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that the symbol (p(R, B) will assert that " The rabbit
swallowed the boa-constrictor."

15. As another example, let t (as usual) denote true,
and let p denote probable. Also let <p(r, p) denote the
implication (A T B T f : (A P B P ) T , which asserts that " If it is
probable that A and B are both true, it is true that A and
B are both probable." Then <p(p, t) will denote the con-
verse (or inverse) implication, namely, " If it is true that
A and B are both probable, it is probable that A and B are
both true." A little consideration will show that <£(t, p)
is always true, but not always <p(p, t).

16. Let <p denote any function of one or more con-
stituents ; that is to say, let (p be short for <p(x), or for
(p(x, y), &c. The symbol <p e asserts that (p is certain,
that is, true for all admissible values (or meanings) of
its constituents ; the symbol (p v asserts that <p is impos-
sible, that is, false for all admissible values (or meanings)
of its constituents ; the symbol <p e means cp^cp' 71 , which
asserts that <p is neither certain nor impossible ; while (p°
asserts that (p is a meaningless statement which is neither
true nor false. For example, let w = whale, h = herring,
c = conclusion. Also let (p(w, h) denote the statement
that " A small whale can swallow a large herring." We



get



(p e (w, h) ■ <p\h, w) • (p\w, c),



a three-factor statement which asserts (1) that it is certain
that a small whale can swallow a large herring, (2) that
it is impossible that a small herring can swallow a large
whale, and (3) that it is unmeaning to say that a small
whale can swallow a large conclusion. Thus we see that
<p(x, y), F(x, y), &c, are really blank forms of more or less



§§ 16, 17] APPLICATION TO GRAMMAR 11

complicated expressions or statements, the blanks being
represented by the symbols x, y, &c, and the symbols or
words to be substituted for or in the blanks being u, /3,
a/3, a + /3, &c, as the case may be.

17. Let <p(x, y) be any proposition containing the
words x and y ; and let <£(»•, z), in which z is substituted
for y, have the same meaning as the proposition (p(x, y).
Should we in this case consider y and z as necessarily
of the same part of speech ? In languages which, like
English, are but little inflected, the rule generally holds
good and may be found useful in teaching grammar to
beginners ; but from the narrow conventional view of
grammarians the rule would not be accepted as absolute.
Take, for example, the two propositions " He talks non-
sense " and " He talks foolishly." They both mean the
same thing ; yet grammarians would call non-sense a
noun, while they would call foolishly an adverb. Here
conventional grammar and strict logic would appear to
part company. The truth is that so-called " general
grammar," or a collection of rules of construction and
classification applicable to all languages alike, is hardly
possible. The complete proposition is the unit of all
reasoning ; the manner in which the separate words are
combined to construct a proposition varies according to
the particular bent of the language employed. In no
two languages is it exactly the same. Consider the
following example. Let S = His son, let A = in Africa,
let K = has been killed, and let (p(S, K, A) denote the
proposition " His son has been killed in Africa!' By
our symbolic conventions, the symbol (p(S, A, K), in
which the symbols K and A have interchanged places,
denotes the proposition " His son in Africa has been
killed." Do these two propositions differ in meaning?
Clearly they do. Let S A denote his son in Africa (to
distinguish him, say, from S c , his son in China), and let
K A denote has been killed in Africa (as distinguished
from K c , has been killed in China). It follows that



12 SYMBOLIC LOGIC [§§17,18

<p(S, A, K) must mean S*, whereas (f)(S, K, A) must
mean S K \ In the former, A has the force of an adjective
referring to the noun S, whereas, in the latter, A has
the force of an adverb referring to the verb K. And
in general, as regards symbols of the form A x , A y , A 2 ,
&c., the letter A denotes the leading or class idea, the
point of resemblance, while the subscripta x, y, z, &c.,
denote the points of difference which distinguish the
separate members of the general or class idea. Hence
it is that when A denotes a noun, the subscripta denote
adjectives, or adjective-equivalents; whereas when A
denotes a verb, the subscripta denote adverbs, or adverb-
equivalents. When we look into the matter closely, the
inflections of verbs, to indicate moods or tenses, have
really the force of adverbs, and, from the logical point
of view, may be regarded as adverb-equivalents. For
example, if S denote the word speak, & x may denote
spoke, S y may denote will speak, and so on ; just as when
S denotes He spoke, S^ may denote He spoke well, or He
spoke French, and S y may denote He spoke slowly, or He
spoke Dutch, and so on. So in the Greek expression
ol totc avOpunroi (the then men, or the men of that time),
the adverb Tore has really the force of an adjective, and
may be considered an adjective- equivalent.



CHAPTER III

1 8. The main cause of symbolic paradoxes is the
ambiguity of words in current language. Take, for
example, the words if and implies. When we say, " If
in the centigrade thermometer the mercury falls below
zero, water will freeze," we evidently assert a general
law which is true in all cases within the limits of tacitly
understood conditions. This is the sense in which the
word if is used throughout this book (see § 10). It
is understood to refer to a general law rather than to



§§ 18, 19] PARADOXES AND AMBIGUITIES 13

a particular case. So with the word implies. Let M z
denote " The mercury will fall below zero',' and let W F
denote " Water will freeze." The preceding conditional
statement will then be expressed by M z : W F , which
asserts that the proposition M z implies the proposition
W F . But this convention forces us to accept some
paradoxical-looking formulas, such as >/ : x and x : e, which
hold good whether the statement x be true or false.
The former asserts that if an impossibility be true any
statement x is true, or that an impossibility implies any
statement. The latter asserts that the statement x
(whether true or false) implies any certainty e, or (in
other words) that if x is true e is true. The paradox
will appear still more curious when we change x into e
in the first formula, or x into rj in the second. We
then get the formula r\ : e, which asserts that any im-
possibility implies any certainty. The reason why the
last formula appears paradoxical to some persons is
probably this, that they erroneously understand >/ : e to
mean Q^ : Q e , and to assert that if any statement Q is
impossible it is also certain, which would be absurd. But
}} : e does not mean this (see § 74) ; by definition it
simply means (>/e / )'', which asserts that the statement
t]e is an impossibility, as it evidently is. Similarly,
r\ : x means {qx'J*, and asserts that nx is an impossibility,
which is true, since the statement r\x' contains the im-
possible factor n. We prove x : e as follows :

x\e — (xe'y = (x>i) ri = if — e.

For e =■>], since the denial of any certainty is some
impossibility (see § 20). That, on the other hand, the
implication Q 1 : Q e is not a valid formula is evident ;
for it clearly fails in the case Q?. Taking Q = »/, we get

Q" : Q e = rp : >f = * : '/ = («/)" = («0" = >/.

19. Other paradoxes arise from the ambiguity of the
sign of equivalence ( = ). In this book the statement



14 SYMBOLIC LOGIC [§| 1 9, 20

(a = /3) does not necessarily assert that a and /3 are
synonymous, that they have the same meaning, but only
that they are equivalent in the sense that each implies
the other, using the word 'implies' as denned in § 10.
In this sense any two certainties, e x and e 2 , are equivalent,
however different in meaning ; and so are any two im-
possibilities, n l and n 2 \ but not necessarily two different
variables, B x and 6 2 . We prove this as follows. By
definition, we have

(e l = e 2 ) = (e l :e 2 )(e 2 :e 1 ) = (e/ 2 ne/ i y

=(vi) , (Vt) , = , »w=V4= e «;

for the denial of any certainty e x is some impossibility r\ y .
Again we have, by definition,



= *K = Vi=



But we cannot assert that any two variables, 6 X and # 2 ,
are necessarily equivalent. For example, 6 2 might be
the denial of 6 V in which case we should get

( e l = e 2 ) = (0 X = e\) = (0 l : e\)(6\ ■. ej = ( 0A) Wi)"

The symbol used to assert that any two statements,
a and /3, are not only equivalent (in the sense of each
implying the other) but also synonymous, is (a = /3); but
this being an awkward symbol to employ, the symbol
(a = /3), though it asserts less, is generally used instead.

20. Let the symbol it temporarily denote the word
possible, let p denote probable, let q denote improbable, and
let u denote uncertain, while the symbols e, r\, 6, t, i have
their usual significations. We shall then, by definition,
have (A 7r = A" ) ) and (A u = A' e ), while A p and A 5 will
respectively assert that the chance of A is greater than
\, that it is less than \. These conventions give us the
nine-factor formula

(*V)W(^WV)W



§ 20] PARADOXES AND AMBIGUITIES 15

which asserts (1, 2) that the denial of a truth is an
untruth, and conversely; (3, 4) that the denial of a
probability is an improbability, and conversely; (5, G) that
the denial of a certainty * is an impossibility, and con-
versely ; (7 ) that the denial of a variable is a variable ;
(8, 9) that the denial of a possibility is an uncertainty,
and conversely. The first four factors are pretty evident ;
the other five are less so. Some persons might reason,
for example, that instead of (tt') w we should have (-n-'y ;
that the denial of a possibility * is not merely an uncer-
tainty but an impossibility. A single concrete example
will show that the reasoning is not correct. The state-
ment " It will rain to-morrow " may be considered a
possibility ; but its denial " It will not rain to-morrow,"
though an uncertainty is not an impossibility. The formula
{tt') u may be proved as follows : Let Q denote any state-
ment taken at random out of a collection of statements
containing certainties, impossibilities, and variables. To
prove {ir') u is equivalent to proving Q 77 : (Q') M . Thus we
get

(Tr'y = Q- : (Q'f = Q e + Q e : (Q'r + (Q7

= Q e + Q e :Q e + Q e = e;

for (Q l , y = Q e , and (Q / f = Q, e , whatever be the statement
Q. To prove that (^y, on the other hand, is not valid,
we have only to instance a single case of failure. Giving
Q the same meaning as before, a case of failure is Q 8 ;
for we then get, putting Q = 6 V

= e 1 ;r 1i = (e/ i y = (e 1 € 2 y = , l2

* By the " denial of a certainty " is not meant (A e )', or its synonym A-*,
which denies that a particular statement A is certain, but (A e )' or its
synonym A' e , the denial of the admittedly certain statement A e . This state-
ment Ae (since a suffix or subscriptum is adjectival and not predicative)
assumes A to be certain ; for both A x and its denial A'x assume the truth of
A* (see §§ 4, 5). Similarly, "the denial of a possibility" does not mean
A-"' but AV, or its synonym (Att)', the denial of the admittedly possible
statement An-.



16 SYMBOLIC LOGIC [§21

21. It may seem paradoxical to say that the pro-
position A is not quite synonymous with A T , nor A' with
A 1 ; yet such is the fact. Let A = It rains. Then A' =
It does not rain ; A T = it is true that it rains ; and A' = it
is false that it rains. The two propositions A and A T are
equivalent in the sense that each implies the other ; but
they are not synonymous, for we cannot always substitute
the one for the other. In other words, the equivalence
(A = A T ) does not necessarily imply the equivalence
(p(A) = (p(A T ). For example, let (p(A) denote A e ; then
<p(A T ) denotes (A T ) € , or its synonym A" (see § 13). Sup-
pose now that A denotes 6 T , a variable that turns out
true, or happens to be true in the case considered, though
it is not true in all cases. We get

ct>(A_) = A< = e;=(e T y = r ,

for a variable is never a certainty, though it may turn out
true in a particular case.
Again, we get

^(A T ) = (A T ) e = (^) e = e« = e;

for 6 T T means (0 T ) T , which is a formal certainty. In this
case, therefore, though we have A = A T , yet (p(A) is not
equivalent to (p(A T ). Next, suppose A denotes t , a
variable that happens to be false in the case considered,
though it is not false always. We get

0(A') = (A') e = A* = 0? = »7;

for no variable (though it may turn out false in a parti-
cular case) can be an impossibility. On the other hand,
we get

</)(A') = (A') € = A" = 6[ e = (d[y = e e = e;

for 6[ means (Oy, which is a formal certainty. In this
case, therefore, though we have A' = A\ yet <£(A') is not
equivalent to </>(A l ). It is a remarkable fact that nearly
all civilised languages, in the course of their evolution,
as if impelled by some unconscious instinct, have drawn



§§ 21, 22] DEGREES OF STATEMENTS 17

this distinction between a simple affirmation A and the
statement A T , that A is true ; and also between a simple
denial A' and the statement A 1 , that A is false. It is the
first step in the classification of statements, and marks a
faculty which man alone of all terrestrial animals appears
to possess (see §§ 22, 99).

22. As already remarked, my system of logic takes
account not only of statements of the second degree, such
as A" 13 , but of statements of higher degrees, such as A a/3y ,
A afiyS , &c. But, it may be asked, what is meant by state-
ments of the second, third, &c, degrees, when the primary
subject is itself a statement ? The statement A a/iy , or its
synonym (A a/3 ) 7 , is a statement of the first degree as re-
gards its immediate subject A a/3 ; but as it is synonymous
with (A a ) Py , it is a statement of the second degree as
regards A tt , and a statement of the third degree as regards
A, the root statement of the series. Viewed from another
standpoint, A a may be called^a revision of the judgment
A, which (though here it is the root statement, or root
judgment, of the series) may itself laave been a revision
of some previous judgment here unexpressed. Similarly,
(A")* 3 may be called a revision of the judgment A a , and so
on. To take the most general case, let A denote any
complex statement (or judgment) of the n tb degree.
If it be neither a formal certainty (see § 109), like
(a/3 : a) e , nor a formal impossibility, like (a/3 : af, it may
be a material certainty, impossibility, or variable, according
to the special data on which it is founded. If it follows
necessarily from these data, it is a certainty, and we write
A* ; if it is incompatible with these data, it is an impossi-
bility, and we write A'' ; if it neither follows from nor is
incompatible with our data, it is a variable, and we write
A". But whether this new or revised judgment be A e or
A^ or A", it must necessarily be a judgment (or state-
ment) of the (w+l) th degree, since, by hypothesis, the
statement A is of the w th degree. Suppose, for ex-
ample, A denotes a functional statement <p(x, y, z) of

B



18 SYMBOLIC LOGIC [§§ 22-24

the n th degree, which may have m different meanings
(or values) <p v (p 2 , (p 3 , &c, depending upon the different
meanings x., x 2 , x y &c, y v y z , y y &c, z v z 2 , z y &c., of x, y, z.
Of these m different meanings of A, or its synonym <p, let
one be taken at random. If A, or its synonym <p(x, y, z),
be true for r meanings out of its m possible meanings,
then the chance of A is rjm, and the chance of its denial
A' is (m — r)lm. When r = m, the chance of A is one,
and the chance of A' is zero, so that we write A e (A / )''.
When r = o, the chance of A is zero, and the chance of A'
is one, so that we write A , (A / ) e . When r is some number
less than m and greater than o, then r/m and(??i — r)/ra
are two proper fractions, so we write A e (A / ) e . But, as
before, whether we get A € or A 11 or A e , this revised
judgment, though it is a judgment of the first degree as
regards its expressed root A, is a judgment of the (n + l) th
degree as regards some unexpressed root ^{x, y, z). For
instance, if A denote \J/ eT,e , then A" will denote x/^ 99 , so
that it will be a judgment (or statement) of the fourth
degree as regards \Ja

23. It may be remarked that any statement A and
its denial A' are always of the same degree, whereas A T
and A', their respective equivalents but not synonyms (see
SS 19, 21), are of one degree higher. The statement
A T is a revision and confirmation of the judgment A;
while A 1 is a revision and reversal of the judgment A.
We suppose two incompatible alternatives, A and A' ', to
be placed before us with fresh data, and we are to decide
which is true. If we pronounce in favour of A, we con-
firm the previous judgment A and write A T ; if we pro-
nounce in favour of A', we reverse the previous judgment
A and write A\

24. Some logicians say that it is not correct to speak
of any statement as " sometimes true and sometimes
false " ; that if true, it must be true always ; and if false,
it must be false always. To this I reply, as I did in my
seventh paper published in the Proceedings of the London



§ 24] VARIABLE STATEMENTS 19

Mathematical Society, that when I say " A is sometimes
true and sometimes false," or " A is a variable," I merely
mean that the symbol, word, or collection of words,
denoted by A sometimes represents a truth and some-
times an untruth. For example, suppose the symbol A
denotes the statement " Mrs. Brown is not at home."
This is not a formal certainty, like 3 > 2, nor a formal
impossibility, like 3<2, so that when we have no data out
the mere arrangement of words, " Mrs. Brown is not at
home," we are justified in calling this proposition, that is
to say, this intelligible arrangement of words, a variable, and
in asserting A 6 . If at the moment the servant tells me
that " Mrs. Brown is not at home " I happen to see
Mrs. Brown walking away in the distance, then / have
fresh data and form the judgment A e , which, of course,
implies A T . In this case I say that " A is certain"
because its denial A' (" Mrs. Brown is at home ") would
contradict my data, the evidence of my eyes. But if,
instead of seeing Mrs. Brown walking away in the
distance, I see her face peeping cautiously behind a
curtain through a corner of a window, I obtain fresh
data of an opposite kind, and form the judgment A v ,
which implies A'. In this case I say that " A is im-
possible," because the statement represented by A,
namely, " Mrs. Brown is not at home," this time contra-
dicts my data, which, as before, I obtain through the
medium of my two eyes. To say that the proposition
A is a different 'proposition when it is false from
what it is when it is true, is like saying that Mrs. Brown
is a different person when she is in from what she is
when she is out.



20 SYMBOLIC LOGIC [§25

CHAPTER IV

25. The following three rules are often useful: —

(1) A'</>(A) = A'tf>(e).

(2) A"4>(A) = A*0(>;).

(3) A e <p(A) = A. e <j)(O x ).

In the last of these formulae, 6 X denotes the first variable
of the series 6 6 , 6 &c, that comes after the last-named
in our argument. For example, if the last variable that
has entered into our argument be 6 then X will denote
6 . In the first two formulae it is not necessary to state
which of the series e , e 2 , e y &c, is represented by the e in
(p(e), nor which of the series rj , rj 2 , y &c, is represented
by the r\ in (p(>i); for, as proved in § 19, we have always
(e x = e y ), and (t] x = r] y ), whatever be the certainties e x and
e and whatever the impossibilities rj x ana " %• Suppose,
for example, that \j/ denotes

A'B'C^C : AB + CA).

= A £ B T, C fl e = A e B"C 9 ;

so that the fourth or bracket factor of \j/- may be omitted
without altering the value or meaning of \f/. In this
operation we assumed the formulas

(1) (ariz=r]); (2) (ae = a); (3) (*i + a==a).

Other formulae frequently required are

(4) (AB)' = A' + B'; (5) (A + B)' = A'B';

(6) e + A = e; (7) AA' = >7; (8) A + A' = e;

(9) / = *,; (10) >/ = *; (11) A + AB = A;

(12) (A + B)(A + C) = A + BC.



We get



§§ 26, 27] FORMULAE OF OPERATION 21

26. For the rest of this chapter we shall exclude the
consideration of variables, so that A, A T , A* will be con-
sidered mutually equivalent, as will also A', A', A''. On
this understanding we get the formulas

(1) A<£(A) = A<p(e) ; (2) Aty( A) = Aty(i) ;
(3) A<£(A') = A<^); (4) A^(A') = A'(/)(e).

From these formulas we derive others, such as

(5) AB'(/)(A, B) = AB'<£(e, rj) ;

(6) AB'<J>(A', B) = AB'^)(»;, n)\

(7) AB'<£(A', B') = AB'<£(>/, e),

and so on; like signs, as in A(p(A) or A / ^)(A / ), in the
same letter, producing (p(e) ; and unlike signs, as in
B'(p(B) or B^>(B / ), producing <jf>(>/)- The following ex-
amples will show the working of these formulas : —

Let </)(A, B) = AB'C + A'BC'. Then we get

AB'<£( A, B) = AB'(AB'C + A'BC)

= AB / («-C + wC')
= AB'(C + >/)=AB'C.

A'B0'(A, B) = A / B(AB / C + A / BC / /
= A / B( w C + eeC / y
= A'B(C')' = A'BC.

Next, let cp(B, D) = (CD' + CD + B / C / ) / .
Then, B'D'0(B, D) = B'D'(CD' + CD + B'C')'

= B / D , (Ce + C / >; + eC'/
= B'D'(C + C)'

= B f D'e , = B'D'>i = > ] .

The application of Formulas (4), (5), (11) of § 25 would,
of course, have obtained the same result, but in a more
troublesome manner.

27. If in any product ABC any statement-factor is
implied in any other factor, or combination of factors,
the implied factor may be omitted. If in any sum (i.e.,



2% SYMBOLIC LOGIC [§§ 27, 28

alternative) A + B + C, any term implies any other, or
the sum of any others, the implying term may be omitted.
These rules are expressed symbolically by the two
formulae —

(1) (A:B):(AB = A); (2) (A:B):(A + B = B).

By virtue of the formula (x : a)(x : /3) = x : a/3, these two
formulae may be combined into the single formula —

(3) (A:B):(AB = A)(A + B = B).

As the converse of each of these three formula? also
holds good, we get

(4) A:B = (AB = A) = (A + B = B).

Hence, we get A + AB = A, omitting the term AB, because
it implies the term A; and we also get A(A + B) = A,
omitting the factor A + B, because it is implied by the
factor A.

28. Since A : B is equivalent to (AB = A), and B is a
factor of AB, it follows that the consequent B may be
called a factor of the antecedent A, in any implication A : B,
and that, for the same reason, the antecedent A may be
called a multiple of the consequent B. The equivalence
of A : B and (A = AB) may be proved as follows : —

(A = AB) = (A : AB)(AB : A) = ( A : AB)e

= A:AB = (A:A)(A:B) = e(A:B) = A:B.

The equivalence of A : B and (A + B = B) may be proved
as follows : —

(A + B = B) = (A + B:B)(B:A + B) = (A + B:B)e
= A + B:B = (A:B)(B:B) = A:B.

The formula? assumed in these two proofs are

(x : aft) = (x : a)(x : /3), and a + /3 : x = (a : x)(fi : x),

both of which may be considered axiomatic. For to
assert that " If x is true, then a and /3 are both true " is
equivalent to asserting that " If x is true a is true, and



§§ 28, 29] REDUNDANT TERMS 23

if x is true /5 is true." Also, to assert that " If either a
or |8 is true x is true " is equivalent to asserting that " If
a is true x is true, and if /3 is true x is true."

29. To discover the redundant terms of any logical
sum, or alternative statement.

These redundant terms are easily detected by mere
inspection when they evidently imply (or are multiples of)
single co-terms, as in the case of the terms underlined in
the expression

a fty + a'y + aft/ + ft/,

which therefore reduces to a!y + fiy'. But when they
do not imply single co-terms, but the sum of two or
more co-terms, they cannot generally be thus detected
by inspection. They can always, however, be discovered
by the following rule, which includes all cases.

Any term of a logical sum or alternative may be
omitted as redundant when this term multiplied by the
denial of the sum of all its co-terms gives an impossible
product rj ; but if the product is not rj, the term must not
be omitted. Take, for example, the alternative statement


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