A primer of logic

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UC-NRLF.

PRIMER o/ LOGIC

SMITH

GIFT OF

Primer of Logic

BY

HENRY BRADFORD SMITH

Assistant Professor of Philosophy in the
University of Pennsylvania

B. D. SMITH A BROS.

PULASKI, VA.

1917

PREFACE

In the pages which follow will be found the outlines
of the logic of a set of categorical forms which are not
Aristotle's own. It is only the fragment of a general
theory, but the content of all the chapters of the old logic,
which are commonly regarded as essential, except that
one which deals with the calculus of classes, will be found
included. In the first appendix the relation of these new
forms to the traditional ones has been pointed out in detail.

I have at least one debt, which calls for a definite
acknowledgement. It is to my friend and teacher, Mr.
Edgar A. Singer, Jr., that I owe what training I have had
in the science. He has never failed, through hints thrown
out in conversation, to correct my misapprehensions.
But my indebtedness is more specific than this. I have so
far employed his own method, a method developed in his
academic lectures, that I could scarcely have ventured
upon the publication of these outlines of a theory without
his express permission.

H. B. S.

TABLE OF CONTENTS

CHAPTER I PAGES

§1-3. The fundamental properties of the categorical

forms. Exercises 1-9

CHAPTER II

§4-8. The relationships of "better" and "worse."
Deduction of the moods of immediate infer-
ence and of the syllogism by rule. Exercises. . . 10-17

CHAPTER III

§9-14. The relationships of "better" and "worse"
defined. Symbolic deduction of the moods
of immediate inference and of the syllogism.
Exercises 18-27

CHAPTER IV
§15. The general solution of the sorites. Exercises.. .28-34

APPENDIX I

On the simplification of categorical expression

and the reduction of the syllogistic figures 35-44

APPENDIX II

Historical note on De Morgan's new preposi-
tional forms . . . . 45-48

CHAPTER I

§1. In 1846 Sir William Hamilton published the
prospectus of an essay on a "New Analytic of Logical
Forms,"* which revived the question as to whether or
not the quantity of the predicate of the categorical forms
shquld be stated explicitly. The chief difficulties of his
system result from the ambiguity of the meaning of some,
from the impossibility of making every form of categorical
expression simply convertible, and from the seemingly
curious effort to establish an order of better and worse
between the relations connecting subject and predicate.
Four of Hamilton's eight forms are redundant. The four
that remain will be represented here by the letters, a,

ft 7, e-

Accordingly let

aab = all a is all b,
/3ab = some a is some b,
7ab = all a is some b,
eab=no a is b.

Here some, the some expressed explicitly in ft and 7,
means some at least, not all. This meaning of the word
is established unambiguously by the properties of the
forms. In addition to these abbreviations we will employ
the notation:

Xab =xab (is true),
x'ab=xab (is false),

xab ' y&b =Xab (is true) and yab (is true),
Xab + y&b =xab (is true) or yab (is true),
xab Z yab =xab (is true) implies yab (is true),
(xab Z. yab)' =xab (is true) does not imply yab (is true).

*Lectures on Logic, ed. by Mansel and Veitch, Boston,
Gould and Lincoln, 1863.

2 A PRIMER OF LOGIC

§2. The implications, which are given below, express
the chief characteristics of the forms. The theorems
follow by the principle of the denial of the consequent,
which may be written in the two forms:

(x Z y') Z (y Z x') and (x' Z y)' Z (y' Z x)'.

Postulates:*

ttab Z 0'ab 0ab Z 7'ab (o'ab ^ £ab)' (0'ab / Tab)'

dab Z T'ab /?ab Z e'ab (d'ab Z Tab)' (jS'ab ^ Cab)'

«ab Z e'ab 7ab Z e'ab (tt'ab Z €ab)' (7'ab ^ Cab)'

7ab Z 7'ba (7'ab Z 7ba)'

Theorems:

€abZa'ab 7abZd'ab (c'abZttab)' (7'ab Z aab)'

€ab Z jS'ab 7ab Z jS'ab ( «'ab / jSabX (7'ab Z jSab)'

Cab Z 7'ab 0ab Z a'ab (e'ab ^ Tab)' (0'ab Z ttab)'

Let us postulate in addition that:

aab Z (a'ab)' (o'ab)' ^ aab

/3ab Z (/3'ab)' O'l*)' ^ £ab

7ab Z (T'ab)' (7'.b)' ^ Tab

6abZ (e'ab)' (c'ab)'Z eab

Then, if k^rjiWafo and if kab and wab represent only
the unprimed letters, aab, /3ab, 7ab, eab, a complete induction
of the propositions given above yields the general result:

kabZ (k'ab)', (k'ab)'Zkab, I

kab Z W'ab, (W'ab Z kab)', II

*These assumptions are in accord with those of the common
logic, but no longer hold when the terms are allowed to take on
the limiting values 0 and 1 ; for 7OI and e0i are both true prop-
ositions. The assumptions (7abZ e'ab)' and (eabZ 7'ab) will
be characteristic of a more general logic, which will include
the classical logic as a special case. (See the concluding remarks
of chapter III.)

A PRIMER OF LOGIC

and by denial of the consequent,

(xZ y) Z(y'Z x'),
(xZy)'Z(y'Zx')',

it follows that:

(k'ab)'Z W'ab, (w'abZ (k'ab)')'. HI

If now we postulate:

(dab Z tt'ab)' (a'ab / aab)'

(/3ab Z /3'ab)' (/3'ab^0ab)'

(7ab^7'ab)' (7'ab/7ab)'

(eab^ €'ab)' (€'ab/ eab)'

and, consequently,

(kab / k'ab)', (k'ab/kab)',

it will follow by*

(xZ y) (xZ z)'Z(yZ z)',

(xZz)/(yZz)Z(xZy)/,
that (kab Z (w'ab)')', ((w'ab)' Z kaby. IV

Finally if we assume

dab £ dba> ^ab -^ ^ba» Tab Z 7&b, Cab

it will follow, by

(x Z y) (y Z z) Z (x Z z) and (x Z y) Z (y' Z x'),
that

kab / kab, k'ab Z k'ab.

Definitions.

If xab Z y'ab and y'ab Z xab, xab is said to be contra-
dictory to yab- By I, kab is contradictory to k'ab and, by
I', k'ab is contradictory to kab.

"These are obtained from (x Z y) (y Z z) Z (x Z z) by
(xy Z z) Z (xz; Z yO and xy Z yx.

4 A PRIMER OF LOGIC

If xab Z y'ab and (y'ab Z xab)', xab is said to be contrary
to yab. By II, kab is contrary to wab, and conversely,
since kab and wab are interchangeable.

If (xab Z y'ab)' and y'ab Z xab, xab is said to be subcon-
trary to yab. By III, k'ab and w'ab are subcontrary pairs.

If (xab Z y'ab)' and (y'ab Z xab)', xab is said to be sub-
alternate to yab. By IV, k'ab and wab are subalternate
pairs.

§3. Having classified the categorical forms under
these heads, it remains to differentiate them by means of
their formal properties. If we assume as valid,

a aa Z aaa, e aa Z eafi, (aaa Z a aa) , (€aa Z e aa) ,

where a represents the class contradictory to a, (non-a),
the other propositions given below may be derived.*

a aa Z aaa (ctaa Z 0, aa) tt aa Z tt aa (tt aa Z O,aa)

0aa Z /3'aa (/3'aa Z £„)' ft afi Z iS'a, (^afi Z /3aa)' V

7aa Z T'aa (Vaa Z Taa)' 7 aa Z ^aa (Vafi Z 7aa)'

€aa Z e'aa (e'aaZ 6aa)' e'aa Z €aa ( €aa Z e'aa)'

The results of V, together with the non-convertible
character of 7**, are enough to establish the definitions
of the four forms.

*Under the conditions mentioned above, in note, p. 2, we
shall have to write (7aa Z 7'aa)'. Implications V are an extension
of the meaning of implication, made necessary by our having
to call aaa and eaa true propositions. (See Boole, Investigation
of the Laws of Thought, chap. XI, p. 169.)

**The operation of simple conversion consists in interchang-
ing subject and predicate. By the principle, (xZ z/(yZ z) Z
(xZ y)', and what has gone before, we have:

(TabZ Vab/CTbaZ T'ab) Z (Tab Z Tba)'.

A PRIMER OF LOGIC 5

Definitions.

A form which is the contrary of itself is called a null-
form. By V, 0aa, Taa, caa, aa&, /3aa, 7&a, are null-forms.

A form which is the subcontrary of itself is called a
one-form. By V, aaa, eaa, are one-forms.

If xab is unprimed and xaa a one-form then xab is
called an a-form.

If xab is unprimed and simply convertible and if
xaa and xaa are null-forms, then xab is called a /3-form.

If xab is unprimed and not simply convertible, then
xab is called a 7-form.

If xab is unprimed and xaa a one-form, then xab is
called an €-form.

A PRIMER OF LOGIC

EXERCISES

(1) Assuming kab = kab * kab, kabZ w'ab, show that,

OfabZ j'ab 7'ab ab T'ba

aab 7ab eab 7ba
7abZ a'ab /3'ab e'ab 7'ba
eab Z a'ab |8 ab 7 ab 7 ba
by the aid of,

(xZy) (yZz)Z(xZz),
(xZy)Z(zxZzy).

Equality is defined by

(xZy)(vZx)Z(x = y),
(x = y)Z(xZ y) (yZ x).

If now

0'ab 7ab c'ab T'ba Z «ab

^

a^ab ^'ab C^ab Vba Z 7ab
« ab jS'ab 7 ab 7'ba Z €ab

it will follow that

aab = /^ab T^b €^ab 7^ba «^ab=/3ab + 7ab + eab + 7ba

j8ab =«'ab T'ab «'ab 7'ba /3'ab =«ab + 7ab + €ab + 7ba

7ab=«'ab ^ab €^ab 7^ba 7/'ab=«ab + /3ab + €ab + 7ba

6ab = «'ab j8 ab 7 ab T ba e'ab =QJab + /3ab + 7ab + 7ba

The second set of equations follows from the first by the
principle, that the contradictory of a product is the sum of the
contradictories of the separate factors, and by substituting kab
directly for (k'ab)'.

(2) Show by the method of the last example that

aab=«ab /3'ab=aab 7'ab=«ab c'ab

0ab = /3ab a'ab = /3ab 7'ab = |8ab e'ab

7ab=7ab ^ab=7ab ^ab = 7ab e'ab
Cab = Cab «'ab = «ab ^'ab = €ab 7'ab

Show too that

«ab=«ab 0'ab 7'ab=«ab j^'ab t/ab=«ab ?'ab c'ab, etc., etc.

and that aab=aab /3'ab 7'ab c'&b, etc., etc.
Derive the analogues of the first set:

a'ab = C/ab + /3ttb = «'ab + Tab =«'ab + Cabi etc., etc,

A PRIMER OF LOGIC 7

Accordingly, since kab = kab ' w'ab and k/ab = k/ab + wab, any
primed letter is a modulus of multiplication with respect to
any unprimed letter not itself and any unprimed letter is a
modulus of addition with respect to any primed letter not itself.

The prepositional zero is defined by

(xyZ (xy)')Z(xyZ 0),
(xyZ 0)Z(xyZ (xy)'),

and the prepositional one by

((xy/Zxy^lZxy),
(1Z xy)Z((xy)'Z xy).

From the principles,

(xZ y')Z(xyZ (xy)'),
(xyZ(xy)')Z(xZy'),
it follows that

(xZyOZ(xyZO)
(xjZO)Z(xZ y')

(3) Derive:

kab ' k'ab / (kab ' k'ab)'; (kab + k'ab)' Z kab + k'ab;
kab ' wab Z (kab ' wab)'; ((kab + wab)' Z kab + wab)';
(k'ab ' w'ab Z (k^ ' w'ab)')'; (k7ab + w'ab)' Z. k'ab + w'ab;
(kab * w'ab ^ (kab ' w'ab)') '; ((kftb + wf&bYZ. kab + w'.b)'.

(4) Show that

«ab |8ab ^ 0, «ab 7ab Z 0, etc.

1 Z a'ab + 0'ab, 1 ^ a'ab + 7'ab, etc.
and that

«'ab #'ab T'ab «'ab T'ba -^ 0

1 Z a:'ab + j3'ab + 7'ab + e'ab + T'ba

(5) Assuming xab = xab ' xab, xab = xab + xabf
show that

a'ab ^'ab=7ab + «ab +^7ba, «'ab 7'ab = /3ab + «ab + 7ba, CtC.
«ab + /3ab=7'ab C ab 7'ba, «ab + 7ab = /3'ab c'ab 7'ba» etc.

(6) Derive the general result:

(ka.b/ Wa,b)'.

The comma between the terms indicates that the term order is
not fixed. Thus ka,b stands for either kab or kba.

8 A PRIMER OF LOGIC

(7) From the principle,

(xZz)'(yZz)Z(xZy)',
and the postulate (aaa Z a'aa)',
derive (aaaZ /3aa)', («aaZ Taa)', («aaZ eaa)'.

(8) From the principles,

(xZy)(yZz)Z(xZz),
(x'Z x)Z(yZ x),
(xZxOZ(xZ y),

and the postulate, c/aaZ aaa, show that all propositions of the
form xaaZ yaa, except the three cases in the last example, are
valid implications, xaa and yaa representing only the unprimed
letters.

(9) By the method of the last example, prove that
(aaa Z a'aay is the only invalid implication of the form xaa Z y'aa.

(10) Derive seven valid implications in each one of the
forms x'aa Z yaa and x'aa Z yaa and nine invalid implications of
each one of the same forms.

(11) From (€aa Z e'aa)' by the method of example 7 derive
(eafl Z aaa)', (€aa Z 0aa)', (caa Z

(12) From e'aa Z €a5 by the method of example 8 derive
thirteen valid implications.

(13) Show that (6aa Z e'aa)' is the only invalid implication
of the form xaa Z y'aa.

(14) Derive the following implications:

iz«aa

0/aaZ 0

1 Z a'aa

«aa Z 0

1 Z 0'aa

j8aa Z 0

1 Z j8'aa

/3aa Z 0

izyaa

Taa Z 0

1 Z T'aa

Taa Z 0

1Z e'aa

6aa Z 0

1Z eaa

€'afiZ 0

(lZo/aay

(«aa / 0)'

(izaaay

(«'aaZO)'

03'aazoy

az/3aay

tf'aaZ 0)'

(IZT.a)'

(VaaZO)'

(14 7s.)

(T'aaZO)'

(1Z 6aa)'

(e'aaZO)'

(1Z 6^)'

(eaaZ 0)'

Some of the postulates in the text (p. 2) of the form,
(k'abZ wab)', (kabZ k'ab)', (k'ab Z kab)', may be established by

A PRIMER OF LOGIC 9

reducing them to one of the forms, (k'aaZ waa)', (k'aaZ waa)'
(kaaZ k'aa)', etc., cases already considered in preceding exercises

(15) Establish the invalidity of

«'ab Z £ab «'ab Z Tab Vab ^ Tba

/3'ab^ Tab /3'ab^ €ab Vab ^ «ab

«'ab ^ «ab jS'ab ^ l^ab V ab ^ Tab

'ab €ab Z c'ab C ab Z €ab

CHAPTER II

§4. At this point in our theory it will be necessary
to introduce certain indefinables, which we shall call the
distinctions of better and worse, following a suggestion of
Sir William Hamilton's.* For our immediate purpose
it will be enough to define better than and worse than de-
notatively, establishing an order among the four forms by
a simple enumeration. Better than and worse than are not
transitive relations. When we wish to express the rules
for the deduction of the moods symbolically, we shall
have to invent symbols to represent worse than (/), doubly
worse than (//), and trebly worse than (///). This necessity
is avoided in the verbal statement of the principles of de-
duction by the words "in the same degree" (see rule 1
below).

Definitions. — An 6-form is worse than an a-, a 0-, or
a 7-form.

A 7-form is worse than an a- or a /3-form.
A /3-form is worse than an a-form.

Best. a~ /3-- 7— e Worst.

§5. Immediate inference is a form of implication
belonging to one of the types:

1. xab/ yab- 2. xabZ yba.

These differences are known as the first and the sec-
ond figures of immediate inference respectively.

The part to the left of the implication sign is called
the antecedent; the part to the right is called the consequent.

*Lectures on Logic, Appendix, p. 536.

A PRIMER OF LOGIC 11

Since x and y may take on any of the forms, a, /3,
7, c, there will be sixteen propositions of each type, ob-
tained from the permutations of the letters two at a time
and by taking each letter once with itself. Each one of
the sixteen distinct propositions in each one of the two
figures is called a mood of immediate inference. The
rules which follow below, applied to the postulates, will
yield all the true and all the false propositions of each type.

Valid Moods.

1. In any valid mood of the first figure make ante-
cedent and consequent worse in the same degree.

2. In any valid mood convert simply in any form
but 7.

Postulate: aab Z. aab. Theorems: The other (6) valid

moods.
Invalid Moods.

1. In any invalid mood of the first figure make
antecedent and consequent worse in the same degree.

2. In any invalid mood of the first figure interchange
antecedent and consequent.

3. In any invalid mood convert simply in any form
but 7.

Postulates:*

{aab/0ab}'; {dab/Tab}'; (aab/ Cab}'; {7ab/7ba}'.

Theorems: The other (21) invalid moods.

§6. We may also formulate rules for the detection
of the invalid moods. These are:

*The mark 0 over the bracket is interred to indicate that
the mood is invalid,

12 A PRIMER OF LOGIC

1. If the antecedent be worse than the consequent,
the mood is invalid.

2. If the antecedent be better than the consequent,
the mood is invalid.

Definition. — Distributed terms are those modified,
either implicitly or explicitly, by the adjective all, i. e.
the subject of the a-, 7- and e-form, and the predicate of
the a- and e-form.

3. it a term be distributed in the consequent but
undistributed in the antecedent, the mood is invalid.

§7. A syllogism is a form of implication belonging
to one of the types:

1. xbaycbZ zca=(xy z)x

2. xabycbZ zca=(xy z)2

3. xba ybc Z zca =(x y z)3

4. xabybcZ zca=(xyz)4

These differences are known as the first, second, third,
and fourth figures of the syllogism respectively. The two
forms conjoined in the antecedent are called the premises
and the consequent is called the conclusion. The predi-
cate of the conclusion is called the major term and points
out the major premise, which by convention is written
first, and the subject of the conclusion is called the minor
term and points out the minor premise. The term common
to both premises and which does not appear in the con-
clusion is called the middle term.

Since x, y and z may have any one of the values,
a, ft 7, e, there will be sixty-four ways in each one of the
four figures, called the moods of the syllogism, in which
x y Z z can be expressed. There will be consequently

A PRIMER OF LOGIC 13

two hundred and fifty-six cases to consider. Twenty-
nine of these are valid implications; the remaining two
hundred and twenty-seven are invalid. From the rules
and postulates below, all the moods, valid and invalid,
may be deduced.

Valid Moods.

1. In any valid mood of the third figure make a like
major premise and conclusion worse in the same degree.

2. In any valid mood of the second figure make a
like minor premise and conclusion worse in the same degree.

3. In any valid mood convert simply in any form
but 7.

Postulates : Theorems :

(aaa) x (aaa)2,3(4 (/3aj8)If2f3i4

(7^)2,4

Invalid Moods.

1. In any invalid mood of the third figure make a
like major premise and conclusion better in the same degree.

2. In any invalid mood of the second figure make
a like minor premise and conclusion better in the same
degree.

3. If the premises and conclusion of an invalid mood
in the fourth figure are all alike, make them all worse in
the same degree.

4. If the premises and conclusion of an invalid mood
are all alike make the conclusion any degree better or any
degree worse.

14 A PRIMER OF LOGIC

5. If the premises and conclusion of an invalid mood
are all unlike, interchange them in any order.

6. In any invalid mood convert simply in any form
but 7.

Postulates:*

(aa/3)', (tfc)', tfry)', (777)'. W,

(aa«)',
(aPy)',

Theorems:
The other (208) invalid moods.

§8. As in the case of immediate inference we may
formulate rules for the detection of the invalid moods of
the syllogism. These are five in number.

1. A mood is invalid if the conclusion differ from the
worse premise.

2. A mood is invalid if an a- and a 7- premise be
conjoined in the antecedent and the middle term be un-
distributed in the major premise.

3. A mood is invalid if the middle term be undis-
tributed in both premises.

4. A mood is invalid if a term which is distributed
in the conclusion be undistributed in the premise.

5. A mood is invalid if each premise be in the
e-form.**

*The mark (') over the bracket is intended to indicate that
the mood is invalid.

**These rules are, of course, not sufficient to declare (766)2,4
and (€7c)I)2 invalid, in case we decide to so regard them. See
the concluding remarks of chap. III.

A PRIMER OF LOGIC 15

EXERCISES

If xa,b^ ya,b be denoted simply by (xy), (the comma be-
tween the terms indicating that the term order and so the figure,
is not determined), the array of sixteen propositions may be
constructed thus:

aa fia yy. ea

"^ 00 70 $

or? 07 77 €7

ae j8c 76 ee

the moods valid in both figures being underlined twice, the one
valid only in the first figure being underlined once. Applying
the first rule to the postulate, we obtain in succession, j3/3, 77,
ee, in the first figure; and converting simply in the consequent
of those valid in the first figure, except 77, we obtain aa, /3/3, ce
in the second figure.

(1) From the rules and postulates for the derivation of the
invalid moods deduce the remaining twenty-one invalid moods.

The rules for the immediate detection of the invalid moods
are all necessary if we can point to at least one example which
falls uniquely under each rule. They are sufficient if they de-
clare all the invalid moods to be invalid.

(2) Construct the set of propositions of immediate inference
and place after each invalid mood the number of a rule which
declares it to be invalid.

(3) Make a list of moods which are declared invalid by the
first rule and by no other rule, and a list of moods which are
declared invalid by the second rule and by no other rule.

|)

(4) Find an invalid mood which is declared invalid by the
third rule and by no other rule and prove that it is the only
unique illustration of this rule.

For those who approach the study of the syllogism for the
first time, it may facilitate manipulation to point out the general
effect of conversion in the form of certain rules.

1, Simple conversion in the major premise changes the
first figure to the second and conversely, the third figure to the
fourth and conversely.

16 A PRIMER OF LOGIC

2. Simple conversion in the minor premies, changes the
first figure to the third and conversely, the second figure to the
fourth and conversely.

3. Simple conversion in the conclusion changes the first
figure to the fourth and conversely and leaves the second and
third figure unchanged.

It must of course not escape the beginner's notice that the
effect of converting simply in the conclusion is to interchange the
premises, since the major term then becomes the minor term and
the minor term becomes the major term. The conjunctive
relation of logic being commutative, the order of the premises
is indifferent, but we agree, as a matter of convention, always
to write the major premise first.

(5) From (eye)i, and the third rule under the valid moods
alone, deduce (eye)2, (7ee)2 and (yee)4.

(6) From the rules and postulates deduce the remaining
valid moods.

(7) Assuming only the third rule under the valid moods
and the rule : in any valid mood of the first figure make a like
major premise and conclusion worse in the same degree, deduce
all the remaining valid moods from (aaa)z, (777)1 and (077)1.

(8) Assuming only the second and third rules under the
valid moods deduce the remaining valid moods from (aaa)lt
(777)i, (7<*7)i and

(9) From (efieYt alone deduce seventy-eight other invalid
moods.

(10) From (afieYt alone deduce twenty-three other invalid
moods.

(11) Deduce the invalid moods in the first figure which have
a 7-minor premise.

The rules for the immediate detection of the invalid moods
are sufficient, if they declare all the moods not already found
to be valid to be invalid. They are all necessary if we can point
to at least one example which falls uniquely under each rule.

(12) Construct the array of the syllogism and place after
each invalid mood the number of a rule that declares it to be
invalid,

A PRIMER OF LOGIC 17

(13) Show that it follows from one of the rules alone that
two jS-premises do not imply a conclusion.

(14) Prove that there are only two moods which illustrate
the second rule uniquely.

(15) Make a list of examples which fall uniquely under
each one of the rules.

18 A PRIMER OF LOGIC

CHAPTER III

§9. In this third chapter it is proposed to completely
define the relationships of better and worse by deducing all
the true and all the false propositions into which these
relationships may enter and then to give a complete ex-
pression in the language of symbols of the rules for the
deduction of the moods of immediate inference and the
syllogism.

§10. Let us, first of all, invent symbols to denote
worse than, doubly worse than, and trebly worse than, i. e.

x / y =x is worse than y,

x // y =x is doubly worse than y,

X///Y =x is trebly worse than y,

and let us add the following:

Definition. — In the propositions, x / y, x//y, and
x///y, x is called the inferior, y the superior form.

Since x and y may take on any of the four forms
a, /3, 7, e, there will be sixteen possible propositions of each
type, x / y, x //y and x///y, obtained by permuting the
letters two at a time and by taking each letter once with
itself. The following postulates and principles will yield
all the valid moods of each type. We have assumed four
principles here because the principles for the deduction of
the invalid moods may be derived from these four as
theorems.

Principles:

i. (x / y) (y //z) Z (x///z) iii. (x / z) (y///z) Z (y //x)
ii. (x / y) (z //x) Z (z///y) iv. (x / z) (y // z) Z (y / x)

Postulates: /3/a; 6/7; 7//a.

Theorems: 7/0; e//0; e///a.

A£PRIMER OF LOGIC w

We may also formulate rules for the derivation of
the moods. It will then be necessary to assume one pos-
tulate only.

Definition: — In the propositions, x / y, x //y and
x///y, the relation connecting x and y is known as the
worse-relation.

Definition: — Trebly worse (///) is worse than doubly
worse (//) and doubly worse than worse (/). Doubly
worse (//) is worse than worse (/).

The rules are:

1. In any valid mood make superior and inferior
form one degree better or one degree wrorse.

2. In any valid mood make inferior form and worse-
relation one degree better or one degree worse.

Postulate : Theorems :

/3 / a. The other (5) valid moods.

The invalid moods of each type may be derived from
the following postulates and principles:

Principles:*

i. (x / y) (x///z)'Z(y//z)'

(x///z)'(y//z) Z(x / y)'
ii. (x / y) (z///y)'Z(z//x)'

(z///y)' (z//x) Z(x / y)'
iii. (x / z) (y//x)'Z(y///z)'

(y//x)'(y///z) Z(x / z)'

iv. (x / z) (y / x)'Z(y//z)'
(y / x)'(y//z) Z(x / z)'

These principles follow from those used for the deduction
of the valid moods by (xyZ z) Z (xz'Z y').

20 A PRIMER OF LOGIC

Postulates: Theorems:

(a/VAX; (0//A)'; (7///e)'; The other (3 6) invalid moods.

(€///€)'; (e///7)';

As in the case of the valid moods, rules may be form-
ulated for the derivation of the invalid moods. Here it
will be necessary to assume only three postulates. The
rules are:

1. In any invalid mood make superior and inferior
form one degree better or one degree worse.

2. In any invalid mood make inferior form and worse-
relation one degree better or one degree worse.

3. In any invalid mood make superior form three
degrees worse.

Postulates: Theorems:

0 / 7)'; (7 / a)'; (e / a)'. The other (39) invalid moods.

§11. Having now completely defined the relationships
of better and worse by deducing all the prepositional forms
into which these relationships may enter, there remains
for this chapter only one other task, which is to deduce
symbolically the moods of immediate inference and the
syllogism.

§12. From the postulates and principles, which are
given below, all the moods, valid and invalid, of immediate
inference may be deduced.

Principles:

i. (y / x) (x» b Z xa b) Z (ya b ^ ya b).
iv. (xZ y) (yZ z)Z (xZ z).

Postulates : Theorems :

aab^ aba*, jSabZ /3ba; €&b Z. eba. The other valid moods.

A PRIMER OF LOGIC 21

Principles:*

ii. (x / y) (xa b / xa b) Z (xa b Z ya b)'
(x//y) (xa b ^ xa b) Z (xa b Z ya b)'

iii. (y / x) (xa b / xa b) Z (xa b Z ya b)'
(y//x) (xa b / xa b) Z (xa b Z ya b)'

v. (xZ y) (xZ z)'Z (yZ z)'
(xZ z)' (yZ z) Z (xZ y)'

Postulates: Theorems:

(dab^ cab)'; (eabZ aab)'; (Tab^ 7ba)'. The other invalid

moods.

§13. All the valid and invalid moods of the syllogism
may be deduced from the assumptions which follow.
The right to convert simply in any form but 7 is ex-
pressed under v and vi. It will be evident that some of
the postulates might have been saved at the expense of
introducing new principles, and conversely. The first
two principles for the deduction of the invalid moods
under iv are theorems from the ones that have gone be-
fore under i, by (xy Z z) Z (xz' Z y').

Principles:

i. (y / x) (xba zbc / xca) Z (yba zbc Z yca)
(y / x) (zab xcb Z xca) Z (zab ycb ^ yca)

v. (xy Z z) (z Z w) Z (xy Z w)
(xy Z z) (w Z x) Z (wy Z z)
(xy Z z) (w Z y) Z (xw Z z)

*These principles, except the second under iii are really
special cases of principles i and iii under the syllogism, obtained
from the latter by making b = c, the primed part of the ante-
cedent in iii becoming unprimed in the special case.

22 A PRIMER OF LOGIC

Postulates: (aaa)z; (777) r»

Theorems: The other (26) valid moods.

Principles:

ii. (y / x) (z I y) (xyZ z)' Z (xz Z y)'
(y / x) (z//y) (xyZz)'Z(xzZ y)'
(y//x) (y / z) (xyZz)'Z(xzZy)'
(y///x) (y / z) (xyZ z)'Z (xzZ y)'
(y /x) (z / y) (xyZz)'Z(zyZx)'
(y / x) (z//y) (xyZz)'Z(zyZx)'
(y//x) (y / z) (xyZz)/Z(zyZx)/
(y///x) (y / z) (xyZ z)'Z (zyZ x)'.

iii. (x / y) (xa,b xb,c Z xca)' Z (xa,b xb,c Z yea)'
(x // y) (xa,b xbtC Z Xca)' Z (xa,b xb,c Z yca)'
(y / x) (xa,b xb,o Z xcay Z (xa,b xb,c Z yca)'

iv. (x / z) (yab xcb Z Xca)' Z (yab zcb Z zca)'
(x / z) (xba ybc Z xca)' Z (zba ybc Z zca)'
(y / x) (xab xbc Z xca)' Z (yab ybc Z yca)'

vi.* (xy Z z)' (w Z z) Z (xy Z w)'
(xy Z z)' (x Z w) Z (wy Z z)'
(xy Z z)' (y Z w) Z (xw Z z)'

(aetf)', (a/37)'t (ae7)'x

(aa7)'x (ajSe)', (^a7)'x (jSeyX, (777)'2 (e^7)'x (e7e)'3

(aac)'z (a77)'3 (jSjSe)', (7a7)'2 (7c7)'

The other (206) invalid moods.

*Principles v and vi are of course not independent. The
first under v is a variation of transitivity, the third a variation
of the second by xyZ yx. Those under vi follow from those
under v by (xy Z z) Z (xz' Z y') . Principles v under immediate
inference follow from transitivity by the same principle.

A PRIMER OF LOGIC 23

§14. We have already pointed out, (note p. 2), that
the product 7a,b ea,b does not vanish in general if we allow
the possibility of the limiting values 0 and 1 for the terms.
Under these conditions, (7ee)2,4 and (ej€)I>2 are not valid
moods of the syllogism, for they become 7<>i e0tl/. 0, for
a = c =0 and b = 1. A logic, which recognizes these limiting
values of the terms, will have to postulate (eye)'^ say,
which yields, (eye)'2 and (yee) '2,4.

The only change, which we should then have to make
in chapters II and III, would be to replace (777) '2 among
the postulates by (eye)*!, from which (777) '2 follows as a
theorem, and to subtract (eye)It2 and (yee)2,4 from the list
of valid moods.

This logic, which might be called non-Aristotelian, or
semi-Aristotelian, or imaginary logic, is more general than
the ordinary or classical logic and includes the latter as
a special case, becoming, in fact, identical with it when the
field of its application is narrowed so as to exclude ''nothing"
and "universe" as limiting values of the terms. One
principle, which is true in the special, but not in the general
case, is:

(y / x) (xba zcb / xca) Z (vba zcb Z yca),

and this principle may be regarded as the differentiating
character of the two cases. If we had chosen to assume
it, instead of the first principle under i, we could have saved
the third postulate, but the second principle under iv
would not then have followed as a theorem.

The definitions of chapter I, §3, hold for both cases;
the only change to be made in implications V in order to
make them true in the more general logic, will be to replace
7aa =0 by 7aa4: 0, and this property has not been made use
of in defining the 7-form,

24 A PRIMER OF LOGIC

The Aristotelian forms, A, E, I, O, (see Appendix I),
will yield only eight valid moods of the syllogism, under the
new condition, instead of the twenty-four valid moods
commonly recognized. They satisfy all the conditions of
maximum simplicity in the special or classical instance —
they are the best possible forms to choose for the con-
struction of an Aristotelian logic — but they fail in the gen-
eral instance, for they then lose their peculiar advantage,
that, corresponding to any member of the set there should
be another member of the set which represents its contra-
dictory.

A PRIMER OF LOGIC 25

EXERCISES

By the aid of the principles,

(xy Z z) (z Z w) Z (xy Z w)
(xy Z z) (w Z x) Z (wy Z z)
(xy Z z) (w Z y) Z (xw Z z)

we are enabled to convert in either premise or the conclusion.
The example which follows will illustrate the method.

(7ba <*cb Z Tea) (o%Z Otj\c) Z (7ba «bc ^ Tea)

(1) From (eye)x derive (7€e)2,4.

(2) From the principles and the postulates in the text de-
duce the remaining valid moods.

(3) From the postulates, (aaa)lt (777)1, (OTY)I and the prin-
ciple (y/x) (xba zcb Z xca) Z (yba zcb ^ yca) deduce the remaining
valid moods.

If we identify the subject and predicate of the conclusion
in the mood, (Pafi)3, we obtain |3ba«baZ 0, (chapter I, impli-
cations v). By the aid of (xyZ 0) Z (xZ y') it follows that

«abZ /3'ab-

(4) Deduce as many as possible of the propositions of the
form, xabZ y'ab, (chapter I, p. 2) by identifying subject and
predicate in the conclusion of the valid moods of the syllogism.

The principles,

(xyZ z)'(wZ z)Z (wyZ w)'
(xyZ z)'(xZ w)Z (wyZ z)'
(xyZ z)'(yZ w) Z (xwZ z)'

enable us to convert in either premise or the conclusion.

(5) From (fi8/3)'x, derive the invalidity of this mood in the
other figures.

(6) From (e/3e)'i alone and principles iii, iv and vi deduce
seventy other invalid moods.

(7) From (ajfryX, alone and principles ii deduce nineteen
other invalid moods,

26 A PRIMER OF LOGIC

(8) Deduce the invalid moods in the third figure, whose
conclusion is in the 7-form.

(9) Deduce the invalid moods in the fourth figure, whose
major premise is in the 7-form.

(10) From 7771 alone, deduce forty-six valid implications
of the form La,b Mb.c £ N'ca, — L, M and N representing only
the unprimed letters.

(11) Assuming eyelt2 and yee2t4 to be invalid, show that
777! yields only thirteen valid implications of the form given
in the last exercise.

(12) Assuming (/3ftJ')'i (B^')\ (/S/Se7)'. (770')'= (rtfO'j (777')'.

(77*')'* (erf% («rO'i ("0')'. («O'.

deduce sixty-nine other non-implications of the same form.

Any non-implication of the form, Lb,a Mc,b ^ N'ca, which
contains an a-form may be proven invalid by identifying terms
in the a-form. Thus /3ba Pbc £ oi c& reduces to /3ba / /3'ba for
c = a; 7baacb^ 7'ca reduces to 7baZ 7rba for c = b, etc.

(13) Establish the invalidity of the thirty-four non-impli-
cations of the form Lb,a Ma,b/ N'ca not accounted for in the
preceding exercise.

(14) Show that there are thirty-six, and only thirty-six,
distinct valid implications of the form La>b Mb,c Nc,aZ 0, —
L, M and N representing only the unprimed letters, a, /3, 7, e.

(15) Derive indirectly the (777') 'i of exercise 12.

A certain number of the postulates for the derivation of the
invalid moods of the syllogism (p. 21) may be shown indirectly
to be invalid by reducing them to invalid moods of immediate
inference. Thus (aa/3)i reduces to a^Z. c/ab when the terms
in the conclusion are identified, and (a&y) i reduces to /3ca^ Tea,
when we identify terms in the major premise and suppress the
part a&& (see chap. IV).

(16) Establish the invalidity of

(€#€)! (« €7) i

(copy)! (0:77)3 (707)

A PRIMER OF LOGIC 27

Most of the other postulates for the deduction of the invalid
moods of the syllogism may be reduced by the method of the
following example:

Suppose that (epy)i is valid. Now (e/3y')i is valid by a
preceding exercise.

• * • ( €ba &b Z Tea) ( €ba 0cb Z T'CB) Z (*ba fob ^ 0)

since (xZ y) (xZ y') Z (xZ 0).

Consequently €ba 0cb ^ cca.

If now we postulate (ej8e)'x, it follows that

(17) Establish the invalidity of

(£77)3 (707)2 (777)4

(7*7)2 (€77)3 (cea)!

The postulates (Pey)'i (ej8e)'x and (eye)'3 that remain (p. 21)
may be reduced by the following method:

(jSba &b / e'ca)' (|8ob ^ Vcb) ^ (^ba 7'cb / e'ca)'

by a principle under vi and the postulate of a preceding exercise

GSba 7'cb/ OVtfba tcaZTcb)'

which yields (Pey)^ by simple conversion in the major premise.
(18) Establish the invalidity of ( c/3 e)z and ( ey e)3.

Any non-implication of the form La,b Mb,c ^ Nca, which
contains an a-premise, may be reduced to an invalid mood of
immediate inference, and so shown to be invalid, by identifying
terms in the a-premise. All of the other invalid moods may be
derived from the postulates of exercise (12), the forms of im-
mediate implication given in chapter I, the principles iv and vi
of chapter III, together with (xy Z z) Z (xz'Z y') and (xy Z z) Z
(z'y^ A

19. From the postulates of exercise (12) deduce all the
non-implications of the form La,b Mb,0^ Nca, without making
use of principles ii and iii of this chapter.

20. Show that there exist no valid implications of the
form L;a,b Mb,c Z Nca or La,b MVC ^ Nca and consequently
none of the form L;a,b M'b.c Z NQa or L'a,b M^.o Z N'ca.

CHAPTER IV

§15. The sorites is a form of implication of the
general type:*

Xx(i|2) x2(2,3) x3(3f4)— xn-x(n7T, n) Z xn(n i),

in which the number of terms is greater than three.

Certain valid moods of the sorites can be constructed
from chains of valid syllogisms. Thus the chain of syl-
logisms:

a(i,2) 0(2,3) Z a(3i),

a(3i) 0(3,4) Z a(4i),

0(41) 0(4,5) Z a(si),

will yield a valid sorites, viz :

0(1,2) 0(2,3) 0(3,4) 0(4,5) Z 0(51), for

{ 0(1,2) 0(2,3) Z 0(31) } Z { 0(1,2) 0(2,3) 0(3,4) Z 0(31) 0(3,4) }

.'. 0(1,2) 0(2,3) 0(3,4) Z 0(41), by the second syllogism and

the principle of transitivity.

{ 0(1,2) 0(2,3) 0(3,4) Z 0(41) } Z

{ 0(1,2) 0(2,3) 0(3,4) 0(4,5) Z 0(41) 0(4,5) }

. *. 0(1,2) 0(2,3) 0(3,4) 0(4,5) Z o(si), as before.

Consequently in general, if

XI(l,2) Xa(2.3) Z X3(3l)
X3(3l) X4(3,4> Z Xs(«)
Xs(4l) X6(4,5) Z X7(si)

X2n-s(n-i i) X2n.4(n-i, n) Z X2n-3(ni)

*In this chapter it will be more convenient to employ the
notation x(ab) for xab or x(i,2) for xlf2. The comma between
the terms means that the term order is not settled.

A PRIMER OF LOGIC 29

be a chain of valid syllogisms, then

xz(i,2) x2(2|3) x4(3,4)— x2n_4(iiT7, n) Z x2n.3(ni)

is a valid mood of the sorites. It remains to be proven
that the only valid moods that exist can be constructed
from chains of valid syllogisms. The proof depends on
the following principles.

Principle i. — A valid mood of the sorites, which has
one premise of the same form as the conclusion, will re-
main valid, when as many of the other premises as we
desire are put in the a-form.

Principle ii. — A valid mood of the sorites will remain
valid, when as many terms have been identified as we desire.

Principle iii. — An a-premise, whose subject and predi-
cate are identical, may be suppressed as a unit multiplier.

Principle iv. — A valid mood of the sorites, none of
whose premises has the same form as the conclusion, will
remain valid, when as many premises as we desire are put
in the a-form.

Theorem i. — There exists no valid mood of the sorites,
in which none of the premises has the same form as the
conclusion.

For (principle iv) put all the premises after the first
in the a-form. Then by identifying terms (principle ii)
the mood of the sorites can be reduced (principle iii) to an
invalid syllogism of the form:

Xx(l,2) a(2,3) Z Xn(3l).

Conclusion in the a-form.

At least one of the premises is in the a-form (theorem i).
If one of the remaining premises, xr(s - i, s), be not in the
a-form, put each one of the other premises in the a-form,

30 A PRIMER OF LOGIC

if all but xr be not already in that form (principle i).
Then by identifying terms (principle ii) the mood of the
sorites will reduce (principle iii) to an invalid syllogism
of the form:

xr(s - i, s ) a(s, s + 1) Z a(s + 1 s - i),
or a(s - 2, s - i) xr(s - i, s) Z a(s s - 2).

Consequently all the premises are in the a-form if the
mood of the sorites is valid and the sorites is of the general
type:

0(1,2) 01(2,3) — a(n- i, n) Z a(ni),

which can be constructed from the chain of valid syllo-
gisms :

a(2,i) a(3,2) Z a(3i),

a(3i) 01(4,3) Z a(4i),

a(4i) 01(5,4) Z a(si),

a(n - 1 i) a(n, n - i) Z a(n i).

Conclusion in the /3-form.

At least one premise, xt, is in the /3-form (theorem i),
and all the other premises are in the a-form. For suppose
one of the other premises xr (s - 1, s) were not in the a-form.
Put all the premises (principle i) except xt and xr in the
a-form. Then by identifying terms (principle ii) the mood
of the sorites will be reducible to an invalid syllogism
(principle iii) of the form:

0(s- i, 3-2) xr(s, s- i) Z |8(s s- 2),
or xr(s, sT~7) j8(s, s + i) Z P(s~+i s- i).

Consequently the sorites must be of the form:
0(2,3) — a(s, s - i) /3(s + i» s) a(s + i, 5+2) — a(n- i, n)

A PRIMER OF LOGIC 31

Z |8(n i), which can be constructed from the chain of
syllogisms :

0(1,2) 0(2,3) Z a(ai)
a(ai) 0(3,4) Z 0(41)

a(s - i i) a(s - i, s) Z a(s i )
a(si) j8(s,

|8(sTi i) o(sTii s+2 ) Z 0(s+2 i)

0(n-i i) a(n-i, n) Z #(n i).

Conclusion in the y-form.

At least one of the premises is in the 7- form (theorem i).
Each 7-form in the antecedent must present its terms in
the order (s s - 1). For suppose that y(s - 1 s) should
appear as one of the premises. Put each one of the
remaining premises in the a-form (principle i). Then by
identifying terms (principle ii) the sorites will reduce to
an invalid syllogism (principle iii) of the form:

T(S - i s) a(s, s +1) Z y(s +1 s -

or a(s - i, s - 2) T(S - 1 s) Z y(s s - 2).

Pursuing the same reasoning as before it can be shown
that no |8- or e- premises can occur. One form of this
sorites may consequently be 7(21)— 7(11 n- 1) Z 7(111), which
can, in fact, be constructed from the chain of valid syl-
logisms:

7(21) 7(32) Z 7(31)

7(31) 7(43) Z 7(41)

7(n~M~ 0 7(n iiTi) Z 7(11 i).

All the other forms of valid sorites with a 7-conclusion
are obtained from the above type by transforming one or

32 A PRIMER OF LOGIC

more of the premises into the a-form in every possible
way under the restrictions of theorem i. Each one of
these types can be built up from a chain of valid syllo-
gisms each member of which has one of the forms :

(77T)i, (a77)i,2, or (7017) i, 3-

Conclusion in the e-form.

At least one of the premises is in the e-form (theorem i),
and there is not more than one e-premise. For, if there
are two or more e-premises, put all the premises but two
of the e-premises in the a-form (principle i). Then by
identifying terms (principle ii) we will come upon an invalid
syllogism (principle iii) of the form:

e(s - i, s) e(s, s+i)Z e(s + i s - i).

There can be present no /3-premise. For suppose
xr (s, s - l) to be a /3-premise. Put all the premises except
xr and the e-premise in the a-form (principle i). By identi-
fying terms (principle ii) we will come upon an invalid
syllogism (principle iii) of the form:

/3(s, s- i) e(s, s+i) Ze(s + i s- i),
or e(s - i, s - 2) /3(s, s - i) / e (s s - 2).

Any 7-premise coming after the e-premise must pre-
sent its terms in the order (s s - i). For suppose 7(5, s - i)
coming after the e-premise to present the term order
(s- i s). Put all the premises except y(s - i s) and the
e-premise in the a-form (principle i). Then by identifying
terms (principle ii) we will come upon an invalid syllogism
(principle iii) of the form:

e(s -1,5-2) 7(s - i s) Z e(s s - 2).

Any 7-premise coming before the e-premise must
present its terms in the order (s - 1 s). For suppose 7(3, s - i)

A PRIMER OF LOGIC 33

coming before the e-premise to present the term order
(s s - i). Put all the premises except 7(5 s - i) and the
c-premise in the a-form (principle i). Then by identifying
terms (principle ii) we will come upon an invalid syllogism
(principle iii) of the form :

T(S s- i) c(s, s + i) Z e(

One form of this sorites may be, consequently,

7(12) 7(23)— 7(3 - 2 «T~i) e(s, s - i) 7(s+i s)— 7(n n~^7) Z
e(n i), which can, in fact, be constructed from the chain
of valid syllogisms:

7(12) 7(23) Z 7(13)
7(13) 7(34) Z 7(14)

7(l S - 2) 7(s - 2 S - l) Z 7(1 S - l)

7(1 s - i) e(s, s- i) Z c(s i)
c(s i) 7(s+7 s) Z e(s~+T 0

e(n- i i) 7(n n - 1) Z e(n i)

All the other forms of valid sorites with an c-conclusion
are obtained from the above type by replacing one or more
of the 7-forms by a-forms in every possible way. Each
new type can be constructed from a chain of valid syllo-
gisms, each member of which has one of the forms:

There exist, consequently, no valid moods of the sorites
which can not be constructed from chains of valid syllo-

gisms.*

*If (7€c)2,4 and (eye)If2 are to be regarded as invalid moods,
(see the concluding remarks of chapter III), then it can be shown
at once that no 7-premise can occur when the conclusion is in
the e-form. The general form of such a sorites will be,

34 A PRIMER OF LOGIC

EXERCISES

(1) Construct a valid sorites from the chain of valid syllo-
gisms :

a2i732^ 73i»

73i<*43^ 74i,

74x ?54 ^ 7si-

(2) By the aid of the principles of chapter IV, reduce the
valid sorites, a21 y32 a43 y54 Z 7SI, successively to each one of the
three valid syllogisms of example 1.

(3) Prove the invalidity of the sorites,

7zi 73* 734 754 ^ 7Si-

(4) From what chain of valid syllogisms can the sorites,
«i,2723 €3,47s4«6f5^ «6i be constructed?

(5) If (€7e)i.2 and (7€e)2,4 be regarded as invalid moods of
the syllogism, (see the concluding remarks of chap. Ill), prove
the invalidity of the sorites,

7i2 7a3"7s-a s-i € s, s-i 7s+i s~7n n-i Z €nr

a(i,2)a(2t3) — a(s - i.s)e (s, s + i)a(s+i, 8+2) — a(n- i, n)/ e(ni)
which can be built up from the chain of syllogisms,

a(i,2)a(2,3)Z a(si)

a(3i)a(s,4)Z a(4i)

a(s - i i)a(s - i, s)Z a(si)
a(si)€(s, sTi) Z e(s+i i)
c(s + i i) a (s-H, s+2)Z e(s+2

c(n - i i )a (n - 1 1 n) Z c (n i)

APPENDIX I

On the Simplification of Categorical Expression and the
Reduction of the Syllogistic Figures

If a and b represent classes, there are four ways in
which they may be related categorically, the one standing
for subject, the other for predicate. These four forms of
relationship are always represented by the letters, A,
E, I, O, i. e.

Aab=all a is b,
Eab=no a is b,
Iab = some a is b,
Oab=not all a is b.

Historical efforts have, been made to reduce the num-
ber of these relationships. If symbols be invented to
denote some a (a) and not-a (ax), the last three may be
represented by means of the first, for:

Eab=Aabi> Iab=Aab> Oab=Aabi-

But an essential difference is here left undistinguished
and the number of necessary forms will not have been
reduced by this device. If a new symbol be employed for
all a (a) and another for the copula is ( Z ), we shall have:

Aab=aZ b,
Eab=aZ bi,
lab = aZb,
Oab=aZ bx.

The four separate categorical forms have, accordingly,
been gotten rid of at the cost of introducing four new unde-
fined symbols, so that no economy of our indefinables has
been effected.

36 A PRIMER OF LOGIC

It is to be observed that the word some, which is im-
plicit or explicit in the meaning of part of each proposition,
means some at least, possibly all. Another set of proposi-
tions, in which some is to mean some at least, not all, may
be used to replace the traditional ones. These other forms
are:

aab = all a is all b,

/3ab=some a is some b,

Tab = all a is some b,

€ab=no a is b.

In addition we shall have to employ
the hypothetical form,

xab / y&b =xab implies yab,
{ xab ^ yab}r = xab does not imply yab,

the conjunctive form,

Xab ' yab = xab and yab,

the disjunctive form,

Xab

Each member of the set, A, E, I, O, may be expressed
in the members of the set, a, /3, 7, c, and conversely, so
that the two are, in fact, logically equivalent, although
each one has certain advantages peculiar to itself.

The members of the second set have this property,
that, if one is true, then all the others are false.* We
assume, accordingly, the

Postulates: a&b L /3'ab 0ab ^ T'ab Tab / T'ba
dab £ T'ab jSab £ c'ab
«ab ^ c'ab Tab ^ c'ab

*Provided we exclude the limiting values 0 and 1 for a and b.
The ordinary definitions of these limits allow 70i and c0i be true
together.

A PRIMER OF LOGIC 37

from which follow, by the principle of the denial of the
consequent, the

Theorems: eab / a'ab Tab ^ o/ab

Cab / 0'ab Tab / 0'ab

Cab / T'ab /3ab ^ a'ab

Consequently*

aab'/3ab=0 /?ab'Tab=0 Tab*Tba=0

ttab *7ab=0 0ab * Cab=0

aab * cab=0 Tab ' €ab=0 I

From the Definitions:**

Aab=aab + Tab

Eab = Cab

lab = ttab + )3ab + Tab + Tba II

Oab = €ab + /3ab + Tba

we obtain immediately***

dab = Aab " Aba
/3ab = lab * Oab ' Oba
Tab =Aab " Oba
Cab = Eab

It is an advantage of the forms of the original set,
an advantage which the set, a, /3, T> c, does not possess,
that the contradictory of any letter is represented by a
single other letter of the set. Suppose that we were to

*a /3 = 0 reads: a(is true) and fi(is true ) is impossible.

**A, E, I, O are simply the sums given in equations II.
That they are the traditional Aristotelian forms, is only an
accident of the reader's application. Hence equations II are
definitions and not postulates.

***Multiplying out the sums in II as if they were ordinary
polynomials, applying the results of I, and assuming that a, J3
and c are simply convertible.

38

A PRIMER OF LOGIC

combine this advantage with that of simple convertibility
in a new set of forms.

To do this it would seem to be enough to subtract
from the meaning of Aab the part 7&b, (equations II), and
to add this part to the meaning of Oab.* Our new set of
forms becomes:

HI

= Cab
liab =Ctab +

+ Tab + 7ba
+ Tab + 7ba

the analogues of the old letters being represented by the
corresponding Greek vowels.

From equations I and III, and remembering that the
sum of dab, /3ab, 7ab> cab, Tba makes up the propositional
"universe," the results of the following table, yielding all
the moods of immediate inference, will easily be seen to
hold.

True

Implies the
truth of
only

Implies the
falsity of
only

False

Implies the
truth of
only

Implies the
falsity of
only

a

a, i

e, 0

a

0

a

e

e, 0

a, i

€

L

e

i

i

€

L

€, 0

a, t

o

o

a

0

a, i

€, 0

An induction of these results shows that a = o't
o = af, e = if, i= e', and that, consequently, contradictory

*Here would seem to be another instance of the manner in
which the language of symbols may free a science from the
accidents imposed upon its development by the language of
speech. The last two members of the new set have apparently
no simple verbal expression.

A PRIMER OF LOGIC

a,

pairs are a, o and c, t. Likewise contraries are
subcontraries are i, 0 ; subalterns are a, t and €, o.

If we define an affirmative form as one that becomes
unity when subject and predicate have been identified
and a negative form as one that becomes unity when sub-
ject and predicate have been made contradictory, then it
is a result of the following

Postulates:* aaa is a true proposition,

Theorems:

18',

»>

>»

» M »

» M >»

>» M »

Qt »

P aa

>» »

/ » >» » M

T aa

and equations III, that a and t are affirmative and that e
and o are negative forms.

If a distributed term be one modified by the quanti-
tative adjective all, it will be seen that a and c distribute
both subject and predicate, while i and o distribute neither.
These results are summarized in the table below, the
distributed terms being underlined.

Affirmative

Negative

,f

Universal

dab

€ab

Particular

tab

0ab

*x is true is to be represented by x=l, x is false by x = 0.
(See Boole, Investigation of the Laws of Thought, ch. XI, p.
169). The theorems follow by equations I, and equations III
become as a result of them, aaa=l, caa = 0, iaa = l, 0aa = 0,
«aa = 0, 6aa = l, iaa = 0, 0aa=L Employing the usual notation,
a^not-a.

40 A PRIMER OF LOGIC

The traditional rules for detecting the invalid moods
of the old syllogism, constructed from the set A, E, J, O,
hold for the new syllogism, built up out of the forms,
a, e, i, o. These rules are:

1. Two negative premises do not imply a conclusion.

EX. €bn €cb Z Cca»

2. Two affirmative premises do not imply a nega-
tive conclusion. Ex. aba acb Z eca.

3. An affirmative and a negative premise do not
imply an affirmative conclusion. Ex. aba «cb ^ ac&.

4. Two premises, in neither of which the middle
term is distributed, do not imply a conclusion.

EX. tba tcb ^ tea.

5. Two premises, in which a given term occurs un-
distributed, do not imply a conclusion, in which that same
term is distributed. Ex. aba tcb Z aca.

The valid moods which remain, and which of course
are valid in all four figures, since each one of the forms
is simply convertible, are twelve in number, viz:

aaa aat aee aeo

au aoo eae eao

tat OQ.O eio teo

It has been previously observed, (note p. 2), that
equations I hold generally only when the limits 0 and 1
are excluded as possible values of a and b. If these pos-
sibilities he included, we shall have to assume:

{Tab Z e'ab }' and .*. { eab Z 7'ab }', since TOI eOI^ 0.
Equations I then become:

Ctab ' 0ab=0 |8ab '7ab=0

0 j8ab ' Cab=0 IV

A PRIMER OF LOGIC 41

Under these conditions, the fact which the old logic
always took for granted, that Eab is the contradictory of
Iab, and Aab the contradictory of Oab, no longer holds
true. For, while the sum of each of these two pairs of
forms is the prepositional universe, their product is not
the prepositional null, (equations II, IV). In order that
Eab * lab and Aab * Oab shall vanish for all values of the
terms, it will be necessary to exclude Tab eab from the
product. A new set of forms, in which part of the meaning
of tab is subtracted from iab and added to eab, will satisfy
this requirement. Let this new set be:

C2ab = €ab + Tab + Tba

02ab = €ab + Pab + Tab + Tba

If a, e, i, o, be replaced by a2, c2, i2, o2, respectively
in the table, (p. 38), all the results of such a new tabulation
will be seen to hold, (equations IV, V). The same defini-
tions as given before will make a2 and i2 affirmative, e2 and
o2 negative forms, (equations V, and the postulates and
theorems, p. 39), but since e2 distributes neither subject
nor predicate, e2 i2 o2 and i2 e2 o2 will not be found among
the valid moods of the syllogism, (see p. 40). The same
rules (p. 40) for the detection of the invalid moods will
hold for the new syllogism, but rule 1 is now redundant,
being a corollary of rule 4.

It might perhaps appear that our original symmetry,
(that of equations I), which was interrupted by the neces-
sity of allowing TOI to stand as a true proposition, could
be saved by assuming that the null class exhausts no part
of the universe, i. e. all of nothing is some of everything,
might be regarded as a false proposition, Now A0» = l,

42 A PRIMER OF LOGIC

or cioa-f 7oa = l, is Schroder's definition of the null class,
and Aoa will be a true proposition for all values of a if
7oi be true, whereas, 7oi=0 involves A0i=0. These con-
sequences lead us to the alternatives of either giving up
our symmetry, (in equations I), or else of regarding the
null class as not essential to our algebra.

It is finally to be noted — what was obvious in the
beginning — that, while the members of the set, a, e, t, 0,
can be expressed in the members of the set, a, 0, 7, e,
the latter can not be expressed in the former. Conse-
quently, an essential difference has been lost, and the
existence of a completed logic of the new forms would not
put aside the necessity of working out the logic of the old.

The attempts of the logician to discover a set of
categorical forms, which establish a complete symmetry
among the moods and figures of the syllogism, are as old
as the science itself. The end would be attained if a new
set could be selected so as to satisfy the following con-
ditions:

1. Each form of the set must be simply convertible.

2. Corresponding to any member of the set, there
must occur another which represents its contradictory.

3. The new set must yield at least one valid mood
of the syllogism.

4. Each member of the new set must be represent-
able in the members of a set already proved necessary
and sufficient to express all differences, (the set A, E, I,
O, say), and conversely.

If Xab be any categorical form, the simplest functions
of x, which are themselves categorical and which are in
general simply convertible, are xab * xba and xab + xba. It
will be enough, therefore, in order to satisfy condition 1,

A PRIMER OF LOGIC 43

to assume as a new set of forms either such a sum or
such a product of each one of the old forms (A, E, I, O,
say).

The equation, {xab * xba }' =x'ab + x'ba, suggests at
once what our manner of satisfying condition 2 must be,
for since the product, xab ' Xba is the contradictory of the
sum x'ab + x'ba and x the contradictory of x', if xab + xba
[respectively xab ' xba] be chosen as one of our new forms,
x'ab ' x'ba [respectively x'ab + x'ba] must be chosen as one
of the others.

Remembering that

lab =Iab * Iba =Iab + Iba,

and that Eab = rab and Iab =E'ab,

it follows that our choice of a new set of forms is limited
to the following two:

AI=Aab 'Aba, Ex=Eab, (1)

Oba, Ix = Iab,

A2=Aab + Aba, E2=Eab, (2)

O2=Oab 'Oba, I2 = lab-

It will be found, however, that the set (2) yields no
valid moods of the syllogism. Consequently, applying
condition 3, our choice is seen to be unambiguously re-
stricted to set (1), which yields twelve moods, valid each
one in each of the four figures. These will be found to
be, (dropping the subscripts):

A A A, A A I, AEE, A E O,
All, AGO, EAE, EAO,
I A I, OAO, EIO, IEO.

It will be impossible, however, to satisfy condition 4,
since every expression involving the new forms will be

44 A PRIMER OF LOGIC

simply convertible. Consequently, an essential difference
has been left undistinguished, and it will not be possible
to substitute the new forms for the old. The new forms
are, in fact, identical with ar, ex, tx, olt considered above.
From this latter discussion and from the discussion
that has gone before, we conclude, that, if it be necessary
to retain in our set of forms at least one that is not simply
convertible, it will be impossible to satisfy the condition
2 above, unless the null-and one-class be excluded or de-
nned in some way other than ordinary.

APPENDIX II

Historical Note on De Morgan's New Prepositional Forms

In introducing the notion of contradictory terms into
logic De Morgan discovered two new prepositional forms,
which cannot be directly expressed by means of the A, E,
I, O relations of traditional logic.* Suppose that we de-
note these two forms by U and V, i. e.

U (ab) =A11 not a is b,

V (ab) =Some not a is not b,

A (ab) =A11 a is b,

E (ab) =No a is b,

I (ab) =Some a is b,

O (ab) =Some a is not b.

U and V are simply convertible, for (if a = not-a)
U(ab) =A (ab) =A (ba) =U (ba), (converting in A by the
principle of contraposition), and

V (ab) = I (ab) = I (ba) = V (ba), (converting simply in I).

V distributes both subject and predicate while U
distributes neither, for V(ab) =O(ab) =O(ba) (converting
in O by contraposition) and, since O distributes its
predicate, both a and b are distributed terms; similarly
U(ab) =A(ab) =A(ba) (converting in A by contraposition)
and, since A does not distribute its predicate, a and
b are undistributed terms.

If an affirmative form be defined as one that becomes
the subcontrary of itself when the subject and predicate

*Formal Logic, p. 61.

46

A PRIMER OF LOGIC

have been identified, and a negative form as one that
becomes the contrary of itself under the same conditions,
it will be seen that A, I and V are affirmative, that E, O
and U are negative forms.

These results are summarized in the following table,
the distributed terms being underlined.

Affirmative

Negative

Universal

A(ab)

E(ab)

Particular

I(ab)

0(ab)

Indefinite

V(ab)

U(ab)

Below are tabulated all the forms of immediate im-
plication which hold among the six propositions A, E, I,
O, U, V. The (144) implications and non-implications
necessary to establish unambiguously the results of the
table can be derived from a certain number of postulates
and the commonly assumed principles of traditional logic.

True

Implies
falsity of

Implies
truth of

False

Implies
falsity of

Implies
truth of

A

E, 0, U.

A, I, V.

A

A.

O.

E

A, I.

E, O.

E

E.

I.

I

E.

I.

I

A, I.

E, 0.

O

A.

O.

0

E, 0, U.

A, I, V.

U

A,V.

0, U.

U

U.

V.

V

U.

V.

V

A,V.

O, U.

An induction of these results will show that:
Contradictory pairs are: A, O; E, I; U, V;
Contrary pairs are: A, E; A, U;

A PRIMER o'LtGJe' •' V: f ; , , 47

Subcontrary pairs are: I, O; O, V;

Subalternate pairs are: A, I; A, V; I, V; I, U; E, O;

E, U; E, V; U, O.

In order to show how the new forms fit the ancient
scheme and as an illustration of method let us solve the
array of the syllogism. We first observe (see table) that
A weakens ambiguously to I or V; that O strengthens am-
biguously to E or U.*

Rules:

1. In any valid mood interchange either premise
and the conclusion and replace each by its contradictory.

2. In any valid mood strengthen a premise or weaken
a conclusion.

3. In any valid mood convert simply in any form
but A or O.

Postulates:

A A A (1st figure) is a valid mood.

AUU ( " " ) " "

EAE ( " » ) » »» »

UEA ( " » ) >' » »

From these rules and postulates will follow sixteen
valid moods in the 1st figure, twenty in each one of the
2nd and 3rd figures, and twenty-one in the 4th figure.

In order to deduce the invalid moods let us assume
the

Rules:

1. In any invalid mood interchange either premise
and the conclusion and replace each by its contradictory.

*If x implies y but y does not imply x, then x is said to be a
strengthened form of y, and y is said to be a weakened form of x.

48 - 'A'PRfMER OF LOGIC

2. In any invalid mood weaken a premise or
strengthen a conclusion.

3. In any invalid mood convert simply in any form
but A or O ; and

Postulates:

A A A (4th figure) is an invalid mood.

AAO ( " " ) " "

AAV (3rd " ) " "

AAI (2nd " ) " "

AAO (1st " ) " "

EEI ( " " ) " "

EEO ( " " ) " "

EU A ( " " ) " "

EUO ( " " ) " "

UUO ( " " ) " "

U U V ( " " ) " " " M

From these postulates and rules follow the remaining
(772) invalid moods.

It will be seen at once that the rules of the old logic
for the immediate detection of the invalid moods of the
syllogism no longer hold. To give only one illustration:
A term may appear distributed in the conclusion of a valid
mood and be undistributed in the premise.

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