sep' = W(p' + V<p' = W$' + Y(p.

(6) W'cp means (W(f>)', the denial of W</>.

(7) S'</> means (S<£)', the denial of Sep.

The symbol Wcp denotes the weakest statement that implies
cp ; while Sep denotes the strongest statement that <p
implies (see § 33, footnote). As A is stronger formally
than A + B, while A + B is formally stronger than
A + B-f-C, and so on, we are justified in calling Wcp
the weakest statement that implies cp, and in calling S(p the
strongest statement that (p implies. 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')'.


present themselves as logical sums or alternatives ; but, in
exceptional cases, they may present themselves as single
terms. From the preceding definitions we get the
formulae, (1) W^S'0; (2) S4>' = W'<£; (3) V°<£ =
(\\ r (^ = S<£ = (£). The last of these three formulas asserts
that to deny the existence of Y(p in our arbitrary uni-
verse of admissible statements, A , A 2 , &c, is equivalent
to affirming that W<^>, Sep, and (p are all three equivalent,
each implying the others. The statement Y°<ft, which
means (V(f)f, is not synonymous with V^> ; the former
asserts that Y<p is absent from a certain list A v A 2 , . . .
A OT , which constitutes our universe of intelligible state-
ments ; whereas Y^cf), which means (Ycpy, assumes the
existence of the statement Y(p in this list, and asserts
that it is an impossibility, or, in other words, that it
contradicts our data or definitions. The statement Y°cp
may be true ; the statement Y v <p cannot be true. The
statement Y°<p is true when, as sometimes happens, every
term of the series A , A 2 , . . . A m either implies (p or
implies <p'. The statement Y v (p is necessarily false,
because it asserts that Yep, which by definition neither
implies <p nor <p', is a statement of the class >/ ; whereas
every statement of the class tj implies both <p and </>', since
(as proved in § 18) the implication >/ : a is always true,
whatever be the statement represented by a. The state-
ment Y^cp also contradicts the convention laid down that
all the statements A , A 2 , . . . A w are possible. Similarly,
we may have W°<£ or W ^/.

68. The following examples will illustrate the mean-
ings of the three symbols Wcp, Y(p, Sc£. Suppose our
total (or " universe ") of possible hypotheses to consist
of the nine terms resulting from the multiplication of
the two certainties A' + A^ + A 9 and B« + B" + B fl . The
product is

A e B e + A^ + A e B" + A^B' + A"B" + A"B*
+ A*B e + A e B" + A 9 B 9 .


Let (p denote (AB) e . We get

(1) W(AB)* = A € B« + A fl B e .

(2) S(AB) e = A*B 9 + A*B f + A e B* = A""B e + A e B"".

(3) W(AB) e = S , (AB) 9 = A" + B T ' + A e B f . (See § 69.)

(4) S( AB)- 9 = W'(AB) 9 = A" + B" + A'B' + A e B fl . (See

§ 69.)

The first of the above formulae asserts that the weakest
data from which we can conclude that AB is a variable
is the alternative A e B 9 + A e B € , which affirms that either
A is certain and B variable, or else A variable and B certain.
The second formula asserts that the strongest conclusion
we can draw from the statement that AB is a variable
is the alternative A _T? B 9 + A^^, which asserts that either
A is possible and B variable, or else A variable and B possible.
Other formulae which can easily be proved, when not
evident by inspection, are the following : —





W<£ : (p : S(f>.

(W(j> = Sep) = (Wdj = <p)(S(p = <j>).
W(AB)« = A e B e = S(ABy.
W(A + B) e = A £ + B e .

S(A + B) e = A 6 + B e + A e B e .

W(A + B)" = A"B'' = S(A + B)" = (A + By.

W(A + B) e = A"B 9 + A e B".

S(A + B) e = A- £ B 9 + A e B^.

W(AB)" = A" + B".

S(AB) I) = A" + B' ! + A e B e .

W(A : B) = W(AB')" = A" + B\

S(A : B) = S( AB')" = A" + B 6 + A e B".

W(A : B/ = S'(A : B) = A £ B e + A"B".

S(A : BY = W'(A : B) = A-"B" 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


conclude that A does not imply B is the alternative that
either A is certain and B uncertain, or else A possible and
B impossible. The formula may be proved as follows :

W(A : B)' = S'(A : B) = (A" + B e + A fl BV = (A") , (B') / (A e B e ) /
= A-^B-^A" 9 + B' e ) = A*B e + A-"B" ;

for, evidently, A^A^M and B e B e = B".

69. All the formulae of § 68 may be proved from first
principles, though some may be deduced more readily
from others. Take, for example, (1), (2), (3). We are
required to find W(AB) fl , S(AB) fl , W(AB)" 9 . We first
write down the nine terms which constitute the product
of the two certainties A e + A" + A fl and B' + B" + B fl , as
in § 68. This done, we underdot every term that implies
(AB) 9 , which asserts that AB is a variable ; we underline
every term that implies (AB)" 5 , which asserts that AB is
not a variable; and we enclose in brackets every term
that neither implies (AB) 9 nor (AB)- . We thus get

A e B e + A'B' 1 + A € B 9 + A"B e + A»B*» + A"B 9
+ A 9 B e + A 9 B" + (A 9 B 9 ).

By our definitions in § 67 we thus have

W(AB) 9 = A £ B 9 + A 9 B e (1)

By definition also Ave have V(AB) 9 = A 9 B 9 , and therefore

S(AB) 9 = W(AB) 9 + V(AB) 9 = A f B 9 + A°B e + A 9 B 9
= A e B e + A 9 B e + A 9 B 9 + A 9 B 9 , for a = a + a
= ( A e + A 9 )B 9 + A*(B' + B 9 ) = A"B 9

+ A fl B-" (2).

We may similarly deduce (3) and (4) from first principles,
but they may be deduced more easily from the two


W((£ + ^) = W(£ + Wxfr .... (a)

S(<f>+x|O = S0 + S^ (£)>

74 SYMBOLIC LOGIC [§§ 69, 70

as follows :

W(AB)-" = W{(AB) f + ( AB)" ( = W(AB)« + W(AB/>

= A C B € + A" + B", from § 08, Formulae 7, 13.
S( AB)- 9 = S { ( AB) £ + ( AB)" } = S( AB) e + S(AB)»

= A e B« + A" + B" + A e B 9 , from § 08, Formulae

7, 14.

70. The following is an example of inductive, or rather
inverse, implicational reasoning (see §§ 11, 112).

The formula (A : x) + (B : x) : (AB : x) is always true ;
when (if ever) is the converse, implication (AB : x) : (A : x) +
(B : x), false ? Let <p denote the first and valid formula,
while <p c denotes its converse formula to be examined.
We get

<p e =(ABxy:(Ax'y + (Bx'y
= (Ax' . Bx'f : {Ax'y + (Baj')"

= (a(3) r < : oP + ffr, putting a for Ax, and (3 for Bx'.

Hence (see § 11), we get

(f>' e I (a/3)Xa" + /3")' ! (a/3)"a-"/3~" ! (a/SjV/S*

! (Ax' . Bx')\kxy(Bxy ! (ABxy(Ax') (Bxy

Thus, the converse implication (p c fails in the case
(a{$) r, a r, fir 7, i which represents the statement

(ABa/yCAa/r^V • • • • ( 1 );

and it therefore also fails in the case (afiy>a fi 9 , which
represents the statement

(ABa/)"(A#')"(Ba/) 6 .... (2) ;

for the second statement implies the first. The failure
of <p c in the second may be illustrated by a diagram as
on opposite page.

Out of the total ten points marked in this diagram,
take a point P at random, and let the three symbols
A, B, x assert respectively (as propositions) that the


point P will be in the circle A, that P will be in the
circle B, that P will be in the ellipse x. It is evident
that the respective chances of the four propositions A, B,
x, AB are T % T %, £>> T 2 o ; so that they are all variables.
It is also clear that the respective chances of the three
statements AB./, Axe', Bx', are 0, i 2 G , ^ ; so that we also
have (ABx'y(Axy(Bx') 9 , which, by pure symbolic reason-
insr, we found to be a case of
failure. We may also show this
by direct appeal to the diagram,
as follows. The implication AB : x
asserts that the point P cannot be W
in both the circles A and B without
being also in the ellipse x, a state-
ment which is a material certainty,
as it follows necessarily from the
special data of our diagram (see § 109). The implication
A : x asserts that P cannot be in A without being in
x, a statement which is a material impossibility, as it is
inconsistent with the data of our diagram ; and B : x is
impossible for the same reason. Thus we have AB : x = e,
A : x = v\, B : x = »/, so that we get

ip = (A : x) + (B : x) : (AB : x) = >i + v : * = e
cf) c = ( AB : x) : (A : x) + (B : x) = e : n + n = > h

The Boolian logicians consider <ft and (p c equivalent,
because they draw no distinction between the true (t)
and the certain (e), nor between the false (i) and the
impossible (>/). Every proposition is with them either
certain or impossible, the propositions which I call
variables (6) being treated as non-existent. The preced-
ing illustration makes it clear that this is a 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 my interpretation of
the word if, arc not equivalent. They are the complex

76 SYMBOLIC LOGIC [§§ 71, 72

conditional, " If A is true, then if B is true x is true" and
the simple conditional, " If A and B are both true, then x
is true!' Expressed in my notation, and with my inter-
pretation of the conjunction if (see § 10), these con-
ditionals are respectively A : (B : x) and AB : x. Giving
to the propositions A, B, x, AB the same meanings as in
§ 70 (all having reference to the same subject, the
random point P), it is evident that B : x, which asserts
that the random point P cannot be in the circle B
without being also in the ellipse x, contradicts our data,
and is therefore impossible. The statement A, on the
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
6 : 1}, which is equivalent to 0", and therefore an im-
possibility. But the simple conditional AB : x, instead of
being impossible, is, in the given conditions, a certainty,
for it is clear from the figure that P cannot be in both
A and B without being also in x. Hence, though
A : (B : x) always implies AB : x, the latter does not always
imply the former, so that the two are not, in all cases,

72. A question much discussed amongst logicians is
the " Existential Import of Propositions." When we
make an affirmation A B , or a denial A" B , do we, at the
same time, implicitly affirm the existence of A ? Do we
affirm the existence of B ? Do the four technical propo-
sitions of the traditional logic, namely, " All A is B,"
" No A is B," " Some A is B," " Some A is not B," taking
each separately, necessarily imply the existence of the
class A ? Do they necessarily imply the existence of the
class B ? My own views upon this question are fully
explained in Mind (see vol. xiv., N.S., Nos. 53-55); here
a brief exposition of them will suffice. The convention
of a "Symbolic Universe" (see §§ 46-50) necessarily
leads to the following conclusions : —


Firstly, when any symbol A denotes an individvM ;
then, any intelligible statement <p(A.), containing the
symbol A, implies that the individual represented by A
has a symbolic existence ; but whether the statement (jf>(A)
implies that the individual represented by A has a real
existence depends upon the context.

Secondly, when any symbol A denotes a class, then,
any intelligible 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
A is wholly real, or wholly unreal, or partly real and partly
unreal, depends upon the context.

As regards this question of " Existential Import," the
one important point in which I appear to differ from
other symbolists is the following. The null class 0,
which they define as containing no members, and which
I, for convenience of symbolic operations, define as con-
sisting of the null or unreal members V 2 , 3 , &c,
is understood by them to be contained in every class, real
or unreal ; whereas I consider it to be excluded from every
real class. Their convention of universal inclusion leads
to awkward paradoxes, as, for example, that " Every
round square is a triangle," because round squares form
a null class, which (by them) is understood to be con-
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 conven-
tion of universal inclusion as regards the null class 0,
is their 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
this is necessarily the case when the class X is null or
non-existent. My convention of a " Symbolic Universe "

78 SYMBOLIC LOGIC [§§ 73, 74

leads, on the contrary, to the common-sense conclusion

1 2 3 4 6 8 9 10 11

Online LibraryHugh MacCollSymbolic logic and its applications → online text (page 6 of 11)