# Book:William Warren Bartley, III/Lewis Carroll's Symbolic Logic

## William Warren Bartley, III: Lewis Carroll's Symbolic Logic

Published $\text {1977}$, Harvester Press

ISBN 0-85527-984-2.

### Contents

Note to the Reader From the Editor and Publisher
Editor's Introduction (W.W.B.)
Editor's Acknowledgements (W.W.B.)
Editor's Bibliography (W.W.B.)

#### Part One: Elementary

Introduction to Learners
Preface to Fourth Edition
BOOK I THINGS AND THEIR ATTRIBUTES
Chapter I Introductory
Things
Attributes
Chapter II Classification
Classificaion
Class
Peculiar Attributes
Genus
Species
Differentia
Real and Unreal, or Imaginary, Classes
Individual
A Class regarded as a single Thing
Chapter III Division
$[\S 1]$ Introductory
Division
Codivisional Classes
$[\S 2]$ Dichotomy
Dichotomy
Arbitrary limits of Classes
Subdivision of Classes
Chapter IV Names
Name
Real and Unreal Names
Three ways of expressing a Name
Two senses in which a plural Name may be used
Chapter V Definitions
Definition
Examples worked as models

BOOK II PROPOSITIONS
Chapter I Propositions Generally
$[\S 1]$ Introductory
Technical meaning of "some"
Proposition
Normal form of a Proposition
Subject, Predicate, and Terms
$[\S 2]$ Normal form of a Proposition
Its four parts :
(1) Sign of Quantity
(2) Name of Subject
(3) Copula
(4) Name of Predicate
$[\S 3]$ Various kinds of Propositions
Three kinds of Propositions :
(1) Begins with "Some." Called a Particular Proposition: also a Proposition in I
(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
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
Chapter III Propositions of Relation
$[\S 1]$ Introductory
Proposition of Relation
Universe of Discourse, or Univ.
$[\S 2]$ Reduction of a Proposition of Relation to Normal form
Rules
Examples worked
$[\S 3]$ A Proposition of Relation, beginning with "All" is a Double Proposition
Its equivalence to two Propositions
$[\S 4]$ What is implied, in a Proposition of Relation, as to the Reality of its Terms?
Propositions beginning with "Some"
Propositions beginning with "No"
Propositions beginning with "All"
$[\S 5]$ Translation of a Proposition of Relation into one or more Propositions of Existence
Rules
Examples worked

BOOK III THE BILITERAL DIAGRAM
Chapter I Symbols and Cells
The Diagram assigned to a certain Set of Things, viz. our Univ.
Univ. divided into the $x$-Class and the $x'$-Class
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. Adjuncts of Classes, and Compartments, or Cells, assigned to them
Chapter II Counters
Meaning of a Red Counter placed in a Cell
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
$[\S 1]$ Introductory
The word "Things" to be henceforward omitted
Uniliteral Proposition
Biliteral Proposition
Proposition in terms of certain Letters
$[\S 2]$ Representation of Propositions of Existence
The Proposition "Some x exist"
Three other similar Propositions
The Proposition "No x exist"
Three other similar Propositions
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"
$[\S 3]$ Representation of Propositions of Relation
The Proposition "Some $x$ are $y$"
Three other similar Propositions
The Proposition "Some $y$ are $x$"
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
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 $x$
Three other similar Trios
The Proposition "All $x$ are $y$"
The Proposition "All $x$ are $y$" is Double, and is equivalent to the two :::: Propositions "Some $x$ are $y$" and "No $x$ are $y'$"
Tables II, III. Representation of Propositions of Existence and Relation
Seven other similar Propositions
Chapter IV Interpretation of Biliteral Diagram, When Marked with Counters
Interpretation of $\begin{array}{|c|c|} \hline \bigodot & \quad \\ \hline \quad & \quad \\ \hline \end{array}$
And of three other similar arrangements
Interpretation of $\begin{array}{|c|c|} \hline \bigcirc & \quad \\ \hline \quad & \quad \\ \hline \end{array}$
And of three other similar arrangements
Interpretation of Work In Progress: Insert table (discuss) You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. When this has been done, the template should be removed from the code.

And of three other similar arrangements
Interpretation of $\begin{array}{|c|c|} \hline \bigodot & \bigodot \\ \hline \quad & \quad \\ \hline \end{array}$
And of three other similar arrangements
Interpretation of $\begin{array}{|c|c|} \hline \bigcirc & \bigcirc \\ \hline \quad & \quad \\ \hline \end{array}$
And of three other similar arrangements
Interpretation of $\begin{array}{|c|c|} \hline \bigodot & \bigcirc \\ \hline \quad & \quad \\ \hline \end{array}$
And of seven other similar arrangements

BOOK IV THE TRILITERAL DIAGRAM
Chapter I Symbols and Cells
Change of Biliteral into Triliteral Diagram
The $xy$-Class subdivided into the $xym$-Class and the $xym'$-Class
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 Quarters similarly assigned
The Inner Square and the Outer Border have thus been assigned to the $m$-Class and the $m'$-Class
Rules for anding readily the Compartment, or Cell, assigned to any given Attribute or Attributes
Table IV. Attributes of Classes, and Compartments, or Cells, assigned to them
Chapter II Representation of Propositions in Terms of $x$ and $m$, or of $y$ and $m$
$[\S 1]$ Representation of Propositions of Existence in terms of $x$ and $m$, or of $y$ and $m$
The Proposition "Some $xm$ exist"
Seven other similar Propositions
The Proposition "No $xm$ exist"
Seven other similar Propositions
$[\S 2]$ 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 $mn$ 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 $m$"
Fifteen other similar Propositions
Tables V, VI, VII, VIII. Representation of Propositions in terms of $x$ and $m$, or of $y$ and $m$
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
Rules
Examples worked
Chapter IV Interpretation, in Terms of $x$ and $y$, of Triliteral Diagram, When Marked with Counters or Digits
Rules
Examples worked

BOOK V SYLLOGISMS
Chapter I Introductory
Syllogism
Premisses
Conclusion
Eliminands
Retinends
Consequent
The Symbol $\therefore$
Specimen-Syllogisms
Chapter II Problems in Syllogisms
$[\S 1]$ Introductory
Concrete and Abstract Propositions
Method of translating a Proposition from concrete into abstract form
Two forms of Problems
$[\S 2]$ 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
Examples worked fully
The same worked briefly, as models
$[\S 3]$ 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
Examples worked briefly, as models

BOOK VI THE METHOD OF SUBSCRIPTS
Chapter 1 Introductory
Meaning of $x_1$, $xy_1$, &c.
Entity
Meaning of $x_0$, $xy_0$, &c.
Nullity
The Symbols $\dagger$ and $\P$
Like and Unlike Signs
Chapter II Representation of Propositions of Relation
The Pair of Converse Propositions
Some $x$ are $y$ $=$ Some $y$ are $x$
Three other similar Pairs
The Pair of Converse Propositions
No $x$ are $y$ $=$ No $y$ are $x$
Three other similar Pairs
The Proposition "All $x$ are $y$"
The Proposition "All $x$ are $y$" is Double, and is equivalent to the two Propositions "Some $x$ exist" and "No $x$ are $y'$"
Seven other similar Propositions
Rule for translating: "All $x$ are $y$ from abstract into subscript form, and vice versa
Chapter III Syllogisms
$[\S 1]$ Representation Syllogisms
Rules
$[\S 2]$ Formulae for solving Problems in Syllogisms
Three Formulae worked out:
Fig. I. $xm_0 \dagger ym'_0 \P xy_0$, its two Variants ($\alpha$) and ($\beta$)
Fig. II. $xm_0 \dagger ym_1 \P x'y_1$
Fig. III. $xm_0 \dagger ym_0 \dagger m_1 \P x'y'_1$
Table IX. Formulae and Rules
Examples worked briefly, as models
Notes
$[\S 3]$ Fallacies
Fallacy
Method of finding Forms of Fallacies
Forms best stated in words
Three Forms of Fallacies:
(1) Fallacy of Like Eliminands not asserted to exist
(2) Fallacy of Unlike Eliminands with an Entity-Premiss
(3) Fallacy of two Entity-Premisses
$[\S 4]$ Method of proceeding with a given Pair of Propositions
Rules

BOOK VII SORITESES
Chapter I Introductory
Sorites
Premisses
Partial Conclusion
Complete Conclusion (or Conclusion)
Eliminands
Retinends
Consequent
The Symbol $\therefore$
Specimen-Soriteses
Chapter II Problems in Soriteses
$[\S 1]$ Introductory
Form of Problem
Two Methods of Solution
$[\S 2]$ Solution by Method of Separate Syllogisms
Rules
Example worked
$[\S 3]$ Solution by Method of Underscoring
Underscoring
Subscripts to be omitted
Example worked fully
Example worked briefly, as model
Seventeen Examination-papers

BOOK VII EXAMPLES, ANSWERS AND SOLUTIONS
Chapter I Examples
$[\S 1]$ Propositions of Relation, to be reduced to normal form
$[\S 2]$ Pairs of Abstract Propositions, one in terms of $x$ and $m$, and the other in terms of $y$ and $m$, to be represented on the same Triliteral Diagram
$[\S 3]$ Marked Triliteral Diagrams, to be interpreted in terms of $x$ and $y$
$[\S 4]$ Pairs of Abstract Propositions, proposed as Premisses.. Conclusions to be found
$[\S 5]$ Pairs of Concrete Propositions, proposed as Premisses: Conclusions to be found
$[\S 6]$ Trios of Abstract Propositions, proposed as Syllogisms: to be examined
$[\S 7]$ Trios of Concrete Propositions, proposed as Syllogisms: to be examined
$[\S 8]$ Sets of Abstract Propositions, proposed as Premisses for Soriteses: Conclusions to be found
$[\S 9]$ Sets of Concrete Propositions, proposed as Premisses for Soriteses: Conclusions to be found
$[\S 1]$
$[\S 2]$
$[\S 3]$
$[\S 4]$
$[\S 5]$
$[\S\S 6, 7]$
$[\S\S 8, 9]$
Chapter III Solutions
$[\S 1]$ Propositions of Relation reduced to normal form
Solutions for $[\S 1]$
$[\S 2]$ Method of Diagrams
Solutions for
$[\S 4]$ Nos. 1-12
$[\S 5]$ Nos. 1-12
$[\S 6]$ Nos. 1-10
$[\S 7]$ Nos. 1-6
$[\S 3]$ Method of Subscripts
Solutions for
$[\S 4]$
$[\S 5]$ Nos. 13-24
$[\S 5]$ Nos. 1-12 and 25-101
$[\S 6]$
$[\S 7]$
$[\S 8]$
$[\S 9]$

BOOK IX SOME ACCOUNT OF PARTS II AND III

BOOK X INTRODUCTORY
Chapter 1 Introductory
Chapter II The Existential Import of Propositions
Letter from Lewis Carroll to T. Fowler, November 13, 1885
Chapter III The Use of "Is-not" (or "Are-not") as a Copula
Chapter IV The Theory that Two Negative Premisses Prove Nothing
Chapter V Euler's Method of Diagrams
Chapter VI Venn's Method of Diagrams
Chapter VII My Method of Diagrams
Chapter VIII Solution of a Syllogism by Various Methods
(1) Solution by ordinary Method
(2) Symbolic Representation
(3) Solution by Euler's Method of Diagrams
(4) Solution by Venn's Method of Diagrams
(5) Solution by my Method of Diagrams
(6) Solution by my Method of Subscripts
Chapter IX My Method of Treating Syllogisms and Sorites
Notes to Book X

BOOK XI SYMBOLS, LOGICAL CHARTS
Chapter I Logical Symbols
Chapter II Figures or Forms
Fig. I
Fig. I$\alpha$
Fig. I$\beta$
Fig. II
Fig. III
Fig. IV
Fig. V
Fig. VI
Chapter III Fallacies
Chapter IV Logical Charts
Logical Chart I
Logical Chart 11
Logical Chart 111
Logical Chart IV
Logical Chart V
Interpretation of Charts I-V
Interpretation of Chart VI
Logical Chart VI
Logical Chart VI*
Logical Chart VI**
Interpretation of Chart VII
Logical Chart VII
Illustrations from Carroll's Workbook of Logical Charts

BOOK XII THE METHOD OF TREES
Chapter I Introductory
Chapter II Sorites-Problems with Biliteral Premisses
Chapter III Sorites-Problems with Triliteral and Multiliteral Premisses
Tree I
Tree 2
Tree 3
Tree 4
Letter from Carroll to John Cook Wilson, November 6, 1896
Tree 5
The Method of Trees: Appendix

BOOK XIII SOME PROBLEMS TO BE SOLVED BY THE METHODS OF PART II
Chapter I Introductory
Chapter II Problems in Sequences
Tree for Problem One
Commentary on the Tree for Problem One
Tree for Problem Three
Addendum: Excerpts from the Eighth and Ninth Papers on Logic
Chapter III The Problem of the School-Boys
Some answers to the Problem of the Schoolboys
Tree I
Tree II
Tree III
Chapter IV The Pork-Chop Problem
Version I
Version 11
Solution to the Pork-Chop Problem
Pork-Chop Problem Dictionary
Pork-Chop Problem in Subscript Form
Pork-Chop Problem Register
Carroll's Letters kowloon Cook Wilson on the Pork-Chop Problem
1. November 12, 1896
11. November 1, 1698
111. November 12, 1896
Chapter V Froggy's Problem
Dictionary for Froggy's Problem
Chapter VI The Members of Parliament Problem
The Solution to the Members of Parliament Problem
The Problem in Abstract Form
Carroll's Letters to John Cook Wilson concerning the Members of Parliament Problem
I. October 29, 1896
II. Undated, probably October 30, 1896
III. November 3, 1896
Carroll's Letters to his sister Miss Louisa Dodgson concerning the Members of Parliament Problem
I. November 16, 1896
II. November 18, 1896
Louisa Dodgson's Attempt to Solve the M.P. Problem
Chapter VII The Problem of Six Friends and their Wives
Version I
Version II
Chapter VIII The Problem of the Five Liars
The Salt and Mustard Problem
The Problem of the Five Liars
Version I
Version II
Carroll's Letters to John Cook Wilson concerning the Five Liars and Salt and Mustard Problems
I. October 25, 1896
II. October 28, 1896
III. November 2, 1896
IV. November 16, 1896
V. December 18, 1896
Chapter IX The Great-Grandson Problem
Solution to the Great-Grandson Problem
I. February 16, 1897
II. May 17, 1897
Chapter X The Jack Sprat Problem
Carroll's letter to his sister, Miss Louisa Dodgson, of September 28, 1896, on the Jack Sprat Problem
An Answer to the Jack Sprat Problem
Chapter XI The Library Problem
Carroll's Letters to John Cook Wilson relevant to the Library Problem
I. November 4, 1896
II. November 11, 1896
III. November 18, 1896
IV. December 26, 1896
Chapter XII The Pigs and Balloons Problem
Chapter XIII The Problem of Grocers on Bicycles
Chapter XIV The Pets Problem
Chapter XV The Winds and Windows Problem

BOOK XIV SOME FURTHER PROBLEMS TO BE SOLVED BY THE METHODS OF PART II
Problems and Exercises 1-83

BOOK XXI LOGICAL PUZZLES
Chapter I Introductory
Chapter II Classical Puzzles
1. Introductory
2. Pseudomenos
3. Crocodilus
4. Antistrephon
5. Achilles
6. Raw Meat
Chapter III Other Puzzles
2. Men Tall and Numerous
3. The Socialist Orator and the Irish Mob
4. Death at Any Moment
5. The Small Girl and Her Sympathetic Friend
6. A Notice at the Seaside
7. On the Way to the Barber-shop
8. What the Tortoise Said to Achilles
Chapter IV Solutions of Classical Puzzles
1. Introductory
2. Pseudomenos
3. Crocodiles
4. Antistrephon
5. Achilles
6. Raw Meat
Chapter V Solutions of Other Puzzles
2. Men Tall and Numerous
3. The Socialist Orator and the Irish Mob
4. Death at Any Moment
5. On the Way to the Barber-Shop
Letter to J. Welton on the Barber-Shop
Appendix A [to Book XXI]
Editor's Note on Carroll's Barber-Shop Paradox
Appendix B [to Book XXI]. Versions of the Barber-Shop Paradox
I. A Disputed Point in Logic. April 1894
II. A Disputed Point in Logic: A Concrete Example. April 11, 1894
III. A Disputed Point in Logic: A Concrete Example. April 16, 1894
IV. A Disputed Point in Logic. May 1, 1894
V. A Theorem in Logic. June 1894
VI. A Logical Paradox. July 1894
Note
VII. A Logical Puzzle. September 1894
Appendix C [to Book XXI]
Editor's Note on Carroll's "What the Tortoise Said to Achilles"
Exchange of Correspondence between Carroll and G. F. Stout
Letter from Carroll to John Cook Wilson, December 14, 1896

BOOK XXII SOLUTIONS TO PROBLEMS SET BY OTHER WRITERS
Chapter I Problems
Taken from the works of George Boole, Augustus DeMorgan, W. B. Grove, W. Stanley Jevons, John Neville Keynes, John Venn, and the Members of the Johns Hopkins University
Illustration from Carroll's Workbook, showing how he worked out some of the problems given