Hugh MacColl.

# Symbolic logic and its applications online

. (page 4 of 11)
Online LibraryHugh MacCollSymbolic logic and its applications → online text (page 4 of 11)
Font size either aft or a'7', which are the two terms of aft + ay',
the denial of the given alternative a + ft'y. The result
will be

aft'y + a'fty + a fty + a ft'y' + a fiy'i
which is, therefore, the fully developed form of the given
alternative a + ft'y.

42. Let (p denote a'cclc + Veil + cd'e + a (Ye. Here we
have 5 elementary constituents a, b, c, d, e ; so that the
product of the five factors (a + a), (b + b'), &c, will contain
2 5 (or 32) terms. Of these 32 terms, 11 terms will
constitute the fully developed form of <p, and the re-
maining 21 will constitute the fully developed form of its
denial (p\ Let \|a denote the fully developed form of (p.
Then the alternatives <p and \J/ will, of course, only differ
in form ; they will be logically equivalent. Suppose the
alternative \f/ to be given us (as in Jevons's " inverse
problem "), and we are required to find the limits of the
5 constituents in the alphabetic order a, b, c, d, e, from
the data e : \^. When we have reduced the alternative
\Jr to its simplest form, we shall find the result to be (p.
Thus we get

e:ylr = e: <p = <p' :t] = ac + bde + c'd + d'e + abe : »/

= (>7 : a : b'c + ce')(>; : b : d' + c)(d : c : e)(e : d : e)(r) :e:e).
This is the final result with every limit expressed. Omit-

§§ 42-44] UNRESTRICTED FUNCTIONS 39

ting the superior limit >/ and the inferior limit e wherever
they occur, and also the final factor >j : c : e because it is a
formal certainty (see § 18), we get

e : \Jr = (a : &'c + ce')(b : ri' + e)(d : c)(e : rf).

Suppose next we arc required to find the limits in the
order d, e, c, a. b. Our final result in this case will be

e : y\$r = (e : d : &'c + <v)(»/ : e : a'c + b'c){a : c : e)(>7 : a : e)(>/ :b:e)
= (e : d : b'c + ce)(e : a'c + b'c)(a : c).

43. When an alternative <p contains n constituents, the
number of possible permutations in the order of con-
stituents when all are taken is 1.2. 3.4... n. In an alter-
native of 5 constituents, like the one in § 42, the number
of possible solutions cannot therefore exceed 1.2.3.4.5,
which = 120. For instance, in the example of § 42, the
solution in the order d, e, c, a, b (the last given), is
virtually the same as the solution in the order d, e, c, b, a ;
the only difference being that the last two factors in the
first case are (as given), n : a : e and r\ : b : e ; while in the
second case they are tj:b:e and >/ : a : e ; that is to say,
the order changes, and both, being certainties, may be
omitted. It will be observed that when the order of
limits is prescribed, the exact solution is prescribed also :
no two persons can (without error) give different solu-
tions, though they may sometimes appear different in
form (see §§39, 40).

CHAPTER VII

44. Let ~F u (x, y, z), or its abbreviated synonym F„, re-
present the functional proposition F(x, y, z), when the
values or meanings of its constituents x, y, z are unre-
stricted ; while the symbol F r (x, y, z), or its abbreviated
synonym F r , represents the functional proposition
F( l i , ) y; z) when the values of x, y, z are restricted. For
example, if x can have only four values. x y x, 2 x. A , x 4 ; y

40 SYMBOLIC LOGIC [§§ 44, 45

the four values y , y 2 , y z , y ; and z the three values
z„ z v z. s ; then we write F r , and not F M . But if each of
the three symbols x, y, z may have any value (or meaning)
whatever out of the infinite series x v x 2 , x 3 , &c, y v y 2 , y 3 ,
&c., z v «„, z , &c. ; then we write F M , and not F r The suffix
r is intended to suggest the adjective restricted, and the
suffix u the adjective unrestricted. The symbols F e , F n , F e ,
as usual, assert respectively that F is certain, that F is
impossible, that F is variable ; but here the word certain
is understood to mean true fur all the admissible values of
.<•, y, z in the functional statement F(x, y, z) ; impossible
means false for every admissible value of x, y, z in the
statement F(x, y, z); and variable means neither certain
nor impossible. Thus F e asserts that Fix, y, z) is neither
always true nor always false ; it is synonymous with
F _e F~", which is synonymous with (F^F"/.

45. From these symbolic conventions we get the three
formulae :

(1)(F-F<); (2)(F? t :F?.); (3)(F?:F? f );

but the converse (or inverse) implications are not neces-
sarily true, so that the three formulae would lose their
validity if we substituted the sign of equivalence ( = )
for the sign of implication (:). The first two formulae
need no proof; the third is less evident, so we will prove
it as follows. Let <p v <p 2 , (p 3 denote the above three
formulae respectively. The first two being self-evident,
we assume <p x (f> 2 to be a certainty, so that we get the
deductive sorites

e:<k4> 2 :(F-F;:)(F£:F?)

: (F; e : F-)(F7 : 1?) [for a : /3 = /3' : «']

: (F-F7 : FfFJ) [for (A : a)(B : b) : (AB : ah)]

: (F*: F*) [for A-'A^ = A e , by definition].

This proves the third formula <p 3 , when we assume the
first two <p x and <p 2 . To give a concrete illustration of
the difference between F M and F r , let the symbol H

§§ 45, 46] SYLLOGISTIC REASONING 41

represent the word horse, and let F(H) denote the state-
ment " The horse has been caught." Then F l (H) asserts
that every horse of the series H r H 2 , &c., has been
caught ; the symbol F' ) (H) asserts that not one horse of
the series H r H 2 , &c., has been caught ; and the symbol
F*(H) denies both the statements F e (H) and F"(H), and
is therefore equivalent to F _e (H) . F" r, (H), which may be
more briefly expressed by F~ 6 E^, the symbol (H) being left
understood. But what is the series H^ H 2 , &c. ? This
universe of horses may mean, for example, all the horses
owned by the horse-dealer ; or it may mean a portion only
of these horses, as, for example, all the horses that had
escaped. If we write F* { we assert that every horse owned
by the horse-dealer has been caught; if we write F* we
only assert that every horse of his that escaped lias been
caught. Now, it is clear that the first statement implies
the second, but that the second does not necessarily
imply the first ; so that we have F' t : F*, but not neces-
sarily F;:F;. The last implication F;:F; is not
necessarily true ; for the fact that all the horses that
had escaped were caught would not necessarily imply
that all the horses owned by the horse-dealer had been
caught, since some of them may not have escaped, and
of these it would not be correct to say that they had
been caught. The symbol F M may refer to the series
V v F 2 , F 3 , . . ., F 60 , while F,. may refer only to the series
i\, F 2 , F 8 , . . ., F 10 . The same concrete illustration will
make evident the truth of the implications F^:F? and
F* : F* , and also that the converse implications F? : F? t and
F* : Ff. are not necessarily true.

46. Let us now examine the special kind of reasoning
called syllogistic. Every valid syllogism, as will be shown,
is a particular case of my general formula

(a : (3)((3 : y) : (a : y),

or, as it may be more briefly expressed,

(a : /3 : y) : (a : y).

42 SYMBOLIC LOGIC [§§ 46, 47

Let S denote our Symbolic Universe, or " Universe of
Discourse," consisting of all the things S v S 2 , &c, real,
unreal, existent, or non-existent, expressly mentioned or
tacitly understood, in our argument or discourse. Let
X denote any class of individuals X , X 2 , &c, forming a
portion of the Symbolic Universe S ; then 'X (with a grave
accent) denotes the class of individuals 'X , 'Xg, &c, that
do not belong to the class X ; so that the individuals
Xj, X 2 , &c, of the class X, plus the individuals X X , 'X 2 .
&c, of the class X X, always make up the total Symbolic
Universe S , S 2 , &c. The class 'X is called the complement
of the class X, and vice versa. Thus, any class A and
its complement 'A make up together the whole Symbolic
Universe S ; each forming a portion only, and both
forming the whole.

47. Now, there are two mutually complementary
classes which are so often spoken of in logic that it is
convenient to designate them by special symbols ; these
are the class of individuals which, in the given circum-
stances, have a real existence, and the class of individuals
which, in the given circumstances, have not a real exist-
ence. The first class is the class e, made up of the
individuals e v e„, &c. To this class belongs every indi-
vidual of which, in the given circumstances, one can
truly say " It exists " — that is to say, not merely sym-
bolically but really. To this class therefore may belong
horse, town, triangle, virtue, vice. We may place virtue
and vice in the class e, because the statement " Virtue
exists " or " Vice exists " really asserts that virtuous
persons, or vicious persons, exist ; a statement which every
one would accept as true.

The second class is the class 0, made up of the
individuals 0^ 2 , &c. To this class belongs every in-
dividual of which, in the given circumstances, we can
truly say " It does not exist " — that is to say, " It does
not exist really, though (like everything else named) it
exists symbolically." To this class necessarily belong

§§47-49] REALITIES AND UNREALITIES 48

centaur, mermaid, round square, fiat sphere. The Symbolic
Universe (like any class A) may consist wholly of realil ies
e v e 2 , &c. ; or wholly of unrealities Oj, 2 , &c, or it may
be a mixed universe containing both. When the members
A v A 2 , &c, of any class A consist wholly of realities, or
wholly of unrealities, the class A is said to be a pure
class ; when A contains at least one reality and also at
least one unreality, it is called a mixed class. Since
the classes e and are mutually complementary, it is clear
that V is synonymous with 0, and v with e.

48. In no case, however, in fixing the limits of the class
e, must the context or given circumstances be overlooked.
For example, when the symbol H|! is read " The horse caught
does not exist," or " No horse has been caught" (see §§ 6, 47),
the understood universe of realities, e v e 2 , &c, may be a
limited number of horses, H , H 2 , &c, that had escaped,,
and in that case the statement Hj! merely asserts that to
that limited universe the individual H c , the horse cauyht,
or a horse caught, does not belong; it does not deny the
possibility of a horse being caught at some other time,
or in some other •circumstances. Symmetry and con-
venience require that the admission of any class A into
our symbolic universe must be always understood to
imply the existence also in the same universe of the
complementary class *A. Let A and B be any two classes
that are not mutually complementary (see § 46) ; if A and
B are mutually exclusive, their respective complements,
V A and 'B, overlap; and, conversely, if 'A and 'B are
mutually exclusive, A and B overlap.

49. Every statement that enters into a syllogism of
the traditional logic has one or other of the following
four forms :

(1) Every X is Y ; (2) No X is Y ;
(3) Some X is Y ; (4) Some X is not Y.

It is evident that (3) is simply the denial of (2), and (4)

44 SYMBOLIC LOGIC [§§ 49, 50

the denial of (1). From the conventions of §§ G, 47, we
get

(1) X° Y = Every X is Y ; (2) X° Y = No X is Y ;

(3) X T = X y ° = Some X is Y ;

(4) X! Y = X:° = Some X is not Y.

The first two are, in the traditional logic, called universals ;
the last two are called particulars ; and the four are respec-
tively denoted by the letters A, E, I, 0, for reasons
which need not be here explained, as they have now only
historical interest. The following is, however, a simpler
and more symmetrical way of expressing the above four
standard propositions of the traditional logic ; and it has
the further advantage, as will appear later, of showing
how all the syllogisms of the traditional logic are only
particular cases of more general formulae in the logic of
pure statements.

50. Let S be any individual taken at random out of
our Symbolic Universe, or Universe of Discourse, and let
x, y, z respectively denote the three propositions S x , S Y ,
S z . Then x', y', z' must respectively denote S~ x , S~ Y ,
S~ z . By the conventions of § 46, the three propositions
x, y, z, like their denials x' , y', z f , are all possible but un-
certain ; that is to say, all six are variables. Hence, we
must always have x e , y e , z\ (x\ {y'y, (z)e ; and never x 71
nor y v nor z v nor x e nor y e nor z\ Hence, when x, y, z
respectively denote the propositions S x , S Y , S z , the proposi-
tions (x : iff, (y : >/)', (z : >/)' (which are respectively synony-
mous with x* 1 , y' 1 *, z" ) must always be considered to form
part of our data, whether expressed or not ; and their
denials, (x : »/), (y : n), (« : »?), must be considered impossible.
With these conventions we get —

(A) Every (or all) X is Y = S x : S Y = (x : y) = {xy'f

(0) Some X is not Y = (S x : S Y / = (x : y)' = (xy'y
(E) No X is Y = S x : S- Y = x : y = (xyY

(1) Some X is Y = (S x : S" T )' = (x : y')' = {xyj*.

§ 50] GENERAL AND TRADITIONAL LOGIC 45

In this way we can express every syllogism of the
traditional logic in 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 r\
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 F r (x, ?/, z), or its abbreviated synonym F r)
and not to the class of unrestricted functional statements
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 F w (x, y, z) assumes not only that each
constituent statement x, y, z may belong to the class
>/ or e, as well as to the class 9, 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

(x : y)(y : z) : (x : z).

This formula asserts that " If x implies y, and y implies
z, then x implies z." The formula holds good whatever
be the statements x, y, z ; whether or not they have
(as in the traditional logic) the same subject S ; and
whether or not they are certainties, impossibilities, or
variables. Hence, with reference to the above formula,
it is always correct to assert F 6 whether F denotes F M
or F r . When x, y, z have a common subject S, then
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 e will mean F^ and will therefore
be a more general formula, of which the traditional
Barbara will be a particular case.

* Barbara asserts that " If every X is Y, and every Y is Z, then every
X is Z," which is equivalent to (S x : S v ) (S v : S z ) : (S x : S z ).

46 SYMBOLIC LOGIC [§§50,51

But now let F, or Y(x, y, z), denote the implication

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

It' we suppose the propositions x, y, z to be limited by
the conventions of §§46, 50, the traditional syllogism
called Darapti will be represented by F r and not by
F M . Now, by the first formula of § 45, we have F,' ( : F, 6 .,
and, consequently, F; 6 : F~ e , but not necessarily F~ e : F; e .
Thus, if F u be valid, the traditional Darapti must be
valid also. We find that F w is not valid, for the above
implication represented by F fails in the case f(xzy, as it
then becomes

(>1 : z){ri : x) : (xz)~ v ,

which is equivalent to ee : if, and consequently to e : »/,
which = {er/f = (ee) 7 ' = rj. But since (as just shown) F; 6
does not necessarily imply F; 6 , this discovery docs not justify
us in concluding that the traditional Darapti is not valid.
The only case in which F fails is y\xz) n , and this case
cannot occur in the limited formula F r (which here repre-
sents the traditional Darapti), because in F r the pro-
positions x, y, z are always variable and therefore possible.
In the general and non-traditional implication F M , the case
x yi y v z r ', 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
some logicians (I formerly thought so myself) to fail when
the class Y is non-existent, while the classes X and Z are
real but mutually exclusive. But this is a mistake, as the
following concrete example will show. Suppose we have

Y = (0 1( 2 , ;i ), Z = (e v e 2 , e 3 ), X = (« 4> e a , e 6 ).

Let P denote the first premise of the given syllogism,
Q the second, and R the conclusion. We get

P = Every Y is Z = > h ; Q = Every Y is X = >; 2 ;

and R = Some X is Z = >/ 3 ; three statements, >/ r »/ 2 , »/ 3 ,

each of which contradicts our data, since, by our data
in this case, the three classes X, Y, Z arc mutually
exclusive. Hence in this case we have

PQ : R = ( V / 2 : >i,) = (>i, : *1 3 ) = {%n^ = e 1 ;

so that, when presented in the form of an implication,
Darapti does not fail in the case supposed. (But see § 52.)
52. Startling as it may sound, however, it is a
demonstrable fact that not one syllogism of the traditional
logic — neither Darapti, nor Barbara, nor any other — is
valid in the form in which it is usually presented in our
text-books, and in which, I believe, it has been always
presented ever since the time of Aristotle. In this form,
every syllogism makes four positive assertions : it asserts
the first premise ; it asserts the second ; it asserts the
conclusion ; and, by the word ' therefore,' it asserts that
the conclusion follows necessarily from the premises,
i.e. that if the premises be true, the conclusion must be
true also. Of these four assertions the first three may be,
and often are, false ; the fourth, and the fourth alone, is
a formal certainty. Take the standard syllogism Barbara.
Barbara (in the usual text-book form) says this :

" Every A is B ; every B is C ; therefore every A is C."
Let \f/(A, B, C) denote this syllogism. If valid it
must be true whatever values (or meanings) we give to
A, B, C. Let A— ass, let B = bear, and let C = camel.
If \J/(A, B, C) be valid, the following syllogism must
therefore be true : " Every ass is a bear ; every bear is a
camel; therefore, every ass is a camel." Is this concrete
syllogism really true ? Clearly not ; it contains three
false statements. 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
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

48 SYMBOLIC LOGIC [§§ 52, 53

turn out false enormously outnumber the cases in which
it will turn out true. But 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 is C."
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 .'. , as is customary , denote the word therefore.
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 the implication P : Q, which, by definition,
means (PQ'y. and is a formal certainty. (See § 10.)

53. Logicians 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 the one really meant — that though
logicians assert " P therefore Q," they only mean " If P
then Q." But why depart from the ordinary common-
sense linguistic convention ? In ordinary speech, when
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 : —

" Why should the linguistic convention be different in logic ? . . .
Where is the necessity ? 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
untried and possibly innocent prisoner, ' He is a spy ; therefore he

* I pointed out this equivalence in Mind, January 1880.

§§ 53, 54] TRADITIONAL SYLLOGISMS 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 un-
fortunate mistake, due to the deplorable ignorance of his subordinates ;
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 say that nearly all fallacies
are due to neglect of the little conjunction, If. Mere
hypotheses are accepted as if they were certainties.

CHAPTER VIII

54. In the notation of § 50, the following are the nine-
teen syllogisms of the traditional logic, in their usual
order. As is customary, they are arranged into four
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-
mises, but not in the conclusion. The constituent z, in
the traditional phraseology, is called the " major term,"
and the constituent x the " minor term." Similarly,
the premise containing z is called the " major premise,"
and the premise containing x the " minor premise."
Also, since the conclusion is always of the form " All
X is Z," or " Some X is Z " or " No X is Z," or " Some
X is not Z," it is usual to speak of X as the ' subject '
and of Z as the ' predicate.' As usual in text-books, the
major premise precedes the minor.

Figure 1

Barbara =(y
Celarent = (y
Darii = (y
Ferio = (y

z)(x :y):(x:z)
z'){x : y) : (x : z)
z)(x : y')' : (x : z 1 )'
z')(, : y')' : (x : z) f

D

50 SYMBOLIC LOGIC [§ 54

Figure 2

Cesare = (z : y'){x : y) : (x : z*)
Camestres = (: : y\x : y') : (x : z)
Festino = («:/)(« :/)':(*: z)'
Baroko = (a : y)(x : y)' : (a: : z)'

Figure 3

Darapti = (y : z)(y : x) : (x : z')'
Disamis = (y : z , )\y : x) : (x : z')'
Datisi = (y : z)(y : a/)' : (a; : z'f
Felapton = (y : z')(y : «) : (x : z)'
Bokardo = ( y : z)\y : x) : {x : z)'
Ferison = (y : z'){y : x')' : (x : z)'

Figure 4

Bramantip = (z : y)(y : x) : (x : z 1 )'
Camenes = (z : y)(y : x') : (x : z')
Dismaris = {z : y')\y : x) : (x : z)'
Fesapo = (z : y')(y : x) : (x : z)'
Fresison = (z : y')(y : x')' : (x : z)'

Now, let the symbols (Barbara),,, (Celarent) M , &c. ; denote,
in conformity with the convention of § 44, these nineteen
functional statements respectively, when the values of
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
as in § 50. The syllogisms (Barbara),., (Celarent),., &c,
with the suffix r, indicating restriction of values, are the
real syllogisms of the traditional logic ; and all these,
without exception, are valid — within the limits of the
understood restriction*. The nineteen syllogisms of general
logic, that is to say, of the pure logic of statements,

§§ 54-5 0] GENERAL LOGIC 51

namely, (Barbara),,, (Celarent),,, &c., in which x, y, z are
a n restricted in values, are more general than and imply

Online LibraryHugh MacCollSymbolic logic and its applications → online text (page 4 of 11)