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

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## William Warren Bartley, III:

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

Published $\text {1977}$, **Harvester Press**

- ISBN 0-85527-984-2.

### Subject Matter

### Contents

- Note to the Reader From the Editor and Publisher

#### Part One: Elementary

- Introduction to Learners
- Preface to Fourth Edition

**BOOK I**THINGS AND THEIR ATTRIBUTES

- Chapter I Introductory
**Things****Attributes****Adjuncts**

- Chapter I Introductory

- 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 II Classification

- Chapter III Division
- $[\S 1]$
*Introductory***Division****Codivisional**Classes

- $[\S 2]$
*Dichotomy***Dichotomy**- Arbitrary limits of Classes
- Subdivision of Classes

- $[\S 1]$

- Chapter III Division

- 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 IV Names

- Chapter V Definitions
**Definition**- Examples worked as models

- Chapter V Definitions

**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

- (1)

- Its four parts :
- $[\S 3]$
*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**

- (1) Begins with "Some." Called 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

- Three kinds of Propositions :

- $[\S 1]$

- Chapter I Propositions Generally

- Chapter II Propositions of Existence
**Proposition of Existence**

- Chapter II Propositions 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

- Its equivalence to
- $[\S 4]$
*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

- $[\S 1]$

- Chapter III Propositions of Relation

**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 I Symbols and Cells

- 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 II Counters

- 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

- $[\S 1]$

- Chapter III Representation of 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

- Chapter IV Interpretation of Biliteral Diagram, When Marked with Counters

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- 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 I Symbols and Cells

- 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 $x$

- The Pair of Converse Propositions
- 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 $m$

- The Pair of Converse Propositions

- $[\S 1]$

- Chapter II 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 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

- Chapter IV Interpretation, in Terms of $x$ and $y$, of Triliteral Diagram, When Marked with Counters or Digits
- Rules
- Examples worked

- Chapter IV Interpretation, in Terms of $x$ and $y$, of Triliteral Diagram, When Marked with Counters or Digits

**BOOK V**SYLLOGISMS

- Chapter I Introductory
**Syllogism****Premisses****Conclusion****Eliminands****Retinends****Consequent**- The Symbol $\therefore$
- Specimen-Syllogisms

- Chapter I Introductory

- 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

- $[\S 1]$

- Chapter II Problems in Syllogisms

**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 1 Introductory

- 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*

- The Pair of Converse Propositions

- Chapter II Representation of Propositions of Relation

- 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

- Three Formulae worked out:
- $[\S 3]$
*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

- $[\S 1]$

- Chapter III Syllogisms

**BOOK VII**SORITESES

- Chapter I Introductory
**Sorites****Premisses****Partial Conclusion****Complete Conclusion**(or**Conclusion**)**Eliminands****Retinends****Consequent**- The Symbol $\therefore$
- Specimen-Soriteses

- Chapter I Introductory

- 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

- $[\S 1]$

- Chapter II Problems in Soriteses

**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*

- $[\S 1]$

- Chapter I Examples

- Chapter II Answers
- Answers to
- $[\S 1]$
- $[\S 2]$
- $[\S 3]$
- $[\S 4]$
- $[\S 5]$
- $[\S\S 6, 7]$
- $[\S\S 8, 9]$

- Answers to

- Chapter II Answers

- 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

- Solutions for
- $[\S 3]$
*Method of Subscripts*- Solutions for
- $[\S 4]$
- $[\S 5]$ Nos. 13-24
- $[\S 5]$ Nos. 1-12 and 25-101
- $[\S 6]$
- $[\S 7]$
- $[\S 8]$
- $[\S 9]$

- Solutions for

- $[\S 1]$

- Chapter III Solutions

#### Part Two: Advanced

**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 II The Existential Import of Propositions

- 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 VIII Solution of a Syllogism by Various Methods

- 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

- Chapter IV 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

- Chapter III Sorites-Problems with Triliteral and Multiliteral Premisses

- 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 II Problems in Sequences

- Chapter III The Problem of the School-Boys
- Some answers to the Problem of the Schoolboys
- Tree I
- Tree II
- Tree III

- Some answers to the Problem of the Schoolboys

- Chapter III The Problem of the School-Boys

- 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 IV The Pork-Chop Problem

- Chapter V Froggy's Problem
- Dictionary for Froggy's Problem

- Chapter V 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

- The Solution to the Members of Parliament Problem

- Chapter VI The Members of Parliament Problem

- Chapter VII The Problem of Six Friends and their Wives
- Version I
- Version II

- Chapter VII The Problem of Six Friends and their Wives

- 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

- The Problem of the Five Liars

- Chapter IX The Great-Grandson Problem
- Solution to the Great-Grandson Problem
- Carroll's Letters to John Cook Wilson about this Problem
- I. February 16, 1897
- II. May 17, 1897

- Chapter IX The Great-Grandson Problem

- 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 X 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

- Carroll's Letters to John Cook Wilson relevant to the Library Problem

- Chapter XI The Library Problem

- 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 II Classical Puzzles

- Chapter III Other Puzzles
- 1. About Less
- 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 III Other Puzzles

- Chapter IV Solutions of Classical Puzzles
- 1. Introductory
- 2. Pseudomenos
- 3. Crocodiles
- 4. Antistrephon
- 5. Achilles
- 6. Raw Meat

- Chapter IV Solutions of Classical Puzzles

- Chapter V Solutions of Other Puzzles
- 1 . About Less
- 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

- Chapter V Solutions of Other Puzzles

- Appendix A [to Book XXI]
- Editor's Note on Carroll's Barber-Shop Paradox

- Appendix A [to Book XXI]

- 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 B [to Book XXI]. Versions of the Barber-Shop Paradox

- 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

- Appendix C [to Book XXI]

**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

- Chapter I Problems