Symbolic Logic and its Applications

= conclusion.

that

"

A

;

a meaningless statement which

is

For example,

true nor false. c

impos-

is

meanings) means cp^cp' which

false for all admissible values (or

Also

let

w = whale, denote the

(p(w, h)

let

while is

h

(p°

neither

= herring, statement

small whale can swallow a large herring."

We

get e

(p

(w, h)

a three-factor statement


which

•

(p\w,

c),

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

unmeaning

to say that a small

whale can swallow a large conclusion.

Thus we see that more or less


it

is

F(x, y), &c, are really blank forms of

§§

APPLICATION TO

16, 17]

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,

&c, as the case may be. containing the
a

17.

+

/3,

Let

;

;

;

languages alike, is hardly unit of all possible. is the reasoning the manner in which the separate words are combined to construct a proposition varies according to

classification applicable to all

The

complete

proposition

;

In no Consider the two languages is it exactly the same. following example. Let S = His son, let A = in Africa, let K = has been killed, and let (p(S, K, A) denote the By proposition " His son has been killed in Africa!' 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 Do these two propositions differ in meaning? killed." Clearly they do. Let S A denote his son in Africa (to distinguish him, say, from S c his son in China), and let KA denote has been killed in Africa (as distinguished from K c has been killed in China). It follows that

the particular bent of the language employed.

,

,

SYMBOLIC LOGIC

12

[§§17,18


referring to the

noun

S,

whereas, in the

the force of an adverb referring to

as regards symbols of the

in general,

latter,

the verb K.

form

A

has

And

Ax Ay A ,

,

2

,

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 &c., the

letter

members of the general or class idea. Hence when A denotes a noun, the subscripta denote adjectives, or adjective-equivalents; whereas when A

separate it

is

that

denotes a verb, the subscripta denote adverbs, or adverbWhen we look into the matter closely, the equivalents. inflections

of verbs,

to

indicate

really the force of adverbs, and,

of view,

or tenses, have logical point

For

be regarded as adverb-equivalents.

S denote the word speak, & x may denote may denote will speak, and so on just as when Sy

example, spoke,

may

moods

from the

if

;

S denotes He spoke French,

may denote He spoke well, may denote He spoke slowly, and S y spoke,

S^

or or

He He

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.

spoke

Dutch,

and so

on.

CHAPTER

III

cause of symbolic paradoxes is the 8. The main Take, for ambiguity of words in current language. When we say, " If example, the words if and implies. 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 This is the sense in which the understood conditions. It word if is used throughout this book (see § 10). is understood to refer to a general law rather than to 1

§§

PARADOXES AND AMBIGUITIES

18, 19]

13

So with the word implies. Let M z F denote " The mercury will fall below zero',' and let The preceding conditional denote " Water will freeze." z F which statement will then be expressed by M a particular case.

W

:

asserts

W

F

that the proposition

But

.

convention

this

M

z

forces

us

W

,

the proposition

implies

accept some

to

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 The latter asserts that the statement x statement. (whether true or false) implies any certainty e, or (in The paradox other words) that if x is true e is true. will appear still more curious when we change x into e in the second. We in the first formula, or x into e, which asserts that any imthen get the formula The reason why the possibility implies any certainty. last formula appears paradoxical to some persons is e to probably this, that they erroneously understand mean Q^ Q e and to assert that if any statement Q is But impossible it is also certain, which would be absurd. paradoxical-looking formulas, such as

>/

:

:

rj

r\

:

>/

:

:

,

by definition it e does not mean this (see § 74) / simply means (>/e )'', which asserts that the statement Similarly, t]e is an impossibility, as it evidently is. x means {qx'J*, and asserts that nx is an impossibility, which is true, since the statement r\x' contains the im}}

:

r\

:

;

We prove x e as follows x\e — (xe'y = (x>i) = if —

possible factor

n.

:

ri

e.

For

e

=>],

since

implication for

Q"

e

:

is

19.

Q = rp e

:

is

some

That, on the other hand, the not a valid formula is evident

§ 20).

clearly fails in the case Q?.

it

sign

1

Q Q

denial of any certainty

the

impossibility (see

:

>f

=

Other paradoxes

of equivalence

(

=

*

:

'/

Q=

= («/)" = («0" =

»/,

we get

>/.

from the ambiguity of the In this book the statement

arise ).

Taking

SYMBOLIC LOGIC

14

[§|

1

20

9,

=

not necessarily assert that a and /3 are /3) does synonymous, that they have the same meaning, but only that they are equivalent in the sense that each implies (a

the other, using the word 'implies' as denned in

In this sense any two

however

meaning

different in

and n 2 62 and variables, B x definition, we have

possibilities, n l

= e = (e 2

l

)

;

x

and

and

e

2

,

so are

§

10.

are equivalent,

any two im-

but not necessarily two different We prove this as follows. By

\

.

(e

e

certainties,

= (e/ ne/ y = »w=V4= e«;

:e )(e :e l

2

2

,

,

=(vi) (Vt) for the denial of any certainty Again we have, by definition,

1)

2

i

,

ex

some

is

impossibility

r\

y

.

= *K = Vi= But we

,

equivalent.

necessarily

are

any two variables, 6 X and # 2 For example, 6 2 might be

cannot assert that

the denial of 6 V in which case (

e l

=

2

)

= (0 = e\) = (0 X

The symbol used a and

/3,

to

:

we should

e\)(6\

.

ej

l

assert

that

get (

0A)Wi)"

any two

are not only equivalent (in

statements,

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 have 6, t, let u denote uncertain, while the symbols e, r\,

We (A u = A'

i

by definition, and A 5 will A = and A" ) ), have (A respectively assert that the chance of A is greater than These conventions give us the \, that it is less than \. nine-factor formula their usual significations. 7r

e

)

shall then, p while

(*V)W(^WV)W

§

PARADOXES AND AMBIGUITIES

20]

15

2) that the denial of a truth is an 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-

which

asserts (1,

untruth,

versely

(7

;

that the denial of a variable

)

that the denial of a possibility

(8, 9)

The

and conversely.

first

the other five are less

a variable

is

an uncertainty,

is

four factors are pretty evident

Some

so.

for example, that instead of

(tt')

that the denial of a possibility *

persons might reason, w

we should have

is

(-n-'y

;

not merely an uncer-

A single concrete example but an impossibility. show that the reasoning is not correct. The statement " It will rain to-morrow " may be considered a

tainty will

possibility

;

but

its

denial " It will not rain to-morrow,"

though an uncertainty may be proved

u {tt')

is

not an impossibility.

Let

Q

equivalent to proving

Q

The formula

denote any statement taken at random out of a collection of statements containing certainties, impossibilities, and variables. To

prove

{ir')

u

is

as follows

:

77 :

(Q')

M

Thus we

.

get (Tr'y

= Q-

(Q'f = Q

Q.

(Q y = Q

e

:

e

e

for

e

+ Q (Q'r + (Q7 = Q + Q :Q + Q = e; :

e

e

and (Q f = Q, whatever be the statement To prove that (^y, on the other hand, is not valid, ,

/

e

l

we have only

e

,

,

to instance a single case of failure.

Q the same meaning as before, a case for we then get, putting Q = 6 V

=e

;r 1

1i

= (e/ y = (e € y = i

1

2

of failure

,

Giving is

Q

8 ;

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 statement 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-.

SYMBOLIC LOGIC

16

may seem

[§21

the proA' with nor position A is not quite synonymous with A' = Then A yet such is the fact. Let A It rains. 21. It

paradoxical

say that

to

A

T

,

=

1

;

A T = it

It does not rain ;

is

true that

it

rains

;

A

and A' = it and AT are

The two propositions false that it rains. equivalent in the sense that each implies the other

is

;

but

they are not synonymous, for we cannot always substitute In other words, the equivalence the one for the other. (A A T ) does not necessarily imply the equivalence e then For example, let (p(A) denote A (p(A T ). (p(A)

=

=

;

T € Sup
,

true, or it is

happens

be true in the case considered, though

to

all cases.

not true in

ct>(A_)

We

get

= A< = e;=(e y = r T

,

never a certainty, though

for a variable is

it

may

turn out

true in a particular case.

Again,

we

get

^(AT ) = (AT )e = (^) = e« = e; e

T

for 6 T

means

T

which is a formal certainty. In this T though we have A = A yet (p(A) is not

(0 T )

case, therefore,

,

,

T equivalent to (p(A ).

A

suppose

Next,

denotes

t

,

a

variable that happens to be false in the case considered, get though it is not false always.

We

0(A') for

no

= (A') = A* = 0? = e

(though

variable

it

may

turn out false in a parti-

cular case) can be an impossibility.

we

»7;

On

the other hand,

get

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

€


for 6[

e

means (Oy, which is a formal certainty. In this though we have A' = A\ yet <£(A') is not

case, therefore,

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

DEGREES OF STATEMENTS

21, 22]

§§

17

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 that A is false. It is the first step in the classification of statements, and marks a this distinction ,

1

,

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 a/3y as A" but of statements of higher degrees, such as A 13

,

A

afiyS

,

may

be asked, what

meant by statements of the second, third, &c, degrees, when the primary subject is itself a statement ? The statement A a/iy or its a/3 7 synonym (A ) is a statement of the first degree as regards 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 and a statement of the third degree as regards A, the root statement of the series. Viewed from another ,

But,

&c.

it

is

,

,

;

,

tt

,

standpoint,

A a may

be called^a revision of the

A, which (though here

judgment, of the

series)

it

is

may

judgment

the root statement, or root itself laave

been a revision

some previous judgment here unexpressed. Similarly, a (A")* may be called a revision of the judgment A and so on. To take the most general case, let A denote any complex statement (or judgment) of the ntb degree. of

3

,

be

neither a formal certainty (see § 109), like nor a formal impossibility, like (a/3 af, it may be a material certainty, impossibility, or variable, according If

it

(a/3

:

a)

e

:

,

If it follows data on which it is founded. from these data, it is a certainty, and we write A* if it is incompatible with these data, it is an impossibility, 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 or A^ or A", it must necessarily be a judgment (or statement) of the (w+l) th degree, since, by hypothesis, the statement A is of the w th degree. Suppose, for ex-

to the special

necessarily ;

;

e

ample,

A

denotes a functional statement


y,

B

z)

of

SYMBOLIC LOGIC

18

[§§

22-24

which may have m different meanings &c, depending upon the different meanings x., x2 x y &c, y v yz y y &c, zv z 2 zy &c., of x, y, z. Of these m different meanings of A, or its synonym
th

degree,

(or values)

,

,

,

,

/

/

e

.

.

11

,

eT,e

,

,

degree as regards \Ja 23. It may be remarked that any statement A and T its denial A' are always of the same degree, whereas A synonyms (see and A', their respective equivalents but not The statement SS 19, 21), are of one degree higher.

A

and confirmation of the judgment A; 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 conT if we profirm the previous judgment A and write A 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 that if true, it must be true always and if false, false " T

is

while

a revision

A

1

is

a revision

',

;

;

;

it

must be

false always.

To

this I reply, as I did in

my

seventh paper published in the Proceedings of the London

§

VARIABLE STATEMENTS

24]

Mathematical

that

Society,

when

false," or "

true and sometimes

"

say

I

A

is

A

19 is

sometimes

a variable," I merely

mean that the symbol, word, or collection of words, denoted by A sometimes represents a truth and someFor example, suppose the symbol A times an untruth. 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

.

Brown

moment the servant tells me home " I happen to see

that

"

Mrs.

Brown walking away in the distance, then / have and form the judgment A which, of course,

Mrs.

is

not at

e

fresh data

,

In this case I say that " A is certain" because its denial A' (" Mrs. Brown is at home ") would But if, contradict my data, the evidence of my eyes. 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

A

implies

T

.

data of an opposite kind, and form the judgment Av which implies A'. In this case I say that " A is im-

,

because

possible,"

dicts

my

medium

A

is

what is

a

a it is

Mrs.

different

when

different

when she

the

Brown

statement

represented

by A,

not at home," this time contradata, which, as before, I obtain through the of my two eyes. To say that the proposition

"

namely,

'proposition

it is true, is

person

is out.

is

when

when

it

is

false

like saying that Mrs.

she

is

in from

from

Brown

what she

is

SYMBOLIC LOGIC

20

[§25

CHAPTER IV The

25.

following three rules are often useful:

= A'tf>(e). A"4>(A) = A*0(>;). A
(1) A'(A) (2)

In the

e

e

(3)

).

denotes the

last of these formulae, 6 X

first

variable

&c, that comes after the last-named For example, if the last variable that in our argument. 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 e2 e y &c, is represented by the e in (p(e), nor which of the series &c, is represented y 2 in (p(>i); for, as proved in § 19, we have always by the e ), and (t] x = (ex whatever be the certainties ex and y y ), e and whatever the impossibilities x ana %• Suppose, of the series 6

6

,

6

.

,

,

rj

,

rj

,

r\

=

r]

"

rj

for

example, that

\j/

denotes

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

We

get

=AB £

T,

C

fl

e

= A B"C e

9 ;

may

be omitted In this without altering the value or meaning of \f/. operation we assumed the formulas

so that the fourth or bracket factor of

(1)

(ariz=r]);

(2) (ae

= a);

(3)

\j/-

(*i

+ a==a).

Other formulae frequently required are (4) (AB)' (6)

(9)

e

= A' + B';

+ A = e; / = *,;

(5) (A + B)' = A'B'; = >7; (8) A + A' = e; AA' (7) = (10) >/ (11) A + AB = A; *;

(12) (A

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

§§

FORMULAE OF OPERATION

26, 27]

we

26. For the rest of this chapter

consideration of variables, so that A,

21

shall exclude the

A

T

A*

,

this

be con-

will

sidered mutually equivalent, as will also A', A',

A''.

On

understanding we get the formulas (1)

A<£(A) = A
(3) A<£(A')

From

;

= A<^);

these formulas

(2)

Aty( A) = Aty(i)

(4)

A^(A') = A'(/)(e).

we derive

(5) AB'(/)(A, B)

;

others, such as

= AB'<£(e,

rj) ;

= AB'^)(»;, n)\ B') = AB'<£(>/,

(6) AB'<J>(A', B) (7) AB'<£(A',

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 examples will show the working of these formulas /

;

:

Let


B) = AB'C

+ A'BC'.

Then we

get

= 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. let cp(B, D) = (CD' + CD + B C B'D'0(B, D) = B'D'(CD' + CD + B'C')' = B D (Ce + C + eC'/ = B'D'(C + C)' = B D'e = B'D'>i = AB'<£( A, B)

/

/

/

/

/

Next,

Then,

/

/

/

/

/

)

/

,

.

/

>;

f

,

>

The

].

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

of course,

troublesome manner. 27. If in any product

ABC

any statement-factor

is

implied in any other factor, or combination of factors, If in any sum (i.e., the implied factor may be omitted.

SYMBOLIC LOGIC

2%

[§§

27, 28

A + B + C, any term implies any other, or any others, the implying term may be omitted. rules are expressed symbolically by the two

alternative)

the

sum

These

of

formulae

(A:B):(AB = A);

(1)

By

(2)

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

virtue of the formula (x a)(x :

may

formulae

:

/3)

=x

:

these two

a/3,

be combined into the single formula

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

(3)

As the converse of each holds good, we get

of

these three formula?

also

A:B = (AB = A) = (A + B = B). Hence, we get A + AB = A, omitting the term AB, because and we also get A(A + B) = A, it implies the term A; omitting the factor A + B, because it is implied by the (4)

factor A.

A B

28. Since

:

AB,

factor of

it

is

(AB = A), and B

equivalent to

called a factor of the antecedent A, in

and

same

that, for the

any implication

reason, the antecedent

called a multiple of the consequent B.

of

and (A = AB)

A B :

may

The equivalence as follows

(A

:

of

A B :

A

:

A may

a

be B,

be

The equivalence

be proved as follows

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

is

B may

follows that the consequent

:

:

and (A

= A:B.

+ B = B) may

be proved

:

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

The formula? assumed (x

:

aft)

= (x

"

If

a)(x

may

both of which assert that

:

x

is

in these :

/3),

two proofs are

and a

+ /3

:

x = (a

:

x)(fi

:

x),

For to be considered axiomatic. then a and /3 are both true " is

true,

equivalent to asserting that

" If

x

is

true a

is

true,

and

x

if

or

a

REDUNDANT TERMS

28, 29]

§§

true

is

|8 is

is

true x

is

Also, to assert that " If either a

true."

/5 is

true x

is

" is

true

true,

23

and

equivalent to asserting that

if /3 is

true x

" If

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 the term must not but if the product is not be omitted. Take, for example, the alternative statement rj

rj,

;

CD' + C'D Beginning with the

CD'(C'D

first

+ B'C' + B'D'.

term we get

+ B'C + B'D')' = CD'(w + B'» + B'e)' 7

= CD (B = BCD /

/

)'

/ .

Hence, the first term CD' must not be omitted. next the second term CD, we get

CTKCD' +

B'C/

Taking

+ B'D'/ = C'D( w + B'e + B',,)'

= C D(BY=BC D. /

/

CD

must not be omitted. Hence, the second term next take the third term B'C, getting

B^CD

7

-I-

C/D

+ B'D'/ = B'C'(>iI)' + eD + eD'/ = B'C'(D + D / = B C'>/ = ,

This shows that the third term

B'C

We

/

>/.

can be omitted as

SYMBOLIC LOGIC

24

29-31

[§§

Omitting the third term, we try the

redundant.

last

term B'D', thus

B'D'(CD'

+ CD)' = B'D'(Ce + C>,)' = B'D'C.

This shows that the fourth term B'D' cannot be omitted But if we retain as redundant if we omit the third term. term B'D', fourth the third term B'C, we may omit the

we then get

for

B'D'(CD'

+ CD + B'C7 = B'D'(Ce + C'n + eC')' = B D (C + C ) =B D )'

/

/ ,

,

/

/

J?

=

i7.

Thus, we may omit either the third term B'C, or else the fourth term B'D', as redundant, but not both. 30. A complex alternative may be said to be in its simplest form* when it contains no redundant terms, and none of its terms (or of the terms left) contains any redundant factor. For example, a + ab + m + m'n is reduced

form when we omit the redundant term ab, term strike out the unnecessary factor m' last of the out and m + m'n m + n, so that the simplest and a, ab For a + to its simplest

=

= +m+

(See § 31.) n. form of the expression is a to its simplest alternative complex a reduce 31. To /3)' a'/3' to the denial of (a formula form, apply the +

=

the alternative. Then apply the formula (a/3/ = a' + ft' to the negative compound factors of the result, and omit Then develop the redundant terms in this new result. and go formulae, same the by product the denial of this result final The before. as process through the same

be the simplest equivalent of the original alternative. Take, for example, the alternative given in § 30, and We get denote it by (p.

will

= a + ab + m + m'n = a + m + m!n. = (a + m + m'n)' = a'm'(m'n)' = a'm'(m + nf) = a'm'n'. = (cp')' = (a' m'n')' = a + m + n.

(p'

here call its " simplest form " I called its " primitive form in my third paper in the Proceedings of the London Mathematical Society but the word " primitive" is hardly appropriate. *

What

"

I

;

METHODS OF SIMPLIFICATION

§§31,32]

As another example take the

25

alternative

AB'C + ABD + A'B'D' + ABD' + A'B'D, and denote it by (p. Then, omitting, as we go along, all terms which mere inspection will show to be redundant, we get r/>

= AB'C + AB(D + D') + A'B'(D' + D) = AB'C + ABe + A'B'e = AB'C' + AB + A'B'.

=(AB C ) (AB)'(A B ) = (A' + B + C)(A' + B')(A + B) = (A' + B'C)( A + B) = A'B + AB'C. p = «p')' = (A'B + AB'C)' = (A + B')(A' + B + C) = AB + AC' + A'B' + B'C. /

/

/ ,

/

, /

<£

i

Applying terms, B'C')

we

may

the

find

test

of

§

29

to

redundant term (AC or

discover

that the second or fourth

be omitted as redundant, but not both.

We

thus get (p

either

of

of

= AB + A'B' + B'C = AB + AC + A'B', which may be

taken

as

the simplest

form

(p.

32. We will now apply the preceding principles to an interesting problem given by Dr. Venn in his " S} mbolic Logic" (see the edition of 1894, page 331). Suppose we were asked to discuss the following set of rules, in respect to their mutual consistency and T

brevity.

Financial Committee a. The amongst the General Committee.

shall

be

chosen

from

No one shall be a member both of the General /3. and Library Committees unless he be also on the Financial Committee. y. No member of the Library Committee shall be on the Financial Committee.

SYMBOLIC LOGIC

26

[§32

Solution.

Speaking of a member taken at random, let the symbols F, G, L, respectively denote the statements " He will be on the Financial Committee," " He will be on the General Committee," " He will be on the Library as usual, for any statement that Putting Committee." contradicts our data, we have >;,

a

= (F:G);

/3

= (GLF'

>,)

:

7 = (LF:#,);

;

so that a/3 .

7 = (F:G)(GLF

/

:i?)(FL:i7)

= (FG':*i)(GLF':T,)(FL:r,) = FG' + GLF + FL:>/. /

Putting

+ GLF' + FL, ^(F' + GXG' + L' + FXF' + I/)

for the antecedent

(J>

(See

we get

FG'

25, Formulae (4) and (5))

§

= (F' + GL')(G' + L' + F) = F'G' + F'L' + GL' + FGL' = F'G' + GL'; term FGL', being a multiple of the term GL', is / / redundant by inspection, and F L is also redundant, because, by § 29, for the

F L (F G + GL')' = F'L'(eG' + /

/

,

Hence,

/

finally, <£

G<)'

= F'L'(G' + G)' = n

.

we get (omitting the redundant term FL)

= (<£')' = (F'G' + GL')' = FG' + GL,

and there ore I

a/3y

That

is

= (f>:ri= (FG' + GL

to say,

:

rf)

= (F

:

the three club rules,

replaced by the two

simple rules

F G :

G)(G u,

:

(3,

and

G

L')-

7 :

,

may

L',

be

which

any member is on the Financial Committee, he must be also on the General Committee," which is rule a in other words and, secondly, that " If any member is on the General Committee, he is not to be on the Library Committee." assert, firstly, that

"

If

;

SOLUTIONS, ELIMINATIONS, LIMITS

§33]

27

CHAPTER V 33.

From

the formula (a

b)(c

:

:

d)

= ah' + cd'

:

»/

number of implications can always be expressed in the form of a single implication, the product of any

«

+ fi + 7 + &c

of which the antecedent

:

i],

a logical

is

sum

(or alternative),

and the consequent an impossibility. Suppose the implications forming the data of any problem that contains the statement x among its constituents to be thus reduced to the form

Ax + B,v' +

A

which

in

is

efficient of x',

the coefficient or co-factor of

and C the term,

contain neither x nor

data

may

C:tj,

or

sum

is

B

the co-

It is easy to see that the

x'.

also be expressed in the

which above

form

(B:asX«:A')(C!:9)

which

x,

of the terms,

J

equivalent to the form

(B^iA'XCm).

When

the data have been reduced to this form, the ^iven

implication, or product of implications,

is

said to be solved

x ; and the statements B and A' (which are generally more or less complex) are called the limits of x; the antecedent B being the strong * or superior limit and Since the the consequent A', the weak or inferior limit. with respect

to

;

*

When from

:

u,

A+B.

we can

our data

we say that AB:A:A + B, we say

|8

el

is

infer a:

that

AB

is

/3,

but have no data for inferring

For example, since we have stronger than A, and A stronger than

stronger than

/3.

SYMBOLIC LOGIC

28 factor

33,

[§§

34

(B x A') implies (B A'), and our data also imply >j), it follows that our data imply :

:

:

the factor (C

:

(B:A')(C:>,),

which

is

AB + C

equivalent to

Thus we get the

: *).

formula of elimination

+ B,/ + C:>,):(AB + C:>;),

(Ac

which asserts that the strongest conclusion deducible from our data, and making no mention of x, is the implication

AB + C

:

>/.

As

the two-factor statement

this conclusion

C^ABy,

ment C and the combination

it

is

equivalent to

asserts that the state-

of statements

AB

arc both

impossible. 34.

From

this

we deduce the

solution of the follow-

Let the functional symbol the symbol (p, denote data simply or a, b), z, <J)(x, y, which refer to any number of constituent statements x, y, z, a, b, and which may be expressed (as in the

ing more general problem.

33) in the form of a single implication + y + &c. rj, the terms a, /3, y, &c, being more or complex, and involving more or less the statements

problem of a

+

§

18

less x, y,

z,

:

a,

firstly, to find

It is required,

b.

successively in

the weakest antecedent

any and strongest consequent) of x, y, z; secondly, to eliminate x, y, z in the same order and, thirdly, to find the strongest implicational statement (involving a or b, but neither x nor y nor z) that remains after this desired order the limits

(i.e.,

;

elimination.

Let the assigned order of limits and elimination be Let A denote the sum of the terms containing the factor z let B denote the sum of the terms containing the factor z and let C denote the sum of the terms Our data being (p, we get containing neither z nor z

z, y, x.

;

,

.

(j,

= kz + B/ + C = (B = (B z A')(C = (B :

:

:

:

17

>;)

:

:

z)(z

z

:

:

A

,

)(C

:

r,)

A')(B A')(C :

:

>/).

§

SOLUTIONS, ELIMINATIONS, LIMITS

34]

The expression represented by Az + to

Bz'

to its simplest

have been reduced

+G

29

understood

is

form

(see §§

30,

The 31), before we collected the coefficients of z and z'. and the result after limits of z are therefore B and A' ;

the elimination of

z is

(B A')(C

:

:

= AB + C

which

>/),

n

:

.

To find the limits of y from the implication AB + C we reduce AB + C to its simplest form (see §§ 30, 31), We thus get, which we will suppose to be By + Ey' + F. >/,

:

as in the previous expression in

AB + C The

:

= By + E/ + F

r\

:

r,

limits of y are therefore

z,

= (E

y D')(E D')(F

:

:

E and

:

:

>,),

is

= ED + F

which

>/)•

:

and the result

D',

and y

after the successive elimination of z

(E D')(F

:

:

>/.

To find the limits of x from the implication ED + F we proceed exactly as before. We reduce ED + F to its simplest form, which we will suppose to be Gx + Hx + K, and get :

ED + F The

:

n

= Gx + Ha/ + K

:

r,

limits of x are therefore

= (H

H

after the successive elimination of

(H G0(K :

:

>/),

which

:

x G')(H G')(K :

:

and

G',

z,

x

y,

and the

:

>/,

>;).

result

is

= HG + K

:

>,.

x having thus been successively +K eliminated, there remains the implication the connecting which indicates the relation (if any) b. Thus, we remaining constituent statements a and

The statements

z,

y,

GH

:

}j,

finally get (/)

=

(B

:

z

:

which A and mention of z)

in

A')(E

:

//

:

D')(H x G')(GH :

:

:

,,).

B

do not contain z (that is, they make no D and E contain neither z nor y G and and the expression K contain neither z nor y nor x ;

;

H

+K

;

SYMBOLIC LOGIC

30

[§§

be destitute of

in the last factor will also

(i.e.,

34, 35

will

make

no mention of) the constitutents x, y, z, though, like G and H, it may contain the constituent statements a and b. a and a e are In the course of this process, since >)

:

:

whatever the statement a may be (see § 18), we can supply for any missing antecedent, and e for any missing consequent. certainties

>/

of the general prob-

35. To give a concrete example lem and solution discussed in § 34, e

We

:

+ xyb +xy z +y

xyza

denote the data

let (p

a

z

.

get, putting (p for these data,
= {xyza + xyb' + xy'z' + y'z'a')' — x'y + + y'z + abz + ax bijz

when

r\

:

»/,

:

the antecedent of this last implication has been its simplest form by the process explained in

reduced to §

Hence we

31.

(j>

get

= (y'+ ab)z + (]jy)z + {x'y + ax')

putting

A

in § 34,

we get

for y'

+ ab, B

n

+ ax'.

As

and the result

after

and C

for by,

:

for x'y

(B:s:A')(AB + C:>7), so that the limits of z are

the

elimination of

is

z

B and

AB + C

A', :

»/.

Substituting their

values for A, B, C, this last implication becomes {ab

which we ab

+ x, E

will for

n,

(f>

+ ,c)y + ax'

denote by *Dy

and F :

z

:

=

:z

:

(B

?/,

+ Ey' + F

:

n,

A')(Dy

+ E/ + F

A0(E

y D')(ED

Having thus found the

putting

J)

for

Thus we get

for ax.

= (B

:

:

limits

:

{ix.,

:

>/)

+F

:

»;).

the weakest ante-

SOLUTIONS, ELIMINATIONS, LIMITS

§§35,36]

cedents and strongest consequents) of z and y, to find the limits of x from the implication

31

we proceed

ED + F

:

n,

the strongest implication that remains after the Substituting for D, E, F the elimination of z and y.

which

is

we

values which they represent,

DE + F in

=

n

:

which G, H,

get

= Gx + BJ + K

{ah

+ J)n + «J

K

respectively denote

:

n

>/,

a,

n,

:

We

n-

thus

get

DE + F

:

= (H

>i

x G')(HG

:

:

+K

tj)

:

;

so that our final result is <$>

= (B = =

To obtain

:

(by (/>//

z

A')(E

:

:

:

//

D')(H

+ b y){n :z:a'y + b'y)(y

:

z

:

f

a'y

y

:

:

:

:

a;

; rt ;«

a'x

G0(HG + K e)(>/ + b'x){a :

:

:

+ b'x)(a

:

£C

:

i,)

:

»/)

x).

we first substituted for A, B, D, E, then we the values we had assigned to them this result

G, H, K in the second factor, omitted the redundant antecedent the redundant consequent e in the third factor, and the ;

>/

redundant certainty

(»/

:

»/),

which constituted the fourth

the fourth factor (HG + K:>/) reduces to the form (n rj), which is a formal certainty (see § 18), indicates that, in this particular problem, nothing can be implicationally affirmed in terms of a or

factor.

The

fact

that

:

z) except formal f &c, which such as (ab a), (aa >;), ab(a + b') are true always and independently of our data (p. 36. If in the preceding problem we had not reduced the alternative represented by As + Bz' + C to its simplest form (see §§ 30, 31), we should have found for the not a'y + b'y, but inferior limit or consequent of z, supposed that the might be it this From b'y). x(a'y + strongest conclusion deducible from z (in conjunction with, or within the limits of, our data) was not A' but But though xh! is formally stronger than A', that xk'.

b

(without mentioning either x or y or

certainties

:

:

:

>i,

SYMBOLIC LOGIC

32 is

36-38

than A' token we have no data but our here we have other data, namely,

say, stronger

to

definitions,

;

we

implies (as lent

[§§

shall prove) that A'

in this case equiva-

is

to xA', so that materially (that

to say, within the

is

limits of our particular data
This

we prove
:

(z

as follows

:

A

7

y

:

:

:

D' x) :

:

(A'

:

x)

:

(A'

= x A')

;

a proof which becomes evident when for A' and D' we substitute their respective values a!y + b'y and a'x + b'x for it is clear that y is a factor of the former, and x a ;

factor of the latter.

37. In the problem solved in § 35, in which our data, namely, the implication e

:

xyza'

(p

denoted

+ xyb' + xy'z' + y'z'a',

y, x as the order of limits and of elimination. taken the order y, x, z, our final result would have

we took

z,

Had we been

(j>

38.

= (z:y:

b'x

+ xz)(z + a

The preceding method

" limits "

my method

x){z

:

a'

of finding

of logical statements

was suggested by,

:

is

+ b').

what

I call

closely allied

to,

the

and

(published in 1877, in the

Lond. Math. Soc.) for successively finding the for the variables in a multiple integration limits of In the next chapter the method integral (see § 138). Proc. of the

will be applied to the solution (so far as solution is possible) of Professor Jevons's so-called

which has given

among

rise

to

logicians but also

"

Inverse Problem,"

much discussion, not among mathematicians.

so

only

PROBLEM"

JEVONS'S "INVERSE

§39]

33

CHAPTER VI Briefly stated, the so-called "inverse problem" of Professor Jevons is this. Let tp denote any alternative, It is required to find an imsuch as abc + a'bc + aVV 39.

'.

plication,

or product of implications,* that implies this

alternative.

Now, any implication whatever implications) that e

of


or

f :

b)((f>

:

alternative

e

or

cp,

:


:

»y,

is

of

a multiple

or (abc

:

ab)(e

:


&c, must necessarily imply the given

rj),

number

that the

so

cp,

any product

(or

equivalent to
example,

as, for

,

(a

is

of possible solutions

But though the problem

as enunthus indeterminate, the number of possible solutions may be restricted, and the really unlimited.

is

ciated

by Professor Jevons

is

problem rendered far more interesting, as well as more and instructive, by stating it in a more modified form as follows Let cp denote any alternative involving any number of

useful

:

constituents,

implication

a,

e

:

c,

b,

cp

&c.

It

required to resolve the

is

that

into factors, so

it

will

take the

form

(M a N)(P :

:

:

b

:

Q)(R

:

c

:

S),

&c,

which the limits M and N (see § 33) may contain &c, but not a; the limits P and Q may contain the limits R and S may neither a nor b c, d, &c, but contain d, e, &c, but neither a nor b nor c and so on When no nearer limits of a conto the last constituent. and e stituent can be found we give it the limits the former being its antecedent, and the latter its conin

b,

c,

;

;

>;

sequent (see * Professor

at

§§ 18, 34).

Jevons

calls these implications

tific

"laws," because he arrives

by which scien" investigators have often discovered the so-called " laws of nature

them by a long tentative inductive

process, like that

(see§ 112).

C

SYMBOLIC LOGIC

34

[§39 *

As a simple example, suppose we have (p

= abc + a'bc + ab'c',

the terms of which are mutually exclusive. form (see §§ 30, 31), we get

to its simplest

Reducing

= be + ab'c',

and therefore e

:

= (f/

<£

= (be)' {ab' )' n = (b' + c')(a' + b + c):r = a!b' + J'c + aV + be' f

:

,,

:

1

>/.

:

This alternative equivalent of § 31) by omitting either the not both so that we get

cp'

first

may

be simplified (see

or the third term, but

;

e

:

= b'c + a'c' + be'

(p

Taking the

rj

= a'b' + b'c + be

equivalent of

first

the limits of a) arranging

we

:

e

:


in the

it

:

17.

and (in order form Aa + Ba'

to find

+C

:

tj,

get (see §§ 33, 34) e

:

(p

V+

= tja + c = (c a

(6'c

+ W)

»/

:

7

:

:

e)(c

:

b

c)(t]

:

:

c

e).

:

Thus, we have successively found the limits of

But

34, 35).

§§

since (a

formal certainties, they that

:

e),

may

(>;

:

c),

and

(c

:

(see

a, b, c e)

are

all

be omitted as factors, so

we get e

:


= (c'

:

«)(c

:

6

c)

:

= (c'

a)(c

:

=

b).

two factors asserts that any term of the given alternative (p which contains c' must also contain a. The second asserts that any term which contains c must also contain b, and, conversely, that any term which con-

The

first

of these

tains b

must

native

(p will

also contain

c.

A

glance at the given alter-

verify these assertions.

denotes an Observe that here and in what follows the symbol denotes a given implication, which In §§ 34, 35 the symbol may take either such a form ase:a + /3 + 7 + &c. or as a + /3 + 7 + &c. 7/. *

alternative.

<j>

<j>

,

:

We

now take

will

the second equivalent of a'b'

and resolve

it

the limits of

a, b,

+ b'e + be'

first

sight e

:

it

/

:rt )( c

(6

tj,

by successively rinding

= &). different

a) in the former result

the factor

(c

factor (b'

a) in the latter.

:

namely,

<jj }

:

might be supposed that the two ways of into factors gave


e

35

Proceeding as before, we get

c.

-:^ =

At

:

into three factors

t

resolving

PROBLEM"

JEVONS'S "INVERSE

§§39,40]

since

results,

replaced by the

is

But

since the second factor informs us that b and c are equivalent, it follows that the two implications c a and b' a are equivalent also. :

= b), common

(c

to

both

results,

:

:

If we had taken the alternative equivalent of
e:(p in

= (p':>] = (b' + c': a)(c = b) = {1/

which either the factor

(b'

:

:

a)(c'

a) or the factor

be omitted as redundant, but not both. the factor yet

= b) alone neither implies = b) implies a), and

(c

(b'

= b),

a)(c

:

{c

:

a)

may

For though :

a) nor

(
:

a),

= b)

(c' implies This redundancy of factors in the result is a necessary consequence of the redundancy of terms in the alternative equivalent of
(b

{b' :a)(c

(c'

:

:a)(c

r

:

a).

omission

of

term

a'c'

the

implicational

factor

(a'b'

:

>/),

or

its

and the omission of the in the alternative leads, in like manner, to the

equivalent

(b'

:

a),

in the result

omission of the factor

(a'c'

:

rf),

;

or

its

equivalent

(c'

:

a),

in

the result. 40. I take the following alternative from Jevons's "Studies in Deductive Logic" (edition of 1880, p. 254, No. XII.), slightly changing the notation, abed

Let

(p

+ abe'd + ab'cd' + a'bed' + a'b'c'd'.

denote this alternative, and

let it

be required to

SYMBOLIC LOGIC

36

find successively the limits of a, b

we

are required to express

(M a N)(P :

:

in

M

which

and

N

:

b

e


Q)(R

:

n

and M.

By

e.

=d +

b
c

;

y

c

:

:

S)(T

d

U),

:

P and Q

;

;

r,,

we

get

V = d, Q = c + d, R =

b'c,

e,

U=

:

A

:

(p

= (d + be' + b'c

:

a

:

bd

glance at the given alternative

+ b'c){d <£>

>,,

6.

Omitting the last two factors R c S and because they are formal certainties, we get e

are

and S are neither to conand T and U must be respectively

N = bd + S= T=

b'c,

:

R

the process of §§ 34, 35,

+

In other words, form

d.

c,

in the

are not to contain a

neither to contain a nor b tain a nor b nor

:

[§40

T d :

:

:b:c

U

:

+ d).

will verify this result,

we have either d or be' or ( 1 that whenever we have a, then we b'c, then we have a (2) have either bd or b'c (3) that whenever we have d, then we have b (4) that whenever we have b, then we have either c or d; and (5) that from the implication e (p we can infer no relation connecting c with c£ without making which

asserts

)

that whenever ;

;

;

-.

mention of a or b or, in other words, that c cannot be e is a expressed in terms of d alone, since the factor c formal certainty and therefore true from our definitions The final factor is alone apart from any special data. for only added for form's sake, for it must always have In other words, when antecedent and e for consequent. we have n constituents, if x be the n th or last in the ;

>/

:

:

>/

must

order taken, the last factor

necessarily be

may

and therefore a formal certainty which understood. of n

:

c

e

:

Others of the factors

may

(as in

taken successively in alphabetic order. reverse order d, c, b, a, our result will be :

:

x

:

e,

left

the case

here) turn out to be formal certainties also, but

not necessarily. We have found the limits of the constituents

e

>;

be

(p

= (ab + ac' + bd

:

d

:

ab)(ab'

+ a'b

:

c

a, b,

c,

d,

we take the

If

:

a

+ b),

§§

ALTERNATIVES

40, 41]

37

b e and a e omitting the third and fourth factors There is one point because they are formal certainties. Since every double in this result which deserves notice. >)

implication a

:

x

:

always implies a

(3

(in the first bracket) ab

+ ac' + he

:

/3,

:

:

>;

it

follows that

:

:

Now, the

implies ab.

formally stronger than the former, since any statement x is formally stronger than the alternative latter

is

x + y. But the formally stronger statement x, though it can never be weaker, either formally or materially, than x + y, may be materially equivalent to x + y; and it must be so whenever y materially (i.e., by the special data of Let us see the problem) implies x, but not otherwise. whether our special data, in the present case, justifies the inferred implication ab tion (/3

:

and

By

\J/-.

x)(y be

:

we

x),

+ ac + be

Call this implica-

ab.

:

virtue of the formula a

+ (3 + y

get (putting ab for a and for

x

:

= (a

ac for

x,

:

x) (3,

for y)

\|z

= (ab al)){ac' = (ac a)(ac = e(ac' b)(bc'

:

:

:

:

ab)(bc b)(bc

:

:

a)e

:

:

ab)

a)(bc

= (ac

:

= e(ac :

:

ab)(bc

:

ab)

b)

b)(bc

:

a).

This asserts that (within the limits of our data in this

problem) whenever we have ac we have also b, and that whenever we have be we have also a. A glance at the given fully developed alternative

+ ac + be

:

ab

the fact that

is,

its

in this problem, legitimate, in spite of

antecedent

is

formally weaker than

its

consequent. 41.

An

alternative

and only when,

it

is

said to be fully developed when,

satisfies

the

conditions

following

Firstly, every single-letter constituent, or its denial,

must

be a factor of every term secondly, no term must be a formal certainty nor a formal impossibility thirdly, all the terms must be mutually incompatible, which means that no two terms can be true at the same time. This last condition implies that no term is redundant or repeated. ;

;

SYMBOLIC LOGIC

38

For example, the

developed form of a+ft is multiply the two

fully

+ aft' + aft. To obtain this we and strike factors a + a and ft + because it is equivalent to (a + As another given alternative a + aft

out the term

/3',

example, let it be developed form of a + ft'y.

ft.

Here we ft +

fully

find the product of the three factors a

first

7 + 7'.

and

ft',

the

find

to

aft',

the denial of the

ft)',

required

41, 42

[§§

equivalent to

a' (ft'y)',

We

next

which

is

that

find

+ a,

{a -{-ft'y)'

equivalent to

is

+ y'),

a'(ft

Then, out of the therefore, finally, to aft + ay'. eight terms forming the product we strike out the three terms a'fty, a'fty, a'/S^', because each of these contains

and

either aft or a'7',

which are the two terms

of aft

+ ay',

+ ft'y.

The

result

the denial of the given alternative a will be

aft'y

which

+ a'fty + a fty + a ft'y' + a fiy'i form of the given

therefore, the fully developed

is,

+ ft'y.

alternative a

42. Let

denote

(p

a'cclc

+ Veil + cd'e + a (Ye.

have 5 elementary constituents

a,

b,

d, e

c,

+ a), (b +

;

Here we so that the

&c, will contain 5 11 terms will terms, Of these 32 (or 32) terms. 2 the reof constitute the fully developed form
Then the

five factors (a

alternatives


and

\J/

will,

b'),

of course, only differ

Suppose the they will be logically equivalent. alternative \f/ to be given us (as in Jevons's " inverse problem "), and we are required to find the limits of the

in

form

5

constituents in the alphabetic order

;

the data \Jr

e

When we

\^.

:

to its simplest form,

a, b,

c,

d,

e,

from

have reduced the alternative shall find the result to be

we

(p.

Thus we get e:ylr

= e:

= This

is

(>7

a

= ; b d' + c)(d c e)(e d e)(r) :e:e). :

:

:

:

:

:

:

»/

:

the final result with every limit expressed.

Omit-

UNRESTRICTED FUNCTIONS

42-44]

§§

ting the superior limit

and the

>/

:

\Jr

= (a

:

&'c

+ ce')(b

:

wherever

inferior limit e

they occur, and also the final factor formal certainty (see § 18), we get e

39

ri'

>j

c

:

+ e)(d

:

because

e

:

c)(e

:

a

it is

rf).

Suppose next we arc required to find the limits in the order e

:

y$r

d,

e,

= (e = (e

d

:

:

&'c

+
:

b'c

+ ce)(e

d

:

Our

a. b.

c,

final result in this case will :

:

e

:

a'c

a'c

+ b'c){a

+ b'c)(a

:

:

c

e)(>7

:

:

a

be :b:e)

e)(>/

:

c).

When

an alternative

number

of possible

solution

in

the order

virtually the

same

d,

e,

c,

a,

(the

b

given),

last

the only difference being that the last first case are (as given), n a e and r\ :

:

:

is

a two factors in the while in the b e

as the solution in the order d,

e, c, b,

;

:

that is to say, a e second case they are tj:b:e and the order changes, and both, being certainties, may be It will be observed that when the order of omitted. >/

limits

is

:

:

prescribed, the exact solution

;

prescribed also

is

no two persons can (without error) give different solutions, though they may sometimes appear different in

form

(see

§§39,

40).

CHAPTER 44.

Let

~F u (x, y, z),

or

values or meanings of ;

,

l

)

y; z)

example,

its

while the symbol

Fr

synonym F( i

abbreviated

synonym F„, rey, z), when the

the functional proposition F(x,

present

stricted

its

VII

,

represents

when the values if

constituents

F r (x, the of x,

y, z),

x, y, z

or its abbreviated

functional y, z

are unre-

proposition

are restricted.

x can have only four values. xy

x,

2

x.

A

,

For x4 y ;

SYMBOLIC LOGIC

40

the four values y y2 z. then we write v s ,

z„ z

;

the three symbols

x,

yz

,

y

,

;

and

44, 45

[§§

the three values

z

F r and not FM But if each of y, z may have any value (or meaning) .

,

whatever out of the infinite series x v x2 x3 &c, y v y 2 y 3 &c., z «„, z &c. then we write F M and not F r The suffix v r is intended to suggest the adjective restricted, and the The symbols F F n F e suffix u the adjective unrestricted. ,

,

;

,

,

,

,

e

,

F

as usual, assert respectively that impossible, that is .<•,

F

z

y,

means

is

variable

mean

understood to

but here the word

;

admissible value of

every z);

y,

and

Thus F e

nor impossible.

asserts that Fix,

,

F

is

certain

;

impossible

in

y, z

the

neither certain

neither

y, z) is

synonymous with

is

;

formulae

x,

means

variable

always true nor always false it F _e F~", which is synonymous with

From

,

that

true fur all the admissible values of

in the functional statement F(x, y, z) false for

statement F(x,

45.

is certain,

(F^F"/.

these symbolic conventions

we get the three

:

(1)(F-F<); (2)(F?

(3)(F?:F? );

:F?.);

t

f

but the converse (or inverse) implications are not necessarily true, so that the three formulae would lose their validity if we substituted the sign of equivalence ( The first two formulae for the sign of implication (:). need no proof; the third is less evident, so we will prove

=

it

as

denote the above three two being self-evident, to be a certainty, so that we get the

Let

follows.

we assume

(f> 2

formulae respectively.

,

The

(p 3

first

deductive sorites e: 2 :(F-F;:)(F£:F?) :

:

:

(F;

e :

F-)(F7 1?) :

(F-F7 FfFJ) :

(F*: F*) [for

[for a

[for

:

/3

= /3'

(A a)(B :

A-'A^ = A e by ,

:

b)

:

:

«']

(AB

:

ah)]

definition].

This proves the third formula

two

,

.

,

§§

SYLLOGISTIC REASONING

45, 46]

represent the word horse, and

ment

"

The

F(H) denote the

let

Then F (H) l

has been caught."

horse

41

H H

state-

asserts

&c., has been r 2 the symbol F' (H) asserts that not one horse of and the symbol the series &c., has been caught r 2 e F*(H) denies both the statements F (H) and F"(H), and

every horse of the series

that

caught

H H

is

,

)

;

;

,

therefore equivalent to

F _e (H)

F" (H), which r,

.

may

be

6

expressed by F~ E^, the symbol (H) being left This &c. ? understood. But what is the series H^ 2 universe of horses may mean, for example, all the horses owned by the horse-dealer ; or it may mean a portion only of these horses, as, for example, all the horses that had

more

briefly

H

If

escaped.

by

we

write F*

{

we

assert that every horse

has been caught;

the horse-dealer

,

if

we

write

owned F*

we

only assert that every horse of his that escaped lias been Now, it is clear that the first statement implies caught.

the second, but that the second does not necessarily imply the first so that we have F' F*, but not necessarily F;:F;. The last implication F;:F; is not :

;

t

all the horses that necessarily imply not had escaped were caught would had been horse-dealer that all the horses owned by the and escaped, caught, since some of them may not have had of these it would not be correct to say that they

necessarily true

;

The symbol F M may

been caught.

V v F2 F3 F 60 i\, F F F 2 8 10 make evident the F* F*

:

:

,

,

.

.

.,

,

,

,

.

.

.,

.

while

may

F,.

refer

to

the series

refer only to the series

The same concrete illustration will truth of the implications F^:F? and

F* and also that the converse implications F? ,

Ff.

:

F? and t

are not necessarily true.

46. Let us called is

that

the fact

for

now examine

syllogistic.

my

a particular case of (a

or, as it

may

the special kind of reasoning will be shown,

Every valid syllogism, as general formula

(3)((3

:

:

y)

:

(a

:

y),

be more briefly expressed, (a

:

/3

:

y)

:

(a

:

y).

SYMBOLIC LOGIC

42

Let S denote our Symbolic

[§§ "

or

Universe,

46, 47

Universe

of

the things S v S 2 &c, real, or non-existent, expressly mentioned or

Discourse," consisting of unreal, existent,

all

,

our argument or discourse. Let denote any class of individuals X X 2 &c, forming a portion of the Symbolic Universe S then 'X (with a grave accent) denotes the class of individuals 'X 'Xg, &c, that do not belong to the class X so that the individuals tacitly understood, in

X

,

,

;

,

;

X

&c, of the class X, plus the individuals X 'X 2 2 &c, of the class X, always make up the total Symbolic Universe S S 2 &c. The class 'X is called the complement of the class X, and vice versa. Thus, any class A and its complement 'A make up together the whole Symbolic Universe S each forming a portion only, and both forming the whole. 47. Now, there are two mutually complementary classes which are so often spoken of in logic that it is convenient to designate them by special symbols these are the class of individuals which, in the given circumstances, have a real existence, and the class of individuals which, in the given circumstances, have not a real existXj,

X

,

.

,

X

,

,

;

;

The

ence.

individuals

class

first e

v

e„,

the class

is

To

&c.

made up

e,

of the

this class belongs every indi-

vidual of which, in the given circumstances, one can "

truly say

"

It exists

— that To

bolically but really.

town, triangle, virtue,

horse,

and

in the class

vice

exists "

or

"

Vice

e,

The second

exists "

class

We may

vice.

merely symmay belong place

because the statement really

persons, or vicious persons, exist

one would accept as

to say, not

is

this class therefore

;

asserts

that

"

virtue

Virtue

virtuous

a statement which every

true. is

the

class

0,

made up

of

the

individuals 0^ To this class belongs every in&c. 2 dividual of which, in the given circumstances, we can ,

truly say not exist exists

" It

does not exist

"

—

that

is

to say, " It does

though (like everything else named) it symbolically." To this class necessarily belong really,

REALITIES AND UNREALITIES

§§47-49]

48

mermaid, round square, fiat sphere. The Symbolic Universe (like any class A) may consist wholly of realil ies or wholly of unrealities Oj, e &c, or it may 2 v e 2 &c. centaur,

;

,

,

When

be a mixed universe containing both.

Av A 2

,

&c, of any

A

class

wholly of unrealities, the class class least

;

when A

the

members

consist wholly of realities, or

A

said to be a pure

is

contains at least one reality and also at

one unreality,

it

mixed

a

called

is

class.

Since

and are mutually complementary, it is clear that V is synonymous with 0, and with e. 48. In no case, however, in fixing the limits of the class e, must the context or given circumstances be overlooked. For example, when the symbol H|! is read " The horse caught does not exist," or " No horse has been caught" (see §§ 6, 47), the understood universe of realities, e v e 2 &c, may be a limited number of horses, H H 2 &c, that had escaped,, and in that case the statement Hj! merely asserts that to the classes

e

v

,

,

,

that limited universe the individual or a horse caught, does not belong;

H

it

c

,

the horse cauyht,

does not deny the

caught at some other time, Symmetry and conor in some other •circumstances. venience require that the admission of any class A into our symbolic universe must be always understood to imply the existence also in the same universe of the complementary class *A. Let A and B be any two classes that are not mutually complementary (see § 46) if A and B are mutually exclusive, their respective complements, A and 'B, overlap; and, conversely, if 'A and 'B are mutually exclusive, A and B overlap. 49. Every statement that enters into a syllogism of the traditional logic has one or other of the following four forms possibility of a horse being

;

V

X

(1) Every (3) It is

Some

evident that (3)

X is

is

is

Y

Y ;

;

(2)

(4)

No

Some

X is Y X is not

simply the denial of

;

Y.

(2),

and (4)

SYMBOLIC LOGIC

44

From

the denial of (1). get

the conventions of §§

(1)

X° Y = Every

(3)

XT = Xy = Some X is Y X! = X:° = Some X is not

X

[§§ 49,

is

Y

X°Y

(2)

;

= No X

G,

is

47,

50

we

Y

°

;

(4)

Y.

Y

The

first two are, in the traditional logic, called universals ; the last two are called particulars ; and the four are respectively denoted by the letters A, E, I, 0, for reasons

which need not be here explained, as they have now only

The following is, however, a simpler symmetrical way of expressing the above four more and of the traditional logic and it has propositions standard historical interest.

;

the further advantage,

how

appear

as will

of

later,

showing

all the syllogisms of the traditional logic are only

particular cases of

more general formulae

in the logic of

pure statements. 50. Let S be any individual taken at random out of our Symbolic Universe, or Universe of Discourse, and let respectively denote the three propositions

x, y, z

S

z

S~

Then

.

z

By

.

y',

x',

must

z'

x, y, z,

like their denials x'

certain

;

that

,

,

y', z

(x\

z\

e

f

tions (x

:

iff,

(y

:

>/)',

{y'y, (z)e

:

>/)'

,

Y ,

Hence, we

and never x

71

;

Hence, when

e

(z

S~

are all possible but un-

,

nor y nor z nor x nor y nor z\ respectively denote the propositions v

,

46, the three propositions

§

to say, all six are variables.

is

must always have xe y e v

,

,

denote S~ x

respectively

the conventions of

Sx SY

x, y, z

S x S Y S z the proposi,

,

,

(which are respectively synony-

must always be considered to form and their part of our data, whether expressed or not denials, (x »/), (y n), (« »?), must be considered impossible. With these conventions we get

mous with x*

1

,

y'1*, z"

)

;

:

:

:

X is Y = S x S Y = (x y) = {xy'f x S Y / = (x y)' = (xy'y (0) Some X is not Y = (S Y x S- = x y = (xyY (E) No X is Y = S x T S" )' = (x y')' = {xyj*. (1) Some X is Y = (S

(A) Every (or

all)

:

:

:

:

:

:

:

:

§

GENERAL AND TRADITIONAL LOGIC

50]

In

this

way we can

express

every syllogism

of

45 the

terms of x, y, z, which represent three propositions having the same subject S, but different predicates X, Y, Z. Since none of the propositions x, y, z (as already shown) can in this case belong to the class or e, the values (or meanings) of x, y, z are restricted. Hence, every traditional syllogism expressed in terms of x, y, z must belong to the class of restricted functional statements Fr (x, ?/, z), or its abbreviated synonym Fr) and not to the class of unrestricted functional statements traditional

logic

in

r\

FJx, y, z), or its abbreviated synonym F w as this last statement assumes that the values (or meanings) of the propositions x, y, z are wholly unrestricted (see § 44). ,

The proposition Fw

assumes not only that each

(x, y, z)

statement

may

belong to the class but also that the three statements x, y, z need not even have the same subject. For example, let F (x, y, z), or its abbreviation F, denote the formula constituent >/

or

e,

x,

(x

:

y)(y

then x implies z." be the statements

z)

:

(x

:

z).

x implies

y,

and y implies

The formula holds good whatever

z,

in

:

9,

" If

This formula asserts that

(as

z

y,

as well as to the class

x,

y,

z

;

whether or not they have same subject S and

the traditional logic) the

;

whether or not they are certainties, impossibilities, or variables. Hence, with reference to the above formula, 6 it is always correct to assert F whether F denotes F M When x, y, z have a common subject S, then or F r F e will mean F^. and will denote the syllogism of the traditional logic called Barbara ;* whereas when x, y, z are wholly unrestricted, F will mean F^ and will therefore be a more general formula, of which the traditional Barbara will be a particular case. .

e

*

Barbara asserts that " If every

X is Z,"

which

is

X

is

equivalent to (S x S v ) (S v :

Y, and every :

Sz)

:

(S x

:

S z ).

Y

is Z,

then every

SYMBOLIC LOGIC

46

But now

let F, or Y(x, y,

(y

z)(y

:

denote the implication

z),

x)

:

[§§50,51

(x

:

:

z')'.

suppose the propositions x, y, z to be limited by It' we the conventions of §§46, 50, the traditional syllogism called Darapti will be represented by F r and not by 6

formula of § 45, we have F,' F, e e e but not necessarily F~ F; and, consequently, F; F~ Thus, if F u be valid, the traditional Darapti must be We find that F w is not valid, for the above valid also. implication represented by F fails in the case f(xzy, as it

FM

Now, by the

.

first

:

.,

(

6

:

:

,

.

then becomes (>1

:

z){ri

x)

:

:

(xz)~ v ,

which is equivalent to ee if, and consequently to e But since (as just shown) F; which = {er/f = (ee) = 6 does not necessarily imply F; this discovery docs not justify :

:

»/,

6

7

rj.

'

,

us in concluding that the traditional Darapti

F

is

not valid.

y\xz)n and this case cannot occur in the limited formula Fr (which here represents the traditional Darapti), because in Fr the pro-

The only

case in which

fails

is

,

x, y, z are always variable and therefore possible. In the general and non-traditional implication F M the case x yv zr since it implies [piiczf, is also a case of failure; but it is not a case of failure in the traditional logic. 51. The traditional Darapti, namely, "If every Y is Z, and every Y is also X, then some X is Z," is thought by

positions

,

yi

',

some real

Y

is

non-existent, while the classes

But

but mutually exclusive.

Y = (0

1(

2

Let P denote the Q the second, and

P = Every

),

,

Y R = Some X

;i

Z = (e v

first

R is

is

e

2

,

X

and Z are

this is a mistake, as the

following concrete example will show.

and

when

logicians (I formerly thought so myself) to fail

the class

e

3 ),

Suppose we have

X = («

4>

e

a,

e

6 ).

premise of the given syllogism,

We Q = Every Y

the conclusion.

Z= h Z= 3 >

>/

;

;

get is

X=

three statements,

>/

>;

r

2

;

»/

2,

»/

3

,

TRADITIONAL SYLLOGISMS

§§51,52]

17

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

PQ R = :

that,

so

fail

52. Startling

demonstrable

—

/

:

2

>i,)

when presented

Darapti does not

logic

V

(

=

(>i,

in

:

*1

3

)

= {%n^ = e

form of an

the

;

1

implication,

(But see however, it

in the case supposed.

as

it

may

sound,

§

52.) is

a

fact that not one syllogism of the traditional

—

is neither Darapti, nor Barbara, nor any other which it is usually presented in our

valid in the form in

text-books, and in which, I believe,

it

has been always

In this form,

presented ever since the time of Aristotle.

every syllogism makes four positive assertions it asserts it asserts the it asserts the second the first premise :

;

;

conclusion

i.e.

;

and, by the

conclusion

the

follows

word

'

therefore,'

necessarily

from

it

asserts

the

that

premises,

that if the premises be true, the conclusion must be Of these four assertions the first three may be, also.

true

and often

are, false

the fourth, and the fourth alone, is Take the standard syllogism Barbara.

;

a formal certainty.

text-book form) says this B is C therefore every A is C." every Every A is If valid it this syllogism. denote Let \f/(A, B, C) meanings) we give to (or values must be true whatever

Barbara

(in the usual

B

"

;

;

A—

=

=

camel. bear, and let C ass, let B Let syllogism must following the If \J/(A, B, C) be valid, " bear every bear is is a a Every ass ; therefore be true Is this camel." concrete a camel; therefore, every ass is it contains three Clearly not syllogism really true ?

A, B, C.

:

;

Hence, in the above form, Barbara (here denoted by \|/) is not valid for have we not just adduced a case of failure ? And if we give random values to A, B, C out of a large number of classes taken false

statements.

;

haphazard

(lings, queens, sailors, doctors, stones, cities, horses,

French, Europeans, white things, black things, &c, &c), we shall find that the cases in which this syllogism will

SYMBOLIC LOGIC

48

53

[§§ 52,

turn out false enormously outnumber the cases in which it

But

will turn out true.

it is

always true in the following

form, whatever values we give to A, B, C " If every A is B, and every B is C, then every :

A

C."

is

Suppose as before that A = ass, that B = bear, and that C = camel. Let P denote the combined premises, " Every ass is a bear, and every bear is a camel," and let Q denote the conclusion, " Every ass is a camel." Also, let the symbol denote the word therefore. as is customary The first or therefore -form asserts P Q, which is .'.

,

,

.".

equivalent* to the two-factor statement P(P:Q); the second or if-form asserts only the second factor P Q. The therefore-form vouches for the truth of P and Q, which are both false the if-form vouches only for the :

;

truth

of

P Q, which, by definition, (See § 10.) a formal certainty.

implication

the

means (PQ'y. and 53. Logicians

is

may

:

say (as some have said), in answer

to the preceding criticism, that

my

objection to the usual

form of presenting a syllogism is purely verbal that the premises are always understood to be merely hypothetical, and that therefore the syllogism, in its general form, is not supposed to guarantee either the truth of the ;

premises or the truth of the conclusion. This is virtually an admission that though (P •'• Q) is asserted, the weaker

statement (P

:

Q)

is

P But why

logicians assert "

the one really meant therefore Q,"

—

that though

they only mean

"

If

P

commonIn ordinary speech, when sense linguistic convention ? we say " P is true, therefore Q is true," we vouch for the truth of P but when we say " If P is true, then Q is true," we do not. As I said in the Athenmum, No. 3989 then Q."

depart from the ordinary

;

:

"

Why

should the linguistic convention be different in logic ? ? Where is the advantage 1 Suppose a general, whose mind, during his past university days, had been over-imbued with the traditional logic, were in war time to say, in speaking of an

Where

is

.

.

.

the necessity

untried and possibly innocent prisoner, * I pointed out this equivalence in

'

He

is

a spy

;

therefore

Mind, January 1880.

he

§§ 53,

TRADITIONAL SYLLOGISMS

54]

49

must be shot,' and that this order were carried out to the letter. Could he afterwards exculpate himself by saying that it was all an unfortunate mistake, due to the deplorable ignorance of his subordinates that if these had, like him, received the inestimable advantages of a logical education, they would have known at once that what he really meant was If he is a spy, he must be shot'? The argument in defence of the traditional wording of the syllogism is exactly parallel." ;

'

It

is

no exaggeration

to

are due to neglect of the

say that nearly

hypotheses are accepted as

if

§

If.

Mere

they were certainties.

CHAPTER 54. In the notation of

all fallacies

conjunction,

little

VIII

50, the following are the nine-

teen syllogisms of the traditional logic, in their usual As is customary, they are arranged into four order. divisions, called Figures, according to the position of the

middle term " (or middle constituent), here denoted by y. This constituent y always appears in both pre"

The constituent

mises, but not in the conclusion.

the traditional phraseology,

is

z,

in

the " major term,"

called

Similarly, minor term." " major premise," and the premise containing x the " minor premise." Also, since the conclusion is always of the form " All

and the constituent x the the premise containing

X X

is

Z," or "

Some

is

not Z,"

it

is

X

z is

is

Z

"

called the

" or "

No X

usual to speak of

X

and of Z as the predicate.' As usual major premise precedes the minor.

Barbara

=(y

Celarent

= (y = (y = (y

Darii

Ferio

z)(x

as the

'

1

:y):(x:z)

z'){x

:

y)

(x

:

z)

:

1

z)(x z')(,

y')'

:

:

y')'

(x

:

:

Some

subject

in text-books, the

'

Figure

Z," or "

is

(x

z

:

:

)'

z)

f

D

SYMBOLIC LOGIC

50

Figure

[§

54

2

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

:

(:

:

:

:

:

:

Figure

= (y Disamis = (y = (y Datisi Felapton = (y Bokardo = y Ferison = (y

Darapti

(

z)(y

:

:

:

:

(a:

:

:

3 x)

:

,

:

:

z )\y

:

(x

:

x)

z)(y

:

z')(y

:

:

z)\y

:

x)

:

z'){y

:

x')'

a/)'

:

«)

(x

:

:

z')'

:

:

(x

:

{x

z'f

:

:

z)'

:

z)'

(x

:

z')'

:

(a;

:

z)'

:

Figure 4 Bramantip = (z y)(y :

Camenes Dismaris

Fesapo Fresison

= (z = {z = (z = (z

:

x)

:

x')

y)(y

:

y')\y

:

y')(y

:

x)

:

y')(y

:

x')'

:

x)

z

:

(x

:

:

1

(x

:

:

:

(x (x :

)'

z')

:

:

:

(x

z)' z)' :

z)'

the symbols (Barbara),,, (Celarent) M &c. denote, in conformity with the convention of § 44, these nineteen functional statements respectively, when the values of

Now,

let

,

their constituent statements

x. y, z

;

are unrestricted

;

while

the symbols (Barbara),., (Celarent),., &c, denote the same functional statements when the values of x, y, z are restricted The syllogisms (Barbara),., (Celarent),., &c, as in § 50. with the suffix r, indicating restriction of values, are the real

syllogisms

of

the traditional logic

;

and

all

these,

within the limits of the without exception, are valid The nineteen syllogisms of general understood restriction*. logic, that is to say, of the pure logic of statements,

GENERAL LOGIC

54-5 0]

§§

namely, (Barbara),,,

which

x, y, z

are

more general than and imply nineteen in which x, y, z are restricted as

in values, are

a n restricted

the traditional in § 5

(Celarent),,, &c., in

51

and four of these unrestricted syllogisms, namely, and (Fesapo),,, fail

;

(Darapti),,, (Felapton),,, (Bramantip),,,

certain

in

(Darapti) w

cases.

the

in

fails

7

case

y '(".:)\ /

and (Fesapo) w fail in the case y%ez ) and (Bramantip u fails in the case &(x'yf. 55. It thus appears that there are two Barbaras, two Celarents, two Dai'ii, &c, of which, in each case, the one

(Felapton),,

TI

,

)

belongs to the traditional logic, with restricted values its constituents x, y, z; while the other is a more

of

general syllogism, of which the traditional syllogism

Now,

particular case.

Fw

law

,

as

shown

in § 45,

when

is

a

a general

with unrestricted values of its constituents, implies F,., with restricted values of its constituents,

a general law

the former

if

may

is

true absolutely and never

be said of the

latter.

This

is

fails,

the same

expressed by the

formula F„ F*. But an exceptional case of failure in F„ does not necessarily imply a corresponding case of failure :

in

F,.

FM

e :

for

;

F;

e

though

(which

F r F ,) e

e

e

F,

is

a valid formula, the implication

F;. is

:

,

equivalent to

the converse implica-

For example, the general and non-traditional syllogism (Darapti),, implies the less general and traditional syllogism (Darapti),.. tion

:

The former

is

not necessarily valid.

but y\xzj in the traditional syllogism this case cannot occur because of the restrictions which limit the statement Hence, though this case of y to the class 6 (see § 50). fails

the exceptional

in

case

failure necessitates the conclusion (Darapti);;*,

from

this

conclusion,

conclusion

(Darapti);

infer

6 .

the

i

;

we

cannot,

but incorrect, reasoning applies to

further,

Similar

the unrestricted non-traditional and restricted traditional

forms of Felapton, Bramantip, and Fesapo. 56. All the preceding syllogisms, with many others not recognised in the traditional logic may. by means of the formulae of transposition a j3 = /3 r a! and a/3' \y' ay:f$, :

:

=

SYMBOLIC LOGIC

52

57

[§§ 56,

be shown to be only particular cases of the formula Two or which expresses Barbara.

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

examples

three

make

will

this

§

54,

Lut

clear.

<j)(x, y,

Referring to the

denote this standard formula.

z)

in

list

we get

Baroko = (z

=

(.«

y)(x

:

z){z

:

//)'

:

y)

:

:

(x

:

(x

z)'

:

which, by transposition,

;

y) =

:

(f)(x, z, //).

obtained from the general standard is formula
Thus Baroko

we

get

= (y z)(x z) (x if) x). z){z x) (y x') = (p(y, We get (see § 54) Next, take (Darapti) = (y:zx): (xz n)' z)(y x) (x (Darapti),. = = = xz)(xz n (y (y xz){xz = since, by the for, in the traditional logic, (y:rf) = (y = (y

Darii

:

z)(x

:

yj

:

:

(x

:

:

z)'

:

:

:

:

:

z,

:

r.

(//

(//

:

:

:

:

:

:

>j)

z')'

:

:

:

:

>/)

»/)

:

:

con-

»/,

S 5 0, y must always be a variable, and, thereThus, finally (Darapti),. = (f)(y, xz, n). always possible. We get Lastly, take (Bramantip),..

vention of lore,

(Bramantip),

= (z = (z = (z

:

y)(y x) :

y){z

:

yx')(yx'

:

:

and therefore

By

y

,

:

:

:

:

:

:

:

:

r,)

:

i]

>i

:

:

//)

r),

Hence,

=
yx\

finally,

we

;

a

get

>/).

similar reasoning the student can verify the list

(see §§

54-56):

= Barbara (p(x, y, = Celarent = Cesare y') = Ferio = Festino (p{x, x') = Darii = Datisi = Ferison = Fresison (p(z, y, x) = Camestres = Camenes

(p(x, y, z)

ip(y z

z")'

x)

possible.

(Bramantip),.

following

:

:

>i) :i]

the traditional logic,

for, in

variable

57.

= (z y)(y x ){x z) n = (z yx)(y x) = (z: yx')(yx (z (z:r]) = since z must be (x

:

x')(y

:

z')

;

:

z,

;

;

;

TESTS OF SYLLOGISTIC VALIDITY

§§57-59]
z)

x

v)

(p(y

}

,

(p(y, xz,
58. It

is

53

= Disamis = Dismaris
;

(y,

= (Felapton) = (Fesapo) n) = (Bramantip),.. evident (since x y = y' x) »/)

:

»/)

r

r

;

that

:

(f)(x, y, z)

:

so

that

=

these

in the preceding list cb(z,' y', x) syllogisms remain valid when we change the order ot their constituents, provided we, at the same time, change ;

all

For example, Camestres and Camenes may each be expressed, not only in the form cp(z, y, x'), as in the list, but also in the form (p(x, y, z). 59. Text-books on logic usually give rather complicated rules, or " canons," by which to test the validity These we shall discuss further of a supposed syllogism. on (see §§ 62, 63); meanwhile we will give the following rules, which are simpler, more general, more reliable, and their signs.

more

easily applicable.

Let an accented capital letter denote a non-implication (or " particular "), that is to say, the denial of an impliwhile a capital without an accent denotes a cation Thus, if A denote simple implication (or " universal "). :

Now, let A. B, C denote x y, then A' will denote (x y)' denote their any syllogistic implications, while A', B', Every valid syllogism must have respective denials. one or other of these three forms .

:

:

C

(1)

that

is

AB:C;

AB C r

(2)

to say, either the

:

;

(3)

AB

:

C

;

two premises and the conclusion

three implications (or " universals ") as in (1); or one premise only and the conclusion are both nonimplications (or "particulars") as in (2); or, as in (3), both premises are implications (or " universals "), while are

the

all

conclusion

is

a non-implication

(or

"

particular

").

If any supposed syllogism does not come under form (1) nor under form (2) nor under form (3), it is not valid that is to say, there will be cases in which it will fail.

;

SYMBOLIC LOGIC

54

The second form may be reduced

59

[§

the

to

first

form by

transposing the premise B' and the conclusion C, and is equivalent to AC B, changing their signs for AB' When thus transAB'C >?. to each being equivalent of AC B, may be is, that C, formed the validity of AB' The of AB C. validity the tested in the same way as in z, be x C to conclusion Suppose the test is easy. example, If, for negative. which z may be affirmative or :

;

C

:

:

:

:

:

:

— He

z

is

z—He

is

a soldier; then

AB

C,

:

not a soldier.

is

a

a soldier; then z' being, by hypothesis, x:z,

C

if valid,

(x

= He

— He

not

conclusion

z'

soldier.

But it The

the syllogism

(see § 11) either

becomes

:y:z):(x:

is

or else {x

z),

y'

:

:

z)

:

(x

:

z),

which the statement y refers to the middle class (or term ") Y, not mentioned in the conclusion x z. If any supposed syllogism AB C cannot be reduced to either if it can be reduced of these two forms, it is not valid a concrete example, take To valid. it is form, to either

in "

:

:

;

be required to test the validity of the following implicational syllogism let

it

:

If

no Liberal approves

of fiscal Retaliation, of fiscal Retaliation

it

of Protection,

do not approve of

Protection.

Speaking of a person taken a

Liberal;

R = He

let

P = He

approves of

the syllogism.

though some Liberals approve who approve

follows that some person or persons

We

at

approves

random,

let

L = He

of Protection;

fiscal Retaliation.

Also, let

is

and let Q denote

get

Q=(L:P')(L:R'/:(R:P)'. To get (see

§

rid of the non-implications,

56)

affirmative,

change

and thus

their

signs

we transpose them from negative

transforming them into

This transposition gives us

Q = (L:P

,

)(R:P):(L:R').

to

implications.

TESTS OF SYLLOGISTIC VALIDITY

§§59, 00]

55

Since in this form of Q, the syllogistic propositions are all three implications (or " universale "), the combination of premises, (L P')(R:P), must (if Q be valid) be equi:

valent

L P R'

either to

which P

in

:

the letter

is

L

or conclusion

:

:

:

L

or else to

:

P'

:

R'

new consequent L P and P R' premises L P' and

out in the

left

Now, the

R'.

L P R' are not R P in the second

of

:

factors

equivalent to the

:

:

:

or transposed form of the syllogism but the factors L P' and P' R' (which is equivalent to R P) of L P' R' are equivalent to the premises in the second or transformed form of the syllogism Q. :

Q

:

:

;

:

:

:

Hence Q is valid. As an instance of AB C, we may give

a non-valid syllogism of the form

:

(x:y')(y:z'):(x:z');

two premises have different signs, one being negative and the other affirmative, the combined premises can neither take the form x:y:z nor

for since the y's in the

the

the form x y' :

:

z'

,

which are respective abbreviations

(x>\y){y:z) and (x t y')(y' /). :

The syllogism

is

for

there-

fore not valid.

The preceding process

00.

testing the validity of

for

C

apply to all syllogisms of the forms AB C and AB' syllogisms without exception, whether the values of their :

constituents

x,

y,

z

ments.

But

AB

traditional

be restricted, as in the

or unrestricted, as in

logic,

:

my

general logic

of state-

as regards syllogisms in general logic of the

C

(a form which includes Darapti, Felapton, in the traditional logic), with Bramantip Fesapo, and and a non-implicational conpremises two implicational

form

:

clusion, they can only be true conditionally logic

(as

distinguished from the

syllogism of this type

is

;

for in general

traditional

a formal certainty.

logic)

no

It therefore

becomes an interesting and important problem

to deter-

SYMBOLIC LOGIC

56

mine the

on which syllogisms of this type can We have to determine two things, firstly,

conditions

be held valid. the

61

[§§ GO,

iveakest

premise

(see

when

which,

33, footnote)

§

joined to the two premises given, would render the syllogism a formal certainty ; and, secondly, the weakest condition which, when assumed throughout, would render

As will be seen, the the syllogism a formal impossibility. general one, which may method we are going to explain is a of the syllogism. be applied to other formulae besides those

AB

The given implication

ABC

implication

:

y,

in

:

C

equivalent to the

is

which A, B, C are three impli-

59) involving three constituents x, y, z. Eliminate successively x, y, z as in § 34, not as in finding the successive limits of x, y, z, but taking each cations (see

§

variable independently.

Let a denote the strongest con-

clusion deducible from ABC and containing no reference Similarly, let /3 and y respectively to the eliminated x.

denote the strongest conclusions after the elimination of y alone (x being left), and after the elimination of z alone Then, if we join the factor a or /3' (x and y being left). or y' to the premises (ix. the antecedent) of the given implicational syllogism AB C, the syllogism will become :

a formal certainty,

ABa'

:

and therefore

C will be a formal certainty

and AB?' C.

;

premise needed

to

AB

alternative a'

:

C

be joined to valid

+ fi' + y',

datum needed

to

(a

will

is

to say,

AB/3'

+fi'+ y)

C

:

C

:

is

a

so that, on the one hand, the weakest

formal certainty syllogism

and so

;

AB

Consequently,

:

That

valid.

{i.e.

AB

to render the given

a formal

certainty)

the

is

and, on the other, the weakest

make

+ /?' + y

an example 61. Take as Here we have an implication

the

AB

:

,

:

>;

C

:

that

syllogism

:

x),

= M* + N./ + P

(y

:

r,,

a formal a(3y.

is,

C in which

respectively denote the implications (y By the method of § 34 we get

ABC = yx + yz' + xz

AB

the syllogism

impossibility is the denied of a

:

Darapti.

A, B, z),

say,

(x

:

C z).

CONDITIONS OF VALIDITY

§61]

57

which M, N, P respectively denote the co-factor of x, The %', and the term not containing x. in which strongest consequent not involving x is MN + P hero M = z, N = y, and P = yz' so that we have in

the co-factor of

*),

:

;

MN + P

:

= zy + yz' = ye = y n

>/

:

Thus we get a = y: we eliminate x is (y

:

= //( + z') -

1

:

v\.

so that the premise required

>/,

(

n

:

>;/

:

when

and therefore

;

r.x)(y.z)(y.ri)

f

-(x:z

should be a formal certainty, which rid of the non-implications

by

,

t

)

a fact

is

;

getting

for,

complex

transposition, this

implication becomes (y

x)(y

:

which

and

= (y

z){x

:

:

:

z)

xz)(xz

:

:

(y

17),

:

(y

n)

n)

;

this is a formal certainty, being a particular case of

the standard formula

(f)(x, y, z),

which represents Barbara

both in general and in the traditional logic (see § 55). Eliminating y alone in the same manner from AB C, = x z' so that the complex we find that (3 = xz :

:

:

*i

;

implication

{y:x)(y:z)(x:zy:(x:z')'

That it is so is evident by should be a formal certainty. inspection, on the principle that the implication PQ Q, Finally, for all values of P and Q, is a formal certainty. we eliminate z, and find that y = y: n- This is the same :

we obtained by the elimination of x, as might have been foreseen, since x and z are evidently inter-

result as

changeable.

Thus we obtain the information sought, namely, that «

/

/

+ /3 + 7

/

premise

the weakest

,

premises of Darapti to

make

certainty in general logic /

(y

:

>/)

+ (xz

:

>/)'

+ (//

the formal

be joined

to

this

syllogism

to

a

is

:

•?)',

which

= y*> + (xz)-

1

" ;

SYMBOLIC LOGIC

58

[§§ 61,

62

and that a/3y, the Aveakest presupposed condition that would render the syllogism Darapti a logical impossibility,

therefore

is

'

+

,p

/

(,,.,)--;

j

t

w hich = y\ocz)\

Hence, the Darapti of general values of

constituents

its

x, y,

with

logic,

unrestricted

in the case

fails

z,

y\xzy

;

but in the traditional logic, as shown in § 50, this case The preceding reasoning may be applied cannot arise. to the syllogisms Felapton and Fesapo by simply changing

z into z!

Here we get

Next, take the syllogism Bramantip.

ABC = yx' + zy' + xz and giving

u,

:

>i,

y the same meanings

/3,

we

before,

as

= z\ y = (x'y)\ Hence, a^y — z\xyf, and Thus, in general logic, Braa' + ft' + y' = z~ + (£c'y)~ a

get

=z

r

/3

>,

r

n

'.

mantip is a formal certainty when we assume z~ v + {x'yY*, and a formal impossibility when we assume &{x'yf but ;

assumption

in the traditional logic the latter sible,

z v is

since

inadmissible by

obligatory, since

inadmis-

50, while the former

§

is

implied in the necessary assump-

is

it

is

tion 2f.

The

62.

validity

traditional

the

of

tests

logic

turn

mainly upon the question whether or not a syllogistic In undistributed.' or distributed term or class is to ever, lead rarely, if words these language ordinary logicians thought but of confusion or any ambiguity have somehow managed to work them into a perplexing '

'

'

'

'

;

tangle.

In the proposition

said to be

'

distributed,'

class

Y

position said

to

position '

Some

be "

X

All

X

is

X

'

undistributed,'

X

X

Y," the class

is

and

Y

the class

Y

'

is

X

and the

X

and the

In the proclass

Finally, in

not Y," the class

X

undistributed.'

'

distributed.'

both 'undistributed.'

Some

Y," the class

is

class

Y," the class

is

be both

are said to "

No

"

In the proposition

"

and the

is

distributed.'

Y

are

the pro-

said to be

§ 6

2]

<

— UNDISTRIBUTED

DISTRIBUTED

,

59

<

Let us examine the consequences of this tangle of Take the leading syllogism Barbara, the technicalities. validity of which no one will question, provided it bo expressed in

conditional form, namely, "

its

If

Y

all

is

Z,

Y, then all X is Z." admittedly valid, this syllogism must hold good whatever values (or meanings) we give to its conIt must therefore hold good when stituents X, Y, Z. X, Y, and Z are synonyms, and, therefore, all denote the In this case also the two premises and the same class.

and

X

all

(see §

Being, in this form

is

52),

three truisms which no one would Consider now one of these truisms,

conclusion will be

dream

of denying.

X is Y." Here, by the usual logical convention, X is said to be distributed,' and the class Y But when X and Y are synonyms they undistributed.'

say

"

All

the class 1

'

denote the same class, so that the same class may, at the same time and in the same proposition, be both disDoes not this sound like tributed' and 'undistributed.' '

a contradiction

Speaking of a certain concrete

?

collec-

tion of apples in a certain concrete basket, can we consistently and in the same breath assert that " All the

apples are already distributed are

'still

undistributed "

"

and that

Do we

?

"

All the apples

get out of the

dilemma

and secure consistency if on every apple in the basket we Can we then constick a ticket X and also a ticket Y ? sistently assert that all the

that

all

every apple.

X

the

Y

apple

X

apples are distributed, but Clearly not for ?

apples are undistributed is

Y

also a

apple,

Y

apple an

X

In ordinary language the classes which we can

and

respectively qualify

as

mutually exclusive

in the logic of

is

;

and every

;

evidently not the

distributed

undistributed

are

our text-books this Students of the traditional

case.

minds of the idea necesundistributed and that the words distributed do in they as exclusive, mutually sarily refer to classes forced but a anything is there everyday speech or that and fanciful connexion between the distributed and

logic

should

therefore disabuse their '

'

'

'

;

'

'

SYMBOLIC LOGIC

60 '

undistributed

'

distributed

'

current English and the undisturbed of logicians.

of

and

'

[§

technical

'

'

Now, how came the words tributed to be employed by '

'

distributed

and

'

'

logicians in a sense

plainly does not coincide with that usually given "

Since the statement

statement "All

X

is

No X

"Y," in

is

Y"

which

them

?

46-50) the

(see §§

Y (or non-Y) contains all Symbolic Universe excluded from the

undis-

equivalent to the

is

which

the individuals of the

class

"

02

Some

X

is

not

definitions of

Y

" is

equivalent

distributed

'

'

and

'

to "

and since

class Y,

Some

X

undistributed

is '

*Y," the

in text-

books virtually amount to this that a class X is distributed with regard to a class Y (or *Y) when every individual of the former is synonymous or identical with :

some individual

or other of the latter

;

and that when

then the class X is undistributed with Hence, when in the stateregard to the class Y (or'Y). ment " All X is Y " we are told that X is distributed with regard to Y, but that Y is undistribided with regard to X, this ought to imply that X and Y cannot denote exactly this is not the case,

In other words, the proposition that to imply that " Some Y is not X." But as no logician would accept this implication, it is distributed clear that the technical use of the words

the "

same

X

All

is

class.

Y"

ought

'

and

'

undistributed

lacking

'

to

linguistic

in

be found in logical treatises is In answer to this

consistency.

criticism, logicians introduce psychological considerations

and say that the proposition " All X is Y " gives us information about every individual, X 1; X 2 &c, of the class X, but not about every individual, Y v Y 2 &c, of the class Y and that this is the reason why the term X is said to be To this 'distributed' and the term Y 'undistributed.' ,

,

explanation it may be objected, firstly, that formal logic that its forshould not be mixed up with psychology mulae are independent of the varying mental attitude of individuals and, secondly, that if we accept this information-giving or non-giving definition, then we should

—

'

;

'

'

'

'DISTRIBUTED'— UNDISTRIBUTED

§62]

X

say, not that

X

that

distributed,

is

known or

is

known

not

1

fil

<

Y

and

undistributed, but

Y

inferred to be distributed, while

—

to be distributed

is

that the inference requires

further data.

To throw symbolic light upon the question we may With the conventions of 8 50 we

proceed as follows.

have (1) All

Some X

(3)

The

Y = (x

:

'

'

No X is Y = x // Some X is not Y = (x

(2) (4)

//)';

positive class (or

logicians

the

X is Y = x:y;

is

term

')

:

X

is

predicate.'

It

//)'.

usually spoken of by

the subject'; and the positive class

as

:

Y

as

be noticed that, in the above

will

examples, the non-implications in (3) and (4) are the respective denials of the implications in (2) and (1). The definitions of

'

distributed

and

'

'

undistributed

are

'

as

follows.

The

term ') referred to by the antean implication is, in text-book language, said to distributed and the class referred to by the conse-

(a)

(or

class

'

cedent of

be

'

'

;

quent

is

(/$)

said to be

The

'

undistributed.'

class referred

implication

is

to

said to be

'

by the

antecedent of a non-

undistributed

referred to by the consequent

is

and the

;

said to be

'

class

distributed.'

to (1) and (2); definition and (4). Let the symbol X d assert that X is distributed' and let X u assert that X is undistributed.' The class 'X being the complement of the class X, and vice versa (see 8 46), we get (*X)* = XM and (X)" = X d From the definitions (a) and (/3), since (Y) d = Y", and ( Y) u = Y d we therefore draw the following

Definition

(/3)

applies

applies

(a)

to

(3)

'

'

,

.

y

,

four conclusions

In

XY u

X d Yu

(1)

d .

For

in

:

in

;

Xd Y d

(2)

;

in (3)

XUY U

(2) the definition (a) gives us Similarly, in (3) the definition

:

in

(4)

Xd Yf r

(

and CY) u = Y d (/3) gives us X u CY) d and ( Y)d = YM If we change y into x in proposition (1) above, we .

,

,

.

SYMBOLIC LOGIC

62

[§§ 62,

63

X is X "=x:x. Here, by definition (a), we have which shows that there is no necessary antagonism between X and X" that, in the text-book sense, the same class may be both distributed and undistributed at the same time. get " All

X dX"

;

rf

;

'

'

'

'

63. The six canons of syllogistic validity, as usually given in text-books, are (1) Every syllogism has three and only three terms, namely, the major term,' the minor term,' and the :

'

'

middle term (see § 5 4). (2) Every syllogism consists of three and only three propositions, namely, the major premise,' the minor premise,' and the 'conclusion' (see § 54). (3) The middle term must be distributed at least once in the premises and it must not be ambiguous. (4) No term must be distributed in the conclusion, unless it is also distributed in one of the premises.* (5) We can infer nothing from two negative pre'

'

'

'

;

mises. (6) If one premise be negative, the conclusion must be so also and, vice versa, a negative conclusion requires one negative premise. Let us examine these traditional canons. Suppose The syllogism \//('', y, z) to denote any valid syllogism. being valid, it must hold good whatever be the classes to which the statements x, y, z refer. It is therefore valid when we change y into x, and also z into x that is to ;

;

say,

\|/(.'",

a case

,/',

:>-,)

valid

is

(§

13,

Yet this is and needsimply a definition, and

footnote).

which Canon (1) appears

arbitrarily

Canon (2) is comment. The second part of Canon (3) all arguments alike, whether syllogistic or not.

lessly to exclude.

requires no applies to

*

Violation of

Canon

(4) is called

"Illicit Process."

is

called " Illicit Process of the Major "

tributed in the conclusion

Process of the Minor " (see

is

;

the term

the major term, the fallacy

when

the term illegitimately dis-

the minor term, the fallacy

§ 54).

When

is

illegitimately distributed in the conclusion

is

called " Illicit

'CANONS

§63] It

is

evident that

1

if

OF TRADITIONAL LOGIC we want

ambiguities.

63

we must Canon (3)

avoid fallacies,

to

The

part of

also avoid The rule about cannot be accepted without reservation. distribution does not apply middle-term the necessity of " If every X is syllogism, perfectly valid to the following that is not X something then Y, and every Z is also Y, expressed may be syllogism Symbolically., this is not Z." first

in either of the two forms

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

(1)

{xy'nzyj'.ix'z'r

(2)

Conservative logicians who still cling to the old logic it impossible to contest the validity of this syllogism, refuse to recognise it as a syllogism at all, on the ;

finding

ground that

has four (instead of the regulation three) the last being the class containing all the individuals excluded from the class X. Yet a mere change of the three constituents, x, y, z, of the syllogism Darapti (which they count as valid) into their denials x', //, z' makes Darapti equivalent to the it

terms, namely, X, Y, Z,

above syllogism.

%

For Darapti

is

{y:x\y:z):{x:zy

(3);

_

and by virtue of the formula a (l) in question becomes

:

(3

= /3'

a, the syllogism

:

(/:*')(/ :*'):(*':*)' Thus,

if

\^(f;, y, z)

denote

(4).

Darapti,

then

y\s(x', //', ;')

denote the contested syllogism (1) in its form (4); and, vice versa, if ^(x, y, z) denote the contested syllogism, namely, (1) or (4), then ^(a/, y z') will denote will

',

To

Darapti. class

X

is

that

class 'X.

class,

be read,

Hence,

if

we

is

it is

call

not

in the

in the

com-

the class 'X the

the syllogism in question, namely,

(/:./)(/:/)

may

any individual

equivalent to asserting that

plementary

non-X

assert

"

:(,/:*)'

If every

non-Y

(4), is

a non-X, and every non-

SYMBOLIC LOGIC

64

[§

03

For then some non-X is a non-Z." , z )' which asserts that it is possible for an individual to belong at the same time In both to the class non-X and to the class non-Z. Thus other words, it asserts that some non-X is non-Z.

Y

also a non-Z,

is

(x':z)'

is

r>

equivalent to (./

,

becomes a case of Darapti,

read, the contested syllogism

Z

being replaced by their respective It is evident that complementary classes 'X, 'Y, 'Z. when we change any constituent x into x in any syllothe classes X, Y,

gism, the words

change

'

distributed

and

'

'

undistributed

inter-

'

places.

Canon

(4)

of the traditional logic asserts that "

No

term' must be distributed in the conclusion, unless it is This is another also distributed in one of the premises."

Take the

canon that cannot be accepted unreservedly. syllogism Bramantip, namely, (z

and denote the

within

by

it

:

y)(y x) :

:

z')'

Since the syllogism

\f/(V).

restrictions

(x

:

of

the

traditional

is

logic

valid (see

should be valid when we change z into /, and We should then get consequently z into z. § 50),

it

>},{/)

Here

(see § 02)

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

we get Z w

in the first premise,

and Z

rf

the conclusion, which is a flat contradiction to the Upholders of the traditional logic, unable to deny the validity of this syllogism, seek to bring it

in

canon.

within the application of Bramantip by having recourse to distortion of language, thus " If every non-Z is Y, and every Y is X, then some X :

is

non-Z."

Z" in d premise and Z in the conclusion, which would contradict the canon, would have ( Z)'' in the first premise and ( Z) u in the conclusion, which, though it means exactly the same thing, serves to "save the face" of the canon

Thus

the

treated, the syllogism, instead of having

first

V

y

and

to hide its real failure

and

inutility.

§

TESTS OF SYLLOGISTIC VALIDITY

G3]

Canon

(5) asserts that "

A

two negative premises."

Avhich into

The example

is

:0(^*') :(*':*)',

obtained from Darapti by simply changing

is

and x into x

z',

can infer nothing from show the

single instance will

unreliability of the canon. (2,

We

65

It

.

may

"

be read,

If

Y

no

is

z

X,

and no Y is Z, then something that is not X is not Z." Of course, logicians may " save the face " of this canon " If also by throwing it into the Daraptic form, thus all Y is non-X, and all Y is also non-Z, then some non-X is non-Z." But in this way we might rid logic of all negatives, and the canon about negative premises would then have no raison d'etre. Lastly, comes Canon (6), which asserts, firstly, that " if one premise be negative, the conclusion must be :

negative

and,

;

secondly,

that

requires one negative premise."

negative

a

The

conclusion

objections to the

preceding canons apply to this canon also. In order to give an appearance of validity to these venerable syllogistic tests, logicians are obliged to have recourse to distortion of language, and by this device they manage to

make

their negatives look like affirmatives.

But when

logic has thus converted all real negatives into

affirmatives the canons about negatives

through refer.

want of negative matter to which they can The following three simple formulae are more

easily

applicable and will supersede

canons

:

(1) (a: (2)

(z

:

first

y x)

:

(x

:

the

traditional

Barbara.

Bramantip.

z)'

....

Darapti.

of these is valid both in general logic

the traditional logic

;

and

in

the second and third are only valid

in the traditional logic. all

all

:z):(x:z) :

(3) (y:x)(y:z):(x:z')'

The

seeming

must disappear

Apart from

this limitation, they

three hold good whether any constituent be affirmaE

SYMBOLIC LOGIC

66 tive

or negative,

and

64

[§§ 03,

whatever order we take the

in

Any

syllogism that cannot, directly or by the /3' a and a/3' y' ay fi, formulae of transposition, a /3 letters.

=

:

=

:

:

be brought to one or other of these forms

:

is invalid.

CHAPTER IX Given one Premise and the Conclusion, to find the missing Complementary Premise.* 64. When in a valid syllogism we are given one premise and the conclusion, we can always find the complementary premise which, with the one imply the conclusion. AVhen the given conclusion is an implication (or " universal ") such as x z or x z\ the complementary premise required is found For example, suppose we readily by mere inspection. f have the conclusion x:z and the given major premise The syllogism required must be z y (see § 5 4).

weakest

given,

will

:

:

:

either {x:y :z'): (x

:

z')

or (x y :

r :

z')

:

(x

:

z'),

The major prethe middle term being either y or y'. is which is not equivalent mise of the first syllogism y z' ',

:

Hence, the first syllomajor premise z y. The major premise of the gism is not the one wanted. y' z', and this, by transposition and second syllogism is change of signs, is equivalent to z y, which is the given major premise. Hence, the second syllogism is the one wanted, and the required minor premise is x y' to the given

:

:

:

:

When

the conclusion, but not the given premise, is a non-implication (or " particular "), we proceed as follows. Let P be the given implicational (or " universal ") premise, and

C the given non-implicational (or "particular")

conclusion. *

A

Let

W be the required weakest premise which,

syllogism with one premise thus left understood

enthymeme.

is

called an

§§ G4,

TO FIND A MISSING PREMISE

05]

joined to P, will imply

We

C.

have

shall then

which, by transposition, becomes

PC W. :

67

PW

C,

:

Let S be the We shall then

strongest conclusion dcducible from PC. have both PC S and PC W'. These two implications having the same antecedent PC, we suppose their consequents S and W' to be equivalent. We thus get S = = S'. The weakest 'premise required W', and therefore :

:

W

therefore

is

PC

from

denial of the strongest conclusion dedueible

the

and

{the given premise

the

of the given

denial

conclusion).

For example,

the given premise be y

let

given conclusion (x

We

r

z )'

:

.

:

x,

and the

are to have

(y:x)W:(x:z'y. Transposing and changing signs, this becomes \{y:x){x:z')'.W. But, by our

fundamental

syllogistic

formula,

we have

also (see § 5G)

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

We

therefore assume f

(y (y

:

:

z

f

)

W=

y:z' and, consequently, )

The weakest premise required

.

//, and the required syllogism (//

:

%)(y

*')'

(«

The only formulae needed complementary premise are 65.

The

*

is

W=

therefore

is

«')' :

in finding the weakest

= (3':a'.

(1)

a:(3

(2)

(a:/3)(/3:

7 ):(a: 7 ).

(3) (a:/3)(a:

7 ):(/3 7 r\

two are true universally, whatever be the statethe third is true on the condition a*, (3, y that a is possible a condition which exists in the first

ments

a,

;

—

* The implication y «, since would also answer as a premise footnote, and § 73).

in the traditional logic

:

;

but

it

it

implies (y

would not be the weakest

:

s')',

(see § 33,

SYMBOLIC LOGIC

68

[§§ 65,

any of the statements

traditional logic, as here

66

a, (3,

y

represent any of the three statements x, y, z, or any every one of which six stateof their denials x y', z ments is possible, since they respectively refer to the six

may

,

,

%Y

Z, every one of which classes X, Y, Z, stood to exist in our Universe of Discourse.

is

under-

Suppose we have the major premise z:y with the z')' and that we want to find the weakest complementary minor premise W. We are to have

conclusion (x

:

',

(z:y)W:(x:z'y, which, by transposition and change of signs, becomes

(z:y)(x:z'):W. This,

by the formula a

:

/3

= ft'

:

a

,

becomes

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

But by Formula

(3)

we have

also

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

We therefore assume W' = (yz')' and consequently W = (yx'y = y:x. The weakest minor premise required 71

,

is

therefore y x :

and the required syllogism

;

:

y)(V

.')

-')'-

('• :

is

:

As the weakest which is the syllogism Bramantip. premise required turns out in this case to be an implication, and not a non-implication, it is not only the weakest complementary premise required, but no other complementary premise is possible. (See § 64, second footnote.) 66. When the conclusion and given premise are both non-implications (or " particulars "), we proceed as follows. Let P' be the given non-implicational premise, and

C

W

denotes the the non-implicational conclusion, while shall required weakest complementary premise.

We

C

or then have P'W transposition. obtain by :

its

equivalent

WC

The consequent P

:

P,

which we

of the second

§§66, 66

THE STRONGEST CONCLUSION

(a)]

69

being an implication (or " universal ") we have only to proceed as in § 64 to find W. For example, let the given non-implioational premise be (// z)'\ and implication

:

the given non-implicational conclusion {x

:

z)'.

:

z

We

are

have

to

(yri/W :(*:«)'. By

becomes

transposition this

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

The

missing in the consequent y P must therefore be

letter

syllogism

WC

is

The

x.

:

either (y x z) :

:

(y

:

z)

:

or else (y:x':z):(y:z);

one or other of which must contain the implication C, which the given non-implicational conclusion C, re-

of

presenting (x

:

and not the second contains

that

W=y

Hence

to

position,

for it is the first

;

:

WC

Now,

x.

:

:

P

of these two syllogisms,

first

the implication

WC

The syllogism

the denial.

is

z)',

must therefore denote the

P

and not the second

or

C, is

its

synonym x

:

z.

equivalent, b}r trans-

WP' C, which is the syllogism required. W, P', C, we find the syllogism sought :

Substituting for to be

(//

:

*)'

'<)(>/

(?

:

*)',

and the required missing minor premise to be y x. 66 (a). By a similar process we find the strongest conclusion derivable from two given premises. One Suppose we have the combination example will suffice. Let S denote the strongest of premises (z y)(x y)' :

:

:

conclusion required. (z

:

y){x

The

:

//)'

letter

:

S,

'.

We

get /

which, by transposition,

is

(z

:

//)S

:

(x

:

y).

missing in the implicational consequent of the

second syllogism must be

is

z,

so

that

either x z y or else x :

:

antecedent

its

:

z'

:

>/.

(z

:

y)S

/

SYMBOLIC LOGIC

70 first

so that its other factor x

y,

by

antecedent

is

:

Hence, we get S'=x:z, and S

S'.

G7

(a),

the one that contains the factor z must be the one denoted

The z

:

6G

[§§

strongest * conclusion required

= (#:«)'.

therefore (x

is

The

z)''.

:

CHAPTER X

We

will now introduce three new symbols, Wcp, which we define as follows. Let A v A 2 A 3 A m be m statements which are all possible, but of which Out of these m statements let it be one only is true. A r imply (each sepaunderstood that A r A 2 A 3 A s imply that A r+1 Ar+2 A.,. +3 rately) a conclusion cp cp' and that the remaining statements, A s+1 As+2 A m neither imply cp nor cp'. On this understanding we 6 7.

Yep, Sep,

,

,

.

,

,

,

.

the following definitions

(5) (6)

W'cp means

W(/)

.

2

1

.

.

.

:

=A +A +A + +A W^) = Ar+1 + Ar+2 + ... +A V4> = V<£' = A s+1 + Ag+2 + ... +A m S^ = W^ + V^ = W
(1)

.

,

,

down

.

.

.

;

lay

.

.

.

;

.

,

,

.

.

r

3

,

(2) (3)

S

.

.

,

(4)

(7)

.

means

S'

(W(f>)',

The symbol Wcp denotes the cp

;

while

Sep

than

A+

A + B-f-C,

denotes

weakest statement that implies

the

33, footnote). B, while A + B

implies (see

the denial of W.

(S<£)', the denial of Sep.

strongest

As

§

and

so

on,

we

is

is

statement

formally

that (p implies.

cp,

and

that


stronger formally

stronger

are justified in

the weakest statement that implies strongest statement

A

calling

than

Wcp

in calling S(p the

Generally

Wcp and

Sep

* Since here the strongest conclusion is a non- implication, there is no other and weaker conclusion. An implicationcU conclusion x z would also admit of the weaker conclusion (x z')'. :

:

EXPLANATIONS OF SYMBOLS

68]

§§ 67,

present themselves as logical sums or alternatives

may

exceptional cases, they

From

terms. formulae, r

(\\ (^

= S<£ =

the

(£).

preceding

The

but, in

present themselves as single

W^S'0;

(1)

;

71

we

definitions

= W'<£;

S4>'

(2)

the

get

(3)

V°<£

=

last of these three formulas asserts

that to deny the existence of Y(p in our arbitrary uni-

A A2

verse of admissible statements,

and

to affirming that W<^>, Sep,

,

(p

are

&c,

,

all

is

equivalent

three equivalent,

The statement Y° the former asserts that Y