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92 SYMBOLIC LOGIC [§§91-94

91. Mediate and Immediate Inferences. When from a
proposition (p(x, y, z) we infer another proposition \j/(a?, z)
in which one or more constituents of the first proposition
are left out (or " eliminated "), we call it Mediate Inference.
If all the constituents of the first proposition are also
found in the second, none being eliminated, we have
what is called Immediate Inference. For example, in
Barbara we have mediate inference, since from x : y : z
we infer x : z ; the middle term y being eliminated. On
the other hand, when from x : y we infer y' : x', or ax : y,
we have immediate inference, since there is no elimination
of any constituent.

92. Law of Excluded Middle. This is the name given
to the certainty A B + A~ B , or its equivalent a + a. The
individual A either belongs to the class B or it does not
belong to the class B — an alternative which is evidently
a formal certainty.

93. Intension and Extension, or Connotation and Denota-
tion. Let the symbols (AB), (ABC), &c, with brackets,
as in § 100, denote the collection of individuals, (AB)^
(AB) 2 , &c, or (ABC) r (ABC) 2 , &c, common to the classes
inside the brackets ; so that S (AB) will not be synonymous
with S AB , nor S (ABC) with S ABC (see § 9). With this inter-
pretation of the symbols employed, let S be any individual
taken at random out of our universe of discourse, and
let S X = S (AB) be our definition of the term or class X.
The term X is said to connote the properties A and B,
and to denote the individuals X 1> X 2 , &c, or (AB) r
(AB) 2 , &c, possessing the properties A and B. As a
rule the greater the number of properties, A, B, C, &c,
ascribed to X, the fewer the individuals possessing them ;
or, in other words, the greater the connotation (or inten-
sion), the smaller the denotation (or extension). In A a
the symbol a connotes as predicate, and in A a it denotes

94. Contrary and Contradictory. The two propositions
" All X is Y " (or x : y) and " No X is Y " (or x : y f ) are

§§ 94-98] TECHNICAL WORDS EXPLAINED 93

called contraries, each being the contrary of the other.
The propositions " All X is Y " and " Some X is not Y,"
respectively represented by the implication x : y and its
denial (x : y)' are called Contradictories, each being the
contradictory or denial of the other (see § 50). Similarly
"No X is Y" and "Some X is Y," respectively repre-
sented by the implication x : y' and its denial (x : y') f , are

95. Subcoutraries. The propositions "Some X is Y"
and " Some X is not Y," respectively represented by the
non-implications (x : y') r and (x : y)' ', are called Sub-
contraries. It is easily seen that both may be true, but
that both cannot be false (see § 73).

96. Subalterns. The universal proposition "All X is
Y," or x : y, implies the particular " Some X is Y," or
(x : y') f ; and the universal " No X is Y," or x : y' ' , implies
the particular " Some X is not Y," or (x : y) f . In each
of these cases the implication, or universal, is called
the Subalternant, and the non-implication, or particular, is
called the Subalternate or Subaltern. That x : y implies
{x:y')' is proved in § 73; and by changing y into y'
and vice versa, this also proves that x : y r implies (x : y)' .

97. Contraposition. This is the name given by some
logicians to the formula x : y = ?/ : x, which, with the
conventions of §§ 46, 50, asserts that the proposition
" All X is Y " is equivalent to the proposition " All
non-Y is non-X." But other logicians define the word
differently.

98. Conversion. Let (p(x, y) denote any proposition,
A, E, I, or O, of the traditional logic (see § 50); and
let \j/(y, x) denote any other proposition which the first
implies, the letters x and y being interchanged. The im-
plication <p(x, y) : y]/(y, x) is called Conversion. When
the two implications <p(x, y) and \|/(y, x) are equivalent,
each implying the other, as in x\y r — y:x, and in
(x:y'y = (y:x'y, the conversion is called Simple Con-
version. When the proposition (p(x, y) implies but is not

94 SYMBOLIC LOGIC [§§ 98-100

implied by \^(v/, x), as in the case of (x : y) : (y : .«')', the
conversion is called Conversion by Limitation or Per
accidens. In all these cases, the antecedent <p(x, y) is
called the Convertend ; and the consequent ^{y, x) is
called the Converse.

99. Modality. In the traditional logic any proposition
A B of the first degree is called a pure proposition, while
any of my propositions A BC or A BCU , &c, of a Mr/her degree
would generally be considered a modal proposition ; but
upon this point we cannot speak with certainty, as
logicians are not agreed as to the meaning of the word
' modal.' For example, let the pure proposition A B
assert that " Alfred will go to Belgium " ; then A Be might
be read " Alfred will certainly go to Belgium" which would
be called a modal proposition. Again, the proposition
A" B , which asserts that " Alfred will not go to Belgium"
would be called a pure proposition ; whereas A B ', or its
synonym (A B )\ which asserts that A B is false, would, by
most logicians, be considered a modal proposition (see
§§ 21, 22, and note 2, p. 105).

1 100. Dichotomy. Let the symbols (AB), (AB 7 ), (ABC),
&c., with brackets, be understood to denote classes (as in
Boolian systems) and not the statements AB, AB 7 , ABC, &c.
We get*

A = A(B + B 7 ) = A(B + B 7 )(C + C) = &c.
= (AB) + (AB 7 ) = (ABC) + (ABC 7 ) + (AB 7 C) + (AB 7 C 7 )

= &c.

Thus any class A in our universe of discourse may be
divided, first, into two mutually exclusive divisions ;
then, by similar subdivision of each of these, into four
mutually exclusive divisions ; and so on. This process
of division into two, four, eight, &c, mutually exclusive

* The symbol (AB) denotes the total of individuals common to A and
B ; the symbol (AB') denotes the total number in A but not in B ; and
so on.

§§ 100-10:.] TECHNICAL WORDS EXPLAINED 95

divisions is called Dichotomy. The celebrated Tree of
Porphyry, or Bamean Tree, affords a picture illustration
of this division by Dichotomy. Jeremy Bentham wrote
enthusiastically of " the matchless beauty of the Ramean
Tree."

101. Simple Constructive Dilemma. This, expressed
symbolically, is the implication

(A : aO(B : x)(A + B) : x.

It may be read, " If A implies x, and B implies x, and
either A or B is true, then x is true."

102. Complex Constructive Dilemma. This is the im-
plication

(A:aOCB:yXA + B):s + y.

103. Destructive Dilemma. This is

(A:;r)(B: t y)( t / + //):A' + B'.

It may be read, " If A implies x, and B implies y, and
either x or y is false, then either A or B is false."

104. Modus ponendo ponens (see Dr. Keynes's "Formal
Logic "). There are two forms of this, the one valid, the
other not, namely,

(A : B)A : B and (A : B)B : A.

The first form is self-evident ; the second form fails in
the case A^B"' 1 and in the case A~ e B e ; for, denoting the
second form by <p, we get (see §§ 67—69)

Wc£ / = A r 'B- > ' + A- e B e .

105. Modus tollendo tollens. Of this also there are two
forms ; the first valid, the second not, namely,

(A : B)B' : A' and (A : B)A' : B'.

The first is evident ; the second fails, as before, in the
case A^B"*, and in the case A~ e B e . For, denoting the

96 SYMBOLIC LOGIC [§§ 105-108

second form by (p, Ave get Wc// = A^B" + A" 6 B £ . (See
§§ 67-69.)

106. Modus tollendo ponens. This also has two forms;
the first valid, the other not. They are

(A + B)A / :B and (AB)'B':A.

The first may be proved formally as follows : —

(A + B)A' : B = A'B'( A + B) : r, = (,, + >,) : ,/
= >j :>] = e.

The second is not valid, for

(AB)'B' : A = A'B'(AB)' : n = A'B' : n
= (A + B) e ;

which fails both in the case (A + By and in the case
(A + B)". To prove its failure in the last case, let (p
denote the given implication. We get

(p = ( AB)'B' : A = (A + B) e ,

as already proved. Therefore, putting A + B = 0, we get

(p = e* = n .

107. Modus poncndo tollens. This also has a valid and
an invalid form, namely,

(AB)'A : B' and (A + B)B : A'.

The first is valid, for

(AB)'A : B' = AB(AB)' : n = 1 : 1 = e.

The second is not valid, for

(A + B)B:A' = AB(A + B):>/ = AB:>,,

which fails both in the case (AB) € and in the case (AB) e .
In the first case the given implication becomes e : >;,
which = t] ; and in the second case it becomes 6 : >/, which
also = >].

108. Essential (or Explicative) and Ampliative. Let x
be any word or symbol, and let <p(x) be any proposition

§§ 108-110] TECHNICAL WORDS EXPLAINED 97

containing x (see § 13). When (p(x) is, or follows neces-
sarily from, a definition which explains the meaning of the
word (or collection of words) x ; then the proposition
<p(x) is called an essential, or an explicative, proposition.
Formal certainties are essential propositions (see § 109).
When we have a proposition, such as x a , or x~ a , or
x a + vf, which gives information about x not contained
in any definition of x ; such a proposition is called
ampliative.

109. Formal and Material A proposition is called a
formal certainty when it follows necessarily from our
definitions, or our understood linguistic conventions,
without further data ; and it is called a formal impossi-
bility, when it is inconsistent with our definitions or
linguistic conventions. It is called a material certainty
when it follows necessarily from some special data not
necessarily contained in our definitions. Similarly, it is
called a material impossibility when it contradicts some
special datum or data not contained in our definitions.
In this book the symbols e and n respectively denote
certainties and impossibilities without any necessary
implication as to whether formal or material. When
no special data are given beyond our definitions, the
certainties and impossibilities spoken of are understood
to be formal ; when special data are given then e and n re-
spectively denote material certainties and impossibilities.

110. Meaningless Symbols. In logical as in mathe-
matical researches, expressions sometimes turn up to
which we cannot, for a time, or in the circumstances
considered, attach any meaning. Such expressions are
not on that account to be thrown aside as useless. The
meaning and the utility may come later; the symbol
^/ — 1 in mathematics is a well-known instance. From
the fact that a certain simple or complex symbol x
happens to be meaningless, it does not follow that every
statement or expression containing it is also meaningless.
For example, the logical statement A^ + A'*, which

G

98 SYMBOLIC LOGIC [§110

asserts that A either belongs to the class x or does not
belong to it, is a formal certainty whether A be mean-
ingless or not, and also whether x be meaningless or not.
Suppose A meaningless and x a certainty. We get

A* + A" x = e + (P = >/ + e = e.

Next, suppose A a certainty and x meaningless. We get

A x + A- r = e° + t-° = >; + e = f .

Lastly, suppose A and x both meaningless. We get

A x + A"* = 0° + 0-° = e + >/ = e.

Let A x denote any function of x, that is, any expression
containing the symbol x ; and let <p(A- x ) be any state-
ment containing the symbol A x ; so that the statement
<p(A x ) is a function of a function of x (see § 13). Suppose
now that the symbol A x , though intelligible for most
values (or meanings) of x, happens to be meaningless
when x has a particular value a, and also when x has a
particular value /3. Suppose also that the statement
<p(A. x ) is true (and therefore intelligible) for all values of
x except the values a and /3, but that for these two
values of x the statement <p(A. x ) becomes meaningless, and
therefore neither true nor false. Suppose, thirdly, that
<p(A x ) becomes true (and therefore intelligible) also for
the exceptional cases x = a and x = ft provided we lay
down the convention or definition that the hitherto
meaningless symbol A a shall have a certain intelligible
meaning m., and that, similarly, the hitherto meaning-
less symbol A^ shall have a certain intelligible mean-
ing m 2 . Then, the hitherto meaningless symbols A a and
Ap will henceforth be synonyms of the intelligible
symbols m 1 and m 2 , and the general statement or formula
<p(A x ), which was before meaningless in the cases x = a
and x = (3, will now be true and intelligible for all values
of x without exception. It is on this principle that

§§110,111] MEANINGLESS SYMBOLS 99

mathematicians have been led to give meanings to the
originally meaningless symbols a° and a n , the first
of which is now synonymous with 1, and the second

with — .

a n

Suppose we have a formula, <p{x)=^^r(x), which holds
good for all values of x with the exception of a certain
meaningless value ?. For this value of x we further
suppose (p(x) to become meaningless, while \J/(.r) remains
still intelligible. In this case, since (p(?) is, by hypo-
thesis, meaningless, we are at liberty to give it any
convenient meaning that does not conflict with any
previous definition or established formula. In order,
therefore, that the formula <p(x) = \j/(x) may hold good
for all values of x without exception (not excluding even
the meaningless value 9), we may legitimately lay down
the convention or definition that the hitherto meaning-
less expression (£(?) shall henceforth be synonymous
with the always intelligible expression yf(s). With
this convention, the formula, (j)(x) = y(s(x), which before
had only a restricted validity, will now become true in
all cases.

111. Take, for example, the formula, s /x > /x = x in
mathematics. This is understood to be true for all
positive values of x; but the symbol ^/x, and conse-
quently also the symbol Jxjx, become meaningless
when x is negative, for (unless we lay down further con-
ventions) the square roots of negative numbers or
fractions are non-existent. Mathematicians, therefore,
have arrived tacitly, and, as it were, unconsciously, at the
understanding that when x is negative, then, Avhatever
meaning may be given to the symbol Jx itself, the
combination y/x^x, like its synonym {^/xf, shall be
synonymous with x ; and, further, that whatever meaning
it may in future be found convenient to give to */— 1,
that meaning must not conflict with any previous formula

100 SYMBOLIC LOGIC [§§ 111, 112

or definition. Those remarks bear solely on the algebraic
symbol *J — 1, which we have given merely as a concrete
illustration of the wider general principles discussed
previously. In geometry the symbol *J — 1 now conveys
by itself a clear and intelligible meaning, and one which
in no way conflicts with any algebraic formula of which
it is a constituent.

112. Induction. — The reasoning by which we infer, or
rather suspect, the existence of a general law by observa-
tion of particular cases or instances is called Induction.
Let us imagine a little boy, who has but little experience
of ordinary natural phenomena, to be sitting close to a
clear lake, picking up pebbles one after another, throwing
them into the lake, and watching them sink. He might
reason inductively as follows: "This is a stone" (a); "I
throw it into the water" (/3) ; "It sinks" (7). These
three propositions he repeats, or rather tacitly and as it
were mechanically thinks, over and over again, until finally
he discovers (as he imagines) the universal law a/3 : y, that
a/3 implies y, that all stones thrown into ivatcr sink. He
continues the process, and presently, to his astonishment,
discovers that the inductive law a/3 : y is not universally
true. An exception has occurred. One of the pebbles
which he throws in happens to be a pumice-stone and
does not sink. Should the lake happen to be in the
crater of an extinct volcano, the pebbles might be all
pumice-stones, and the little boy might then have
arrived inductively at the general law, not that all stones
sink, but that all stones float. So it is with every so-
called " law of nature." The whole collective experience
of mankind, even if it embraced millions of ages and
extended all round in space beyond the farthest stars that
can ever be discovered by the most powerful telescope,
must necessarily occupy but an infinitesimal portion of
infinite time, and must ever be restricted to a mere
infinitesimal portion of infinite space. Laws founded upon
data thus confined, as it were, within the limits of an

§ 1 1 2] " LAWS OF NATURE " 101

infinitesimal can never be regarded (like most formulae in
logic and in mathematics) as absolutely certain ; they
should not therefore be extended to the infinite universe
of time and space beyond — a universe which must
necessarily remain for ever beyond our ken. This is a
truth which philosophers too often forget (see § 80).

Many theorems in mathematics, like most of the laws
of nature, were discovered inductively before their validity
could be rigorously deduced from unquestionable premises.
In some theorems thus discovered further researches have
shown that their validity is restricted within narrower
limits than was at first supposed. Taylor's Theorem
in the Differential Calculus is a well-known example.
Mathematicians used to speak of the " failure cases " of
Taylor's Theorem, until Mr. Homersham Cox at last
investigated and accurately determined the exact con-
ditions of its validity. The following example of a
theorem discovered inductively by successive experiments
may not be very important ; but as it occurred in the
course of my own researches rather more than thirty
years ago, I venture to give it by way of illustration.

Let C be the centre of a square. From C draw in a
random direction a straight line CP, meeting a side of
the square at P. What is the average area of the circle
whose variable radius is CP ?

The question is very easy for any one with an
elementary knowledge of the integral calculus and its
applications, and I found at once that the average area
required is equal to that of the given square. I next
took a rectangle instead of a square, and found that the
average area required (i.e. that of the random circle) was
equal to that of the rectangle. This led me to suspect
that the same law would be found to hold good in regard
to all symmetrical areas, and I tried the ellipse. The
result was what I had expected : taking C as the centre
of the ellipse, and CP in a random direction meeting the
curve at P, I found that the average area of the variable

102 SYMBOLIC LOGIC [§112

circle whose radius is CP must be equal to that of the
ellipse. Further trials with other symmetrical figures
confirmed my opinion as to the universality of the law.
Next came the questions : Need the given figure be
symmetrical ? and might not the law hold good for
any point C in any area, regular or irregular ? Further
trials again confirmed my suspicions, and led me to the
discovery of the general theorem, that if there be any
given areas in the same plane, and we take any point C
anywhere in the plane (whether in one of the given areas
or not), and draw any random radius CP meeting the
boundary of any given area at a variable point P, the
average area of the circle whose radius is CP is always
equal to the sum of the given areas, provided we con-
sider the variable circle as positive when P is a point of
exit from any area, negative when P is a point of
entrance, and zero when P is non-existent, because the
random radius meets none of the given boundaries.

Next came the question : Might not the same general
theorem be extended to any number of given volumes
Experiment again led to an affirmative answer — that is
to say, to the discovery of the following theorem which (as
No. 3486) I proposed in the Educational Times as follows :

Some shapeless solids lie about —

No matter where they be ;
Within such solid, or without,

Let's take a centre C.
From centre C, in countless hosts,

And meet a surface each at P,

Or, may be, meet with none.
Those shapeless solids, far or near,

Their total prove to be
The average volume of the sphere

§§112, 113] FINITE, INFINITE, ETC. 103

The sphere, beware, is positive

When out at P they fly ;
But, changing sign, 'tis negative

When entrance there you spy.
One caution more, and I have done :
The sphere is naught when P there's none.

In proposing the question in verse instead of in plain
prose, I merely imitated the example of more dis-
tinguished contributors. Mathematicians, like other
folk, have their moments of exuberance, when they
burst forth into song just to relieve their feelings. The
theorem thus discovered inductively was proved de-
ductively by Mr. G. S. Carr. A fuller and therefore
clearer proof was afterwards given by Mr. D. Biddle,
who succeeded Mr. Miller as mathematical editor of
the Educational Times.

113. Infinite and Infinitesimal. Much confusion of
ideas is caused by the fact that each of those words
is used in different senses, especially by mathematicians.
Hence arise most of the strange and inadmissible para-
doxes of the various non- Euclidean geometries. To
avoid all ambiguities, I will define the words as follows.
The symbol a denotes any positive quantity or ratio too
large to he expressible in any recognised notation, and any
such ratio is called a positive infinity. As we may, in the
course of an investigation, have to speak of several such
ratios, the symbol a denotes a class of ratios called infinities,
the respective individuals of which may be designated
by a a 2 , a g , &c. An immensely large number is not
necessarily infinite. For example, let M denote a million.
The symbol M M , which denotes the millionth power of a
million, is a number so inconceivably large that the ratio
which a million miles has to the millionth part of an
inch would be negligible in comparison ; yet this ratio
M M is too small to be reckoned among the infinities
a , a a y &c, of the class a, because, though inconceivably

104 SYMBOLIC LOGIC [§113

large, its exact value is still expressible in our decimal nota-
tion ; for we have only to substitute 10° or 1,000,000
for M, and we get the exact expression at once. The
symbol /3, or its synonym — a, denotes any negative in-
finity ; so that fi v j3 2 , /3 3 , &c, denote different negative
ratios, each of which is numerically too large to be
expressible in any recognised notation. Mathematicians
often use the symbols oo and — co pretty much in the
sense here given to a and /3 ; but unfortunately they
also employ oo and — oo indifferently to denote expres-

1 3
sions such as -, -, &c. which are not ratios at all, but mire

non-existences of the class (see § 6). Mathematicians
consider oo and — oo equivalent when they are employed
in this sense; but it is clear that a and — a are not
equivalent. They speak of all parallel straight lines
meeting at a point at infinity ; but this is only an
abbreviated way of saying that all straight lines which
meet at any infinite distance a v or a 2 , or a,, &c, or fi v
or /8 or /3 3 , &c, can never be distinguished by any
possible instrument from parallel straight lines ; and
may, therefore, for all practical purposes, be considered
parallel.

The symbol h, called a positive infinitesimal, denotes
any positive quantity or ratio too small numerically to be
expressible in any recognised notation; and the symbol
7c, called a negative infinitesimal, denotes any negative
quantity or ratio too small numerically to be expressible
in any recognised notation. Let c temporarily denote
any positive finite number or ratio — that is to say,
a ratio neither too large nor too small to be expres-
sible in our ordinary notation; and let symbols of the
forms xy, x + y, x — y, &c„ have their customary mathe-
matical meanings. From these conventions we get various
self-evident formula?, such as

§ 113] FINITE, INFINITE, ETC. 105

(1) (cay, (c(3f; (2) (ch)\ (ckf ; (3) (« - c)\ ;
(4) (,±/0 c ; (5) ((3 + cf; (6) (f)", (|) fl ;

(7) Q\ (£)*; (8) («Y, (/S 2 )"; (9) (aflP;

(10) of : afar* ; ( 1 1 ) « a + s^ : ar° ; (12) (M)*.

The first formula asserts that the product of a positive
finite and a positive infinite is a positive infinite ; the tenth
formula asserts that if any ratio x is a positive finite, it
is neither a positive nor a negative infinite. The third

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