Hugh MacColl.

# Symbolic logic and its applications online

. (page 7 of 11)
Online LibraryHugh MacCollSymbolic logic and its applications → online text (page 7 of 11)
Font size of the traditional logic that the two propositions " All
X is Y " and " No X is Y " are incompatible. This may
be proved formally as follows. Let (p denote the pro-
position to be proved. We have

(t> = (x:y)(x:y / ):v = (xy / )\xyy:f ]

= (V + xy : >]) : >/ = {,/•(/ + y) : >/} : n
= (xe : tj) : t] = (x : tj) : tj — (6 : tj) : >/

In this proof the statement x is assumed to be a variable
by the convention of § 46. See also § 5 0. It will be
noticed that (p, the proposition just proved, is equiva-
lent to {x : y) : (x : y')' ', which asserts that " All X is Y "
implies " Some X is Y."

74. Most symbolic logicians use the symbol A~< B, or
some other equivalent (such as Schroeder's A=£ B), to
assert that the class A 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
the statement B. That this is an error may be proved
easily as follows. If the statement A : B be always
equivalent to the statement A -< B, the equivalence must
hold good when A denotes >;, and B denotes e. Now,
the statement >/ : e, by definition, is synonymous with
(ye'y, which only asserts the truism that the impossibility
r\e is an impossibility. (For the compound statement yja,
whatever a may be, is clearly an impossibility because
it has an impossible factor tj.) But by their definition
the statement n -< e asserts that the class >? is wholly
included in the class e; that is to say, it asserts that
every individual impossibility. tj v >/ 2 , >; 3 , &c, of the class >;
is also an individual (either e r or e 2 , or e 3 , &c.) of the
class of certainties e ; which is absurd. Thus, >j : e is a
formal certainty, whereas y -< e is a formal impossibility.
(See 8 18.)

§ 75] CLASS INCLUSION AND IMPLICATION 79

75. Some logicians (see § 74) have also endeavoured
to drag my formula

(A:B)(B:C):(A:C) (1)

into their systems under some disguise, such as

(A -< B)(B -< C) -< (A -< C) .... (2).

The meaning of (1) is clear and unambiguous; but how
can we, without having recourse to some distortion of
language, extract any sense out of (2) ? The symbol
A -< B (by virtue of their definition) asserts that every
individual of the class A is also an individual of the
class B. Consistency, therefore, requires that the com-
plex statement (2) shall assert that every individual of
the class (A -< B)(B -< C) is also an individual of the
class (A -< C). But how can the double-factor compound
statement (A -< B)(B ■< C) be intelligibly spoken of as a
class contained in the single-factor statement (A-<C)?
It is true that the compound statement (A -< B)(B -< C)
implies the single statement (A-<C), an implication
expressed, not by their formula (2) but by

(A-<B)(B-<:C):(A-<C) (3);

but that is quite another matter. The two formulae (1)
and (3) are both valid, though not synonymous; whereas
their formula (2) cannot, without some arbitrary departure
from the accepted conventions of language, be made to
convey any meaning whatever.

The inability of other systems to express the new ideas
represented by my symbols A xy , k xyz , &c, may be shown
by a single example. Take the statement A 80 . This
(unlike formal certainties, such as e T and AB : A, and
unlike formal impossibilities, such as 6 e and 6 : >/) may, in
my system, be a certainty, an impossibility, or a variable,
according to the special data of our problem or investi-
gation (see §§ 22, 109). But how could the proposition
A 09 be expressed in other systems ? In these it could

80 SYMBOLIC LOGIC [§§ 75, 76

not be expressed at all, for its recognition would involve
the abandonment of their erroneous and unworkable
hypothesis (assumed always) that true is synonymous
with certain, and false with impossible. If they ceased to
consider their A (when it denotes a proposition) as
equivalent to their (A= 1), and their A' (or their corre-
sponding symbol for a denial) as equivalent to their
(A = 0), and if they employed their symbol (A=l) in
the sense of my symbol A e , and their symbol (A=0) in
the sense of my symbol A v , they might then express my
statement A ee in their notation ; but the expression
would be extremely long and intricate. Using A^B
(in accordance with usage) as the denial of (A = B), my
statement A e would then be expressed by (A=/=0)(A=/ r l),
and my A 80 by

{(A^0)(A^l)^0}{(A^0)(A=/=l)=£l}.

This example of the difference of notations speaks for
itself.

CHAPTER XI

76. Let A denote the premises, and B the conclusion,
of any argument. Then A .\ B (" A is true, therefore B is
true "), or its synonym B v A (" B is true because A is
true "), each of which synonyms is equivalent to
A(AB / ) r ', denotes the argument. That is to say, the
argument asserts, firstly, that the statement (or collec-
tion of statements) A is true, and, secondly, that the
affirmation of A coupled with the denial of B constitutes
an impossibility^ that is to say, a statement that is incom-
patible with our data or definitions. When the person
to whom the argument is addressed believes in the truth
of the statements A and (AB / )' ) , he considers the argument
valid ; if he disbelieves either, he considers the argument
invalid. This does not necessarily imply that he dis-

§§76,77] 'BECAUSE 1 AND < THEREFORE' 81

believes either the premises A or the conclusion B ; he
may be firmly convinced of the truth of both without
accepting the validity of the argument. For the truth of
A coupled with the truth of B does not necessarily imply
the truth of the proposition (AB') 17 , though it does that
of (AB')'. The statement (AB') 1 is equivalent to (AB')'
(see § 23) and therefore to A' + B. Hence we have

A(AB') 1 = A(A' + B) = AB = A T B\

But A .-. B, like its synonym A(AB / ) > ', asserts more than
A T B T . Like A(AB / ) 1 , it asserts that A is true, but, unlike
A(AB')\ it asserts not only that AB' is false, but that it
is impossible — that it is incompatible with our data or
definitions. For example, let k = He turned yah, and let
B = Ife is guilty. Both statements may happen to be
true, and then we have A T B T , which, as just shown, is
equivalent to A(AB') 1 ; yet the argument A .-. B (" He
turned pale : therefore he is guilty ") is not valid, for
though the weaker statement A(AB')' happens on this
occasion to be true, the stronger statement A(AB')'' is
not true, because of its false second factor (KB'f. I call
this factor false, because it asserts not merely (AB') 1 , that
it is false that he turned pale without being guilty, an
assertion which may be true, but also (AB')'', that it is
impossible he should turn pale without being guilty,
an assertion which is not true.

77. The convention that A .-. B shall be considered
equivalent to A(A:B), and to its synonym A(AB'y,
obliges us however to accept the argument A ,\ B as
valid, even when the only bond connecting A and B
is the fact that they are both certainties. For example,
let A denote the statement 13 + 5 = 18, and let B denote
the statement 4 + = 10. It follows from our symbolic
conventions that in this case A .\ B and B .-. A are both
valid. Yet here it is not easy to discover any bond of
connexion between the two statements A and B ; we
know the truth of each statement independently of

F

82 SYMBOLIC LOGIC [§§ 77, 78

all consideration of the other. We might, it is true,
give the appearance of logical deduction somewhat as
follows : —

By our data, 13 + 5 = 18. From each of these equals take away 9.
This gives us (subtracting the 9 from the 13) 4 + 5 = 9. To each of
these equals add 1 (adding the 1 to the 5). We then, finally, get
4 + G = 10 ; quod end demonstrandum.

Every one must feel the unreality (from a psycho-
logical point of view) of the above argument ; yet much
of our so-called ' rigorous ' mathematical demonstrations
are on lines not very dissimilar. A striking instance is
Euclid's demonstration of the proposition that any two
sides of a triangle are together greater than the third — a
proposition which the Epicureans derided as patent even
to asses, who always took the shortest cut to any place
they wished to reach. As marking the difference be-
tween A .-. B and its implied factor A : B, it is to be
noticed that though A : e and >/ : A are formal certainties
(see § 18), neither of the two other and stronger state-
ments, A .-. e and »/ .*. A, can be accepted as valid. The
first evidently fails when A = j/, and the second is always
false ; for i] .: x, like its synonym >?(>/ : x), is false, because,
though its second factor >j : x is necessarily true, its first
factor 7] is necessarily false by definition.

78. Though in purely formal or symbolic logic it is
generally best to avoid, when possible, all psychological
considerations, yet these cannot be wholly thrust aside
when we come to the close discussion of first principles,
and of the exact meanings of the terms we use. The
words if and therefore are examples. In ordinary speech,
when we say, " If A is true, then B is true," or " A is
true, therefore B is true," we suggest, if we do not
positively affirm, that the knowledge of B depends in
some way or other upon previous knowledge of A. But
in formal logic, as in mathematics, it is convenient, if not
absolutely necessary, to work with symbolic statements

§§ 78, 79] CAUSE AND EFFECT 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 self-
evident or axiomatic, and proofs, arguments, and logic
generally would have no raison d'etre. But when we
lay aside psychological considerations, and define the
word 'therefore,' or its synonym .*. , as in § 7G, it ceases
to be meaningless, and the seemingly meaningless argu-
ment, (7 x 9 = 63)/. (2 + 1 = 3), becomes at once clear,
definite, and a formal certainty.

79. In order to make our symbolic formula? and
operations as far as possible independent of our changing
individual opinions, we will arbitrarily lay down the
following definitions of the word ' cause ' and ' explana-
tion.' Let A, as a statement, be understood to assert
the existence of the circumstance A, or the occurrence
of the event A, while V asserts the posterior or simul-
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 of A ; and the symbol A(A : V), or its synonym
A.*. V, is called an explanation of the event or circum-
stance 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,
which is an explanation of the event V. We do not
call A the cause of V, nor do we call the argument
A .•. V the explanation of V, because we may have also

84 SYMBOLIC LOGIC [§§ 79, 80

B .•. V, in which case B would be another sufficient
cause of V, and the argument B .-. V another sufficient
explanation of V.

80. Suppose we want to discover the cause of an
event or phenomenon x. We first notice (by experiment
or otherwise) that x is invariably found in each of a
certain number of circumstances, say A, B, C. We
therefore provisionally (till an exception turns up) regard
each of the circumstances A, B, C as a sufficient cause of
x, so that we write (A : x)(B : x)(C : x), or its equivalent
A + B + C : x. We must examine the different circum-
stances A, B, C to see whether they possess some cir-
cumstance or factor in common which might alone
account for the phenomena. Let us suppose that they
do have a common factor /. We 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:/,'.

If / be not posterior to x, we may suspect it to be
alone the real cause of x. Our next step should be to
seek out some circumstance a which is consistent with
/, but not with A or B or C ; that is to say, some circum-
stance a which is sometimes found associated with /, but
not with the co-factors of / in A or B or C. If we find
that fa is invariably followed by x — that is to say, if we
discover the implication fa : x — then our suspicion is con-
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 con-
sidered as alone, and independently of its co-factors, a
sufficient cause of x. If, moreover, we discover that
while on the one hand fa implies x, on the other f'a
implies x' ; that is to say, if we discover (fa : %){fa : x' ) :
our suspicion that / alone is the cause of x is confirmed

§ 80] CAUSE AND EFFECT 85

still 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
each other. If we similarly find the same result for
these as for a ; so that

(fa : x)(f'a : x'), which =/a : x :/+ a
(//3 : x)(fp : x'), which = /]8 : x :f + /3'
(/? : x )(f'y '• x ')> which =fy : x :/+ y'
(/<M(/'<S: •<•'), which =fS:x:f+S'

our conviction that / alone is a sufficient cause of x re-
ceives stronger and stronger confirmation. But by no
inductive process can we reach absolute certainty that /
is a sufficient cause of x, when (as in the investigation of
natural laws and causes) the number of hypotheses or
possibilities logically consistent with / are unlimited ; for,
eventually, some circumstance q may turn up such that
fq does not imply x, as would be proved by the actual
occurrence of the combination fqx'. Should this com-
bination ever occur — and in natural phenomena it is
always formally possible, however antecedently improbable
— the supposed law f:x would be at once disproved.
For, since, by hypothesis, the unexpected combination
fqx' has actually occurred, we may add this fact to our
data e , e 2 , e 3 , &c. ; so that we get

e:fqx' :(fqx'r :(fx'r '•(/'■*)'.

This may be read, " It is certain that fqx' has occurred.
The occurrence fqx' implies that fqx is possible. The
possibility of fqx' implies the possibility of fx' ; and the
possibility oifx' implies the denial of the implication /: x."
The inductive method here described will be found,
upon examination, to include all the essential principles
of the methods to which Mill and other logicians have
given the names of ' Method of Agreement ' and ' Method
of Difference ' (see § 112).

86 SYMBOLIC LOGIC [SS 81, 82

CHAPTER XII

We will now give symbolic solutions of a few miscel-
laneous questions mostly taken from recent examination
papers.

81. Test the validity of the reasoning, "All fairies are
mermaids, for neither fairies nor mermaids exist."

Speaking of anything S taken at random out of our
symbolic universe, let/= a It is a fairy" let m = "it is a
mermaid," and let e = " it exists." The implication of the
argument, in symbolic form, is

(f:e){m :/):(/: m)
which = (/: e')(e : m') : (/: ra).

Since the conclusion /: m is a "universal" (or implica-
tion), the premises of the syllogism, if valid, must (see § 59)
be either f:e:m or /: e : m. This is not the case, so that
the syllogism is not valid. Of course, may replace e .

Most symbolic logicians, however, would consider this
syllogism valid, as they would reason thus : " By our
data, /= and m = ; therefore /= m. Hence, all fairies
are mermaids, and all mermaids are fairies" (see § 72).

82. Examine the validity of the argument : " It is not
the case that any metals are compounds, and it is in-
correct to say that every metal is heavy ; it may there-
fore be inferred that some elements are not heavy, and
also that some heavy substances are not metals."

Lete = "it is an element" = " it is not a compound";
let m = " it is a metal " ; and let h = " it is heavy."

The above argument, or rather implication (always
supposing the word " If " understood before the pre-
mises) is

(m : e)(m : K)' : (e : h)\h : m)'.

Let A = m : e, let B = m : h, let C = e : h, let D — h: m , and

§§ 82, 83] MISCELLANEOUS EXAMPLES 87

let <p denote the implication of the given argument.
We then get

</> = AB' : CD' = (AB' : C')(AB' : D'),
since x : yz = (x : y) (x : z).

In order that <p may be valid, the two implications
AB' : C and AB' : D' must both be valid. Now, we have
(see § 59)

AB 7 : C = AC : B = (m :e)(e: h) . (m : h),

which is valid by § 56. Hence, C, which asserts (e:h)',
that " some elements are not heavy " is a legitimate con-
clusion from the premises A and B'. We next examine
the validity of the implication AB 7 : D'. We have

AB' : D' = (m : e)(m : K)' : (h : m)'.

Now, this is not a syllogism at all, for the middle term

m, which appears in the two premises, appears also in

the conclusion. Nor is it a valid

implication, as the subjoined figure

will show. Let the eight points in

the circle m constitute the class m ;

let the twelve points in the circle e

constitute the class e ; and let the

five points in the circle h constitute

the class h. Here, the premises " Every m is e, and some

m is not h " are both true ; yet the conclusion, " Some h

is not m" is false.

Hence, though the conclusion C r is legitimate, the
conclusion D r is not.

83. Examine the argument, " No young man is wise;
for only experience can give wisdom, and experience
comes only with age."

Lety = "he is young") letw = "he is wise" ; and let
e = "he has had experience." Also, let (p denote the
implication factor of the given argument. We have

cj> = (/ : w'){y : e') : (y : w') = (y : e f : w') : (y : w').

88 SYMBOLIC LOGIC [§§ 83-85

The given implication is therefore valid (see §§ 11, 56,
59).

84. Examine the argument, " His reasoning was correct,
but as I knew his conclusion to be false, I was at once
led to see that his premises must be false also."

Let P = " his premises were true," and let C = "his con-
clusion was true." Then P : C = " his reasoning (or rather
implication) was valid." Let (p denote the implication of
the argument to be examined. We get (see | 105)

<£ = (P:C)C':P'

= the valid form of the Modus tollendo tollens.

Thus interpreted (p is valid. But suppose the word
" premises " means P and Q, and not a single compound
statement P. We then get

<£=(PQ:C)C:P'Q' ;

an interpretation which fails in the case CP'Q 1 , and also in
the case C^P^Q 6 . To prove its failure in the latter case,
we substitute for C, P, Q their respective exponential
values r\, t}, e, and thus get

<p = (>/e : rfirf : i/e' = (rj : ?])e : et] = ee : >/ = rj.

85. Supply the missing premise in the argument:
" Not all mistakes are culpable ; for mistakes are some-
times quite unavoidable."

Let m = "it is a mistake," let c = "it is culpable," let
u = " it is unavoidable," and let <p denote the implication
of the argument. Putting Q for the missing premise,
we get (see §§ 59, 64)

cp = (m : m')'Q : (m : c) x = (m : c)Q : (m : u').

For this last implication to be valid (see § 64), we must
have its premises (or antecedent) either in the form

m : c : vf , or else in the form m : c : u .

The first form contains the antecedent premise m : c; the

§§ 85-87] MISCELLANEOUS EXAMPLES 89

second form does not. The first form is therefore the
one to be taken, and the complete syllogism is

(m : c : u) : (m : n),

the missing premise Q being c : vf , which asserts that
" nothing culpable is unavoidable." The original reasoning
in its complete form should therefore be, " Since mistakes
are sometimes unavoidable, and nothing culpable is un-
avoidable, some mistakes are not culpable."

86. Supply the missing promise in the argument,
" Comets must consist of heavy matter ; for otherwise
they would not obey the law of gravitation."

Let c = "it is a comet" let A = "it consists of heavy
matter" and let # = "it obeys the law of gravitation."
Putting <p for the implication of the argument, and Q
for the missing premise understood, we get

(P = (h':g')Q:(c:h)=:(c:g:h):(c:h),

by application of §64; for g:h = h':g', so that the
missing minor premise Q understood is c : g, which asserts
that " all comets obey the law of gravitation." The full
reasoning is therefore (see §11)

(c:h)\(c:g)(g:h),

or its equivalent (see §11)

(c : g){g :h):(c: h).

In the first form it may be read, " Comets consist of
heavy matter ; for all comets obey the law of gravitation,
and everything that obeys the law of gravitation consists
of heavy matter."

87. Supply the missing proposition which will make
the following enthymeme * into a valid syllogism:
" Some professional men are not voters, for every voter
is a householder."

Let P = "he is a professional man," let V = "he is a

* An enthymeme is a syllogism incompletely stated.

90 SYMBOLIC LOGIC [§§ 87-89

voter" and let H = " he is a householder." Let <p denote
the implication of the argument, and W the weakest
additional premise required to justify the conclusion.
We have (see § 11)

<£ = (P : V)' !(V : H)W = (V : H)W : (P : V)'
= (P : V)(V : H) : W' = (P : V : H) : W.

The strongest conclusion deducible from P : V : H is
P : H. We therefore assume P : H = W', and conse-
quently W = (P : H)', which is therefore the weakest
premise required. The complete argument is therefore
this : " Some professional men are not voters, for every
voter is a householder, and some professional men are
not householders."

88. Put the following argument into syllogistic form,
and examine its validity : " The absence of all trace of
paraffin and matches, the constant accompaniments of
arson, proves that the fire under consideration was not
due to that crime."

Let F = " it was the fire under consideration " ; let A =
" it was due to arson " ; let T = " it left a trace of paraffin
and matches " ; and let <fi denote the implication of the
given argument. We get

<P = (¥ : T')(A : T) : (F : A') = (F : T')(T' : A 7 ) : (F : A 7 )

= (F:T' :A'):(F: A 7 ).

The implication of the given argument is therefore valid.
The argument might also be expressed unsyllogisti-
cally (in the modus tollendo tollens) as follows (see § 105).
Let T = " the fire left a trace of paraffin and matches " ;
let A = " the fire was due to the crime of arson " ; and
let (p denote the implication of the argument. We get
(see § 105)

(j) = T'(A : T) : A'

which is the valid form of the Modus tollendo tollens.

89. Put the following argument into syllogistic form :
•' How can any one maintain that pain is always an evil,

§§ 89, 90] TECHNICAL WORDS EXPLAINED 91

who admits that remorse involves pain, and yet may
sometimes be a real good ? "

Let R = " It is remorse " ; let P = " it causes pain " ; let
E = " it is an evil " ; and let (f) denote the implication of
the argument. We get (as in Figure 3, Bokardo)

<£ = (R:P)(R:E) , :(P:E) /
= (R:P)(P:E):(R:E),

which is a syllogism of the Barbara type. But to reduce
the reasoning to syllogistic form we have been obliged to
consider the premise, " Remorse may sometimes be a real
good," as equivalent to the weaker premise (R : E)', which
only asserts that " Remorse is not necessarily an evil."
As, however, the reasoning is valid when we take the
weaker premise, it must remain valid when we substitute
the stronger premise ; only in that case it will not be
strictly syllogistic.

CHAPTER XIII

In this chapter will be given definitions and explana-
tions of some technical terms often used in treatises on
logic.

90. Sorites. — This is an extension of the syllogism
Barbara. Thus, we have

Barbara = (A:B:C):( A: C)
(Sorites)! = (A : B : C : D) : (A : D)
(Sorites), = (A : B : C : D : E) : (A : E)
&c, &c.

Taken in the reverse order (see § 11) we get what may
be called Inverse Sorites, thus : —

Barbara=(A!C)!(A!B!C)
(Sorites^ = (A ! D) ! (A ! B ! C ! D).

Online LibraryHugh MacCollSymbolic logic and its applications → online text (page 7 of 11)