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ȣ mathematical logic, , ɹ symbolic logic .

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2.

BOOK IV.

THE TRILITERAL DIAGRAM.

CHAPTER I.

SYMBOLS AND CELLS.

Change of Biliteral into Triliteral Diagram 39

The xy- Class subdivided into the xym- Class and the xym

- Class 40

pg_xxiii

The Inner and Outer Cells of the North- West Quarter assigned to these

Classes

The xy

- Class, the x

y- Class, and the x

y

- Class similarly subdivided

The Inner and Outer Cells of the North- East, the South- West, and the

South- East Quarter similarly assigned

The Inner Square and the Outer Border have thus been assigned to the

m- Class and the m

- Class

Rules for finding readily the Compartment, or Cell, assigned to any given

Attribute or Attributes

Table IV. Attributes of Classes, and Compartments, or Cells, assigned to

them 42

CHAPTER II.

REPRESENTATION OF PROPOSITIONS IN TERMS OF x AND m, OR OF y

AND m.

1.

Representation of Propositions of Existence in terms of x and m, or of y

and m.

The Proposition Some xm exist 43

Seven other similar Propositions

The Proposition No xm exist 44

Seven other similar Propositions

2.

Representation of Propositions of Relation in terms of x and m, or of y

and m.

The Pair of Converse Propositions Some x are m = Some m are x

Seven other similar Pairs

The Pair of Converse Propositions No x are m = No m are x

Seven other similar Pairs

The Proposition All x are m 45

Fifteen other similar Propositions

Table V. Representations of Propositions in terms of x and m 46

Table VI. Representations of Propositions in terms of y and m 47

Table VII. Representations of Propositions in terms of x and m 48

Table VIII. Representations of Propositions in terms of y and m 49

pg_xxiv

CHAPTER III.

REPRESENTATION OF TWO PROPOSITIONS OF RELATION, ONE IN

TERMS OF x AND m, AND THE OTHER IN TERMS OF y AND m, ON THE

SAME DIAGRAM.

The Digits I and O to be used instead of Red and Grey Counters 50

Rules

Examples worked

CHAPTER IV.

INTERPRETATION, IN TERMS OF x AND y, OF TRILITERAL DIAGRAM,

WHEN MARKED WITH COUNTERS OR DIGITS.

Rules 53

Examples worked 54

BOOK V.

SYLLOGISMS.

CHAPTER I.

INTRODUCTORY.

Syllogism 56

Premisses

Conclusion

Eliminands

Retinends

Consequent

The Symbol š

Specimen- Syllogisms 57

CHAPTER II.

PROBLEMS IN SYLLOGISMS.

1.

Introductory.

Concrete and Abstract Propositions 59

Method of translating a Proposition from concrete into abstract form

Two forms of Problems

2.

Given a Pair of Propositions of Relation, which contain between them a

Pair of codivisional Classes, and which are proposed as Premisses: to

ascertain what Conclusion, if any, is consequent from them.

Rules 60

Examples worked fully

The same worked briefly, as models 64

3.

Given a Trio of Propositions of Relation, of which every two contain a Pair

of codivisional Classes, and which are proposed as a Syllogism: to

ascertain whether the proposed Conclusion is consequent from the

proposed Premisses, and, if so, whether it is complete.

Rules 66

Examples worked briefly, as models

ȣ .Symbolic Logic, by Lewis Carroll

CONTENTS

BOOK I.

THINGS AND THEIR ATTRIBUTES.

CHAPTER I.

INTRODUCTORY.

PAGE

Things 1

Attributes

Adjuncts

CHAPTER II.

CLASSIFICATION.

Classification 1

Class

Peculiar Attributes

Genus

Species

Differentia

Real and Unreal, or Imaginary, Classes 2

Individual

A Class regarded as a single Thing 2

pg_xvi

CHAPTER III.

DIVISION.

1.

Introductory.

Division 3

Codivisional Classes

2.

Dichotomy.

Dichotomy 3

Arbitrary limits of Classes

Subdivision of Classes 4

CHAPTER IV.

NAMES.

Name 4

Real and Unreal Names

Three ways of expressing a Name

Two senses in which a plural Name may be used 5

CHAPTER V.

DEFINITIONS.

Definition 6

Examples worked as models

pg_xvii

BOOK II.

PROPOSITIONS.

CHAPTER I.

PROPOSITIONS GENERALLY.

1.

Introductory.

Technical meaning of some 8

Proposition

Normal form of a Proposition

Subject, Predicate, and Terms 9

2.

Normal form of a Proposition.

Its four parts:?

(1) Sign of Quantity

(2) Name of Subject

(3) Copula

(4) Name of Predicate

3.

Various kinds of Propositions.

Three kinds of Propositions:?

(1) Begins with Some . Called a Particular Proposition: also a

Proposition in I 10

(2) Begins with No . Called a Universal Negative Proposition: also

a Proposition in E

(3) Begins with All . Called a Universal Affirmative Proposition:

also a Proposition in A

pg_xviii

A Proposition, whose Subject is an Individual, is to be regarded as

Universal

Two kinds of Propositions, Propositions of Existence, and Propositions

of Relation

CHAPTER II.

PROPOSITIONS OF EXISTENCE.

Proposition of Existence 11

CHAPTER III.

PROPOSITIONS OF RELATION.

1.

Introductory.

Proposition of Relation 12

Universe of Discourse, or Univ.

2.

Reduction of a Proposition of Relation to Normal form.

Rules 13

Examples worked

3.

A Proposition of Relation, beginning with All, is a Double Proposition.

Its equivalence to two Propositions 17

pg_xix

4.

What is implied, in a Proposition of Relation, as to the Reality of its

Terms?

Propositions beginning with Some 19

Propositions beginning with No

Propositions beginning with All

5.

Translation of a Proposition of Relation into one or more Propositions of

Existence.

Rules 20

Examples worked

BOOK III.

THE BILITERAL DIAGRAM.

CHAPTER I.

SYMBOLS AND CELLS.

The Diagram assigned to a certain Set of Things, viz. our Univ. 22

Univ. divided into the x- Class and the x

- Class 23

The North and South Halves assigned to these two Classes

The x- Class subdivided into the xy- Class and the xy

- Class

The North- West and North- East Cells assigned to these two Classes

The x

- Class similarly divided

The South- West and South- East Cells similarly assigned

The West and East Halves have thus been assigned to the y- Class and

the y

- Class

Table I. Attributes of Classes, and Compartments, or Cells, assigned to

them 25

pg_xx

CHAPTER II.

COUNTERS.

Meaning of a Red Counter placed in a Cell 26

Meaning of a Red Counter placed on a Partition

American phrase sitting on the fence

Meaning of a Grey Counter placed in a Cell

CHAPTER III.

REPRESENTATION OF PROPOSITIONS.

1.

Introductory.

The word Things to be henceforwards omitted 27

Uniliteral Proposition

Biliteral do.

Proposition in terms of certain Letters

2.

Representation of Propositions of Existence.

The Proposition Some x exist 28

Three other similar Propositions

The Proposition No x exist

Three other similar Propositions 29

The Proposition Some xy exist

Three other similar Propositions

The Proposition No xy exist

Three other similar Propositions

The Proposition No x exist is Double, and is equivalent to the two

Propositions No xy exist and No xy

exist 30

pg_xxi

3.

Representation of Propositions of Relations.

The Proposition Some x are y

Three other similar Propositions

The Proposition Some y are x 31

Three other similar Propositions

Trio of equivalent Propositions, viz. Some xy exist = Some x are

y = Some y are x

Converse Propositions, and Conversion

Three other similar Trios 32

The Proposition No x are y

Three other similar Propositions

The Proposition No y are x

Three other similar Propositions

Trio of equivalent Propositions, viz. No xy exist = No x are y = No y

are x 33

Three other similar Trios

The Proposition All x are y is Double, and is equivalent to the two

Propositions Some x are y and No x are y

ǡ

Seven other similar Propositions 34

Table II. Representation of Propositions of Existence 34

Table III. Representation of Propositions of Relation 35

CHAPTER IV.

INTERPRETATION OF BILITERAL DIAGRAM, WHEN MARKED WITH

COUNTERS.

Interpretation of 36

And of three other similar arrangements

pg_xxiiInterpretation of

And of three other similar arrangements

Interpretation of 37

And of three other similar arrangements

Interpretation of

And of three other similar arrangements

Interpretation of

And of three other similar arrangements

Interpretation of

And of seven other similar arrangements 38