Hugh MacColl.

Symbolic logic and its applications online

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. This little volume may be regarded as the final con-
centrated outcome of a series of researches begun in
1872 and continued (though with some long breaks)
until to-day, My article entitled " Probability Notation
No. 2," which appeared in 1872 in the Educational Times,
and was republished in the mathematical " Reprint," con-
tains the germs of the more developed method which I
afterwards explained in the Proceedings of the London
Mathematical Society and in Mind. But the most impor-
tant developments from the logical point of view will be
found in the articles which I contributed within the last
eight or nine years to various magazines, English and
French. Among these I may especially mention those
in Mind and in the Athenceum, portions of which I have
(with the kind permission of these magazines) copied into
this brief epitome.

Readers who only want to obtain a clear general view
of symbolic logic and its applications need only attend to
the following portions: §§ 1 to 18, §§ 22 to 24, §§ 46 to
53, §§ 7G to 80, §§ 112 to 120, §§ 144 to 150.

Students who have to pass elementary examinations
in ordinary logic may restrict their reading to §§ 1 to 18,
§§ 46 to 59, §§ 62 to 0Q, §§ 76 to 109, § 112.

Mathematicians will be principally interested in the
last five chapters, from § 114 to § 156; but readers


who wish to obtain a complete mastery of my symbolic

system and its applications should read the whole. They

will find that, in the elastic adaptability of its notation,

it bears very much the same relation to other systems

(including the ordinary formal logic of our text-books)

as algebra bears to arithmetic. It is mainly this nota-

tional adaptability that enables it to solve with ease and

simplicity many important problems, both in pure logic

and in mathematics (see § 75 and § 15 7), which lie

wholly beyond the reach of any other symbolic system

within my knowledge.


August 17 th, 1905.




1-3. General principles — Origin of language ... 1


4-12. Definitions of symbols — Classification of propositions —

Examples and formulae ...... 4


13-17. Logic of Functions — Application to grammar ... 9


18-24. Paradoxes — Propositions of the second, third, and higher

degrees . . . . . . . . .12


25-32. Formulae of operations with examples worked — Venn's

problem ......... 20


33-38. Elimination — Solutions of implications and equations —

Limits of statements 27


39-43. Jevous's " Inverse Problem " ; its complete solution on

the principle of limits, with examples ... 33





44-53. Tests of Validity — Symbolic Universe, or Universe of

Discourse — No syllogism valid as usually stated . 39


54-63. The nineteen traditional syllogisms all deducible from
one simple formula — Criticism of the technical
words ' distributed ' and ' undistributed ' — The
usual syllogistic ' Canons ' unreliable ; other and
simpler tests proposed 49


64-66 (a). Enthymemes— Uiven one premise of a syllogism and
the conclusion, to find the missing premise-
Strongest conclusion from given premises


67-75. To find the weakest data from which we can prove a
given complex statement, and also the strongest
conclusion deducible from the statement — Some
contested problems — ' Existential Import of Pro-
positions ' — Comparison of symbolic methods . 70


76-80. The nature of inference — The words if, therefore, and

because — Causation and discovery of causes . . 80


81-89. Solutions of some questions set at recent examina-
tions . 86


90-113. Definitions and explanations of technical terms often
used in logic — Meaningless symbols and their uses ;
mathematical examples — Induction: inductive
reasoning not absolutely reliable ; a curious case in
mathematics — ' Infinite ' and ' infinitesimal ' . .91





114-131 Application to elementary algebra, with examples . 106


132-140. Nearest limits — Table of Reference . . . .117


141-143. Limits of two variables — Geometrical illustrations . 123


144-150. Elementary probability — Meaning of 'dependent'
and ' independent ' in probability, with geo-
metrical illustrations . . . . . .128


151-157. Notation for Multiple Integrals — Problems that re-
quire the integral calculus . . . ' . . 132


(The numbers indicate the sections, not the pages.)

Alternative, 7, 41
Anipliative, 108
Antecedent, 28
Cause, 79
Complement, 46
Connotation, 93
Consequent, 28
Contraposition, 97
Contrary, 94
Conversion, 98

Couturat's notation, 132 (footnote)
Dichotomy, 100
Dilemma, 101-103
Elimination, 33-38
Entliymeme, 64
Equivalence, 11, 19
Essential, 108
Excluded Middle, 92
Existential import of proposi-
tions, 72, 73
Factor, 7, 28
Formal, 109
Functions, 13-17
Grammar, 17

Illicit process, 63 (footnote)
Immediate inference, 91
Implication, 10, 18



Induction, 112
Inference, nature of, 76-80
Infinite and infinitesimal, 113
Jevons's 'inverse problem,' 39-43
Limits of statements, 33
Limits of variable ratios,

Major, middle, minor, 54
Material, distinguished

Formal, 109
Meaningless symbols, 110
Mediate inference, 91
Modality, 99
Multiple, 28
Particulars, 49

Ponendo ponens, &c, 104-107
Product, 7
Sorites, 90

Strong statements, 33, 34
Subalterns and subcontraries, 95,

Syllogisms, 54
Transposition, 56
Universals, 49

Universe of discourse, 46-50
Venn's problem, 32
Weak statements, 33, 34



1. In the following pages I have done my best to
explain in clear and simple language the principles of
a useful and widely applicable method of research.
Symbolic logic is usually thought to be a hard and
abstruse subject, and unquestionably the Boolian system
and the more modern methods founded on it are hard
and abstruse. They are, moreover, difficult of application
and of no great utility. The symbolic system explained
in this little volume is, on the contrary, so simple that
an ordinary schoolboy of ten or twelve can in a very
short time master its fundamental conceptions and learn
to apply its rules and formulas to practical problems,
especially in elementary mathematics (see §§ 114, 118).
Nor is it less useful in the higher branches of
mathematics, as my series of papers published in the
Proceedings of the London Mathematical Society abundantly
prove. There are two leading principles which separate
my symbolic system from all others. The first is the
principle that there is nothing sacred or eternal about
symbols ; that all symbolic conventions may be altered
when convenience requires it, in order to adapt them
to new conditions, or to new classes of problems. The
symbolist has a right, in such circumstances, to give a
new meaning to any old symbol, or arrangement of
symbols, provided the change- of sense be accompanied
by a fresh definition, and provided the nature of the


problem or investigation be such that we run no risk
of confounding the new meaning with the old. The
second principle which separates my symbolic system
from others is the principle that the complete state-
ment or proposition is the real unit of all reasoning.
Provided the complete statement (alone or in connexion
with the context) convey the meaning intended, the
words chosen and their arrangement matter little. Every
intelligible argument, however complex, is built up of
individual statements ; and whenever a simple elementary
symbol, such as a letter of the alphabet, is sufficient to
indicate or represent any statement, it will be a great
saving of time, space, and brain labour thus to repre-
sent it.

2. The words statement and proposition are usually
regarded as synonymous. In my symbolic system,
however, I find it convenient to make a distinction,
albeit the distinction may be regarded as somewhat
arbitrary. I define a statement as any sound, sign, or
symbol (or any arrangement of sounds, signs, or symbols)
employed to give information ; and I define a proposition
as a statement which, in regard to form, may be divided
into two parts respectively called subject and predicate.
Thus every proposition is a statement ; but we cannot
affirm that every statement is a proposition. A nod,
a shake of the head, the sound of a signal gun, the
national flag of a passing ship, and the warning " Caw "
of a sentinel rook, are, by this definition, statements but
not propositions. The nod may mean " I see him " ; the
shake of the head, " I do not see him " ; the warning
" Caw " of the rook, " A man is coming with a gun," or
" Danger approaches " ; and so on. These propositions
express more specially and precisely what the simpler
statements express more vaguely and generally. In thus
taking statements as the ultimate constituents of sym-
bolic reasoning I believe I am following closely the
gradual evolution of human language from its primitive


prehistoric forms to its complex developments in the
languages, dead or living, of which we have knowledge
now. There can be little doubt that the language or
languages of primeval man, like those of the brutes
around him, consisted of simple elementary statements,
indivisible into subject and predicate, but differing from
those of even the highest order of brutes in being
uninherited — in being more or less conventional and
therefore capable of indefinite development. From their
grammatical structure, even more than from their com-
munity of roots, some languages had evidently a common
origin; others appear to have started independently;
but all have sooner or later entered the propositional
stage and thus crossed the boundary which separates
all brute languages, like brute intelligence, from the

3. Let us suppose that amongst a certain prehistoric
tribe, the sound, gesture, or symbol S was the understood
representation of the general idea stag. This sound or
symbol might also have been used, as single words are
often used even now, to represent a complete statement
or proposition, of which stag was the central and leading
idea. The symbol S, or the word stag, might have
vaguely and varyingly done duty for "It is a stag," or
" I see a stag," or " A stag is coming," &c. Similarly,
in the customary language of the tribe, the sound or
symbol B might have conveyed the general notion of
bigness, and have varyingly stood for the statement " It
is big" or " I see a big thing coming," &c. By degrees
primitive men would learn to combine two such sounds
or signs into a compound statement, but of varying
form or arrangement, according to the impulse of the
moment, as SB, or BS, or S B , or S B , &c., any of which
might mean "I see a big stag," or "The stag is big" or
" A big stag is coming," &c. In like manner some varying
arrangement, such as SK, or S K , &c, might mean " The
stag has been killed," or " I have killed the stag" &c.


Finally, and after many tentative or haphazard changes,
would come the grand chemical combination of these
linguistic atoms into the compound linguistic molecules
which we call propositions. The arrangement S B (or
some other) would eventually crystallize and permanently
signify " The stag is big," and a similar form S K would
permanently mean " The stag is killed" These are two
complete propositions, each with distinct subject and
predicate. On the other hand, S B and S K (or some
other forms) would permanently represent " The big stag "
and " The killed stag." These are not complete pro-
positions ; they are merely qualified subjects waiting
for their predicates. On these general ideas of linguistic
development I have founded my symbolic system.


4. The symbol A B denotes a proposition of which the
individual A is the subject and B the predicate. Thus,
if A represents my aunt, and B represents brown-haired,
then A B represents the proposition " My aunt is brown-
haired." Now the word aunt is a class term ; a person
may have several aunts, and any one of them may be
represented by the symbol A. To distinguish between
them we may employ numerical suhixes, thus A 1} A 2 ,
A 3 , &c, Aunt No. 1, Aunt No. 2, &c. ; or we may
distinguish between them by attaching to them different
attributes, so that A B would mean my brown-haired aunt,
A R my red-haired aunt, and so on. Thus, when A is a
class term, A B denotes the individual (or an individual)
of whom or of which the proposition A B is true. For
example, let H mean " the horse " ; let w mean " it won
the race " ; and let s mean " I sold it," or " it has been sold
by me." Then H£, which is short for (H w ) s , represents
the complex proposition " The horse which won the race
has been sold by me," or " I have sold the horse which


won the race." Here we are supposed to have a series
of horses, H r H 2 , H 3 , &c, of which H vv is one; and we
are also supposed to have a series, S 1; S 2 , S 3 , &c, of things
which, at some time or other, I sold ; and the proposition
H* asserts that the individual H w , of the first series H,
belongs also to the second series S. Thus the suffix w
is adjectival; the exponent s predicative. If we inter-
change suffix and exponent, we get the proposition H^,
which asserts that "the horse which I have sold won
the race." The symbol H w , without an adjectival
suffix, merely asserts that a horse, or the horse, won
the race without specifying which horse of the series
H x , H 2 , &c.

5. A small minus before the predicate or exponent, or
an acute accent affecting the whole statement, indicates
denial. Thus if H° means " The horse has been caught " ;
then H~° or (H c )' means " The horse has not been caught."
In accordance with the principles of notation laid down,
the symbol H_ c will, on this understanding, mean " The
horse which has not been caught" or the " uncaught horse " ;
so that a minus suffix, like a suffix without a minus, is
adjectival. The symbol H c (" The caught horse ") assumes
the statement H c , which asserts that " The horse has been
caught." Similarly H_ c assumes the statement H~°.

6. The symbol denotes non-existence, so that , 2 ,
3 , &c, denote a series of names or symbols which
correspond to nothing in our universe of admitted
realities. Hence, if we give H and C the same meanings
as before, the symbol H° will assert that " The horse
caught does not exist," which is equivalent to the statement
that "No horse has been caught." The symbol H~ ,
which denies the statement H°, may therefore be read
as " The horse caught does exist," or " Some horse has been
caught." Following the same principle of notation, the
symbol H° c may be read "An uncaught horse does not
exist," or " Every horse has been caught," The context
would, of course, indicate the particular totality of horses


referred to. For example, H° c may mean " Every horse
that escaped has been caught," the words in italics being
understood. On the same principle H:° denies H° c , and
may therefore be read " Some uncaught horse does exist"
or " Some horse has not been caught."

7. The symbol A B x C D , or its usually more convenient
synonym A B -C r> , or (without a point) A B C D , asserts two
things — namely, that A belongs to the class B, and that C
belongs to the class D ; or, as logicians more briefly express
it, that " A is B " and that " C is D." The symbol
A B + C D asserts an alternative — namely, that " Either A
belongs to the class B, or else C to the class D " ; or, as it is
more usually and briefly expressed, that " Either A is B,
or C is D." The alternative A B + C D does not necessarily
imply that the propositions A B and C D are mutually
exclusive ; neither does it imply that they are not. For
example, if A B means " Alfred is a barrister," and C D
means "Charles is a doctor"; then A B C D asserts that
" Alfred is a barrister, and Charles a doctor" while
A B + C D asserts that "Either Alfred is a barrister, or
Charles a doctor," a statement which (apart from context)
does not necessarily exclude the possibility of A B C D , that
both A B and C D are true. # Similar conventions hold
good for A B C D E F and A B + C p + E r , &c. From these con-
ventions we get various self-evident formulae, such
as (1) (A B C D )' = A- B + C- D ; (2) ( A B + C p )' = A- B C- D ; (3)
(A B C- D )' = A" B + C B ; (4) ( A B + C^)' = A" B C D .

8. In pure or abstract logic statements are represented
by single letters, and we classify them according to
attributes as true, false, certain, impossible, variable, respec-
tively denoted by the five Greek letters t, i, e, >/, 9.
Thus the symbol A T B l C e D r 'E 9 asserts that A is true,
that B is false, that C is certain, that D is impossible, that

* To preserve mathematical analogy, A B and O may be called factors
of the product A B C D , and terms of the sum A B +C D ; though, of course,
these words have quite different meanings in logic from those they
bear in mathematics.


E is variable (possible but uncertain). The symbol
A T only asserts that A is true in a particular case or
instance. The symbol A e asserts more than this: it
asserts that A is certain, that A is always true (or true
in every case) within the limits of our data and defini-
tions, that its probability is 1. The symbol A' only
asserts that A is false in a particular case or instance ;
it says nothing as to the truth or falsehood of A in
other instances. The symbol A 71 asserts more than
this ; it asserts that A contradicts some datum or defini-
tion, that its probability is 0. Thus A T and A 1 are simply
assertive; each refers only to one case, and raises no
general question as to data or probability. The symbol
A e (A is a variable) is equivalent to A -7, A~' ; it asserts
that A is neither impossible nor certain, that is, that A
is p>ossible but uncertain. In other words, A 6 asserts that
the probability of A is neither nor 1, but some proper
fraction between the two.

9. The symbol A BC means (A B ) C ; it asserts* that the
statement A B belongs to the class C, in which C may
denote true, or false, or possible, &c. Similarly A BCD means
(A BC ) D , and so on. From this definition it is evident
that A VL is not necessarily or generally equivalent to
A" 1 , nor A" equivalent to A' £ .

10. The symbol A B : C D is called an implication, and
means (A B C" D )^, or its synonym (A" B + C D ) € . It may be
read in various ways, as (1) A B implies C D ; (2) If A
belongs to the class B, then C belongs to the class D ;
(3) It is impossible that A can belong to the class B
without C belonging to the class D ; (4) It is certain that
either A does not belong to the class B or else C belongs
to the class D. Some logicians consider these four
propositions equivalent, while others do not ; but all
ambiguity may be avoided by the convention, adopted

* The symbol A BC must not be confounded with the symbol A BC , which
I sometimes use as a convenient abbreviation for A B A C ; nor with the
symbol A" r , which I use as short for A B + A c .

8 SYMBOLIC LOGIC [§§ 10, 11

here, that they are synonyms, and that each, like
the symbol A B : C D , means (A B C" D )' 7 , or its synonym
(A" B + C D ) e . Each therefore usually asserts more than
(A B C- D )' and than (A- B + C D ) T , because A" and A e (for
any statement A) asserts more than A' and A T respec-
tively (see § 8).

11. Let the proposition A B be denoted by a single
letter a ; then a will denote its denial A~ B or (A B ) .
When each letter denotes a statement, the symbol
A : B : C is short for (A : B)(B : C). It asserts that A
implies B and that B implies C. The symbol (A = B)
means (A : B)(B : A). The symbol A ! B (which may
be called an inverse implication) asserts that A is implied
in B ; it is therefore equivalent to B : A. The symbol
A ! B ! C is short for (A ! B)(B ! C) ; it is therefore
equivalent to C : B : A. When we thus use single letters
to denote statements, we get numberless self-evident or
easily proved formulae, of which I subjoin a few. To
avoid an inconvenient multiplicity of brackets in these and
in other formulae I lay down the convention that the sign
of equivalence ( = ) is of longer reach than the sign of
implication ( : ), and that the sign of implication ( : ) is of
longer reach than the sign of disjunction or alternat ion( + ).
Thus the equivalence a = ft : y means a = (ft: y), not
(a = ft):y, and A + B : x means (A + B) : x, not A + (B : x).

(I) x(a + ft)=xa + xP; (2) (aft)' = a' + ft' ;
(3) (a + ft)' = a f ft f - (4) a:ft = ft':a';

(5) (x:a)(x:ft) = x:aft; (6) « + ft : x = (« : x)(ft :x) ;
(7) (A:B:C):(A:C); (8) (A ! B ! C) : (A ! C) ;
(9) (A!C)!(A!B!C); (10) (A: C) !(A:B: C) ;

(II) (A + A r ) e ; (12) (A T + A') f ; (13) (AA r )\
(14) (A f + A» + A") e ; (15) A f :A T ;

(16) A": A 1 ; (17) A e = (A'y; (18) A" = (A / ) f ;
(19) A e = (A') 9 ; (20) e : A = A e ; (21) A : r, = A" ;
(22) Ae = A; (23) A*i = *i.


These formulae, like all valid formulae in .symbolic
logic, hold good whether the individual letters represent
certainties, impossibilities, or variables.

12. The following examples will illustrate the working
of this symbolic calculus in simple cases.

( 1 ) ( A + B'C)' = A'(B'C)' = A'(B + C) - A'B + A'C.

(2) ( A e + B f C e )' = A^B-'C 6 / = A-(B* + C~ 6 )

= (A" + A e XB e + C + C).

(3) (A" + A 9 B e ) f = A (A e B e Y = A e (A" e + B' 9 )

= A 9 A" 9 + A e B- = A e (B< + B") ;
for A e A- e = r] (an impossibility), and B e = B e + B".


13. Symbols of the forms F(x), f(x), (p(x), &c, are
called Functions of x. A function of x means an expression
containing the symbol x. When a symbol <p(x) denotes a
function of x, the symbols (p(a), (j)(P), &c, respectively
denote what <p(x) becomes when a is put for x, when /3
is put for x, and so on. As a simple mathematical ex-
ample, let (p(x) denote 5a; 2 — Sx + 1 . Then, by definition,
(p(u) denotes 5a 2 — 3 a + 1 ; and any tyro in mathematics
can see that <£(4) = 69, that cp(l) = 3, that <£(0) = 1, that
<p( — 1) = 9, and so on. As an example in symbolic logic,
let (p(x) denote the complex implication (A : B) : (A* : B*).
Then 0(e) will denote (A : B) : (A e : B e ), which is easily
seen to be a valid # formula ; while (f)(6) will denote
(A : B) : (A : B B ), which is not valid.

14. Symbols of the forms F(x, y), (f)(x, y), &c, are
called functions of x and y. Any of the forms may be
employed to represent any expression that contains both the
symbols x and y. Let <p(x, y) denote any function of x
and y ; then the symbol (p(a, /3) will denote what (p(x, y)

* Any formula <f>{x) is called valid when it is true for all admissible
values (or meanings), ar x , x- 2 , x 3 , &c , of x.

10 SYMBOLIC LOGIC [§§ 14-16

becomes when a is put for x and /3 for y. Hence, <p{y, x)
will denote what cp(x, y) becomes when x and y inter-
change places. For example, let B = boa-constrictor, let
R = rabbit, and let (p(B, R) denote the statement that
" The boa-constrictor swallowed the rabbit." It follows

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