Symbolic Logic

CHAPTER III.. _propositions of Relation._

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

" " "No" "

" " "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._

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' 24

=Table I.= Attributes of Classes, and Compartments, or Cells, assigned to them 25

pg-xx

CHAPTER II.. _counters._

COUNTERS.

Meaning of a Red Counter placed in a Cell 26

" " " " on a Partition "

American phrase "=sitting on the fence=" "

Meaning of a Grey Counter placed in a Cell "

CHAPTER III.. _representation of Propositions._

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'" "

Seven other similar Propositions 34

=Tables II, III.= Representation of Propositions of Existence and Relation 34, 35

CHAPTER IV.. _interpretation of Biliteral Diagram, When Marked With Counters._

INTERPRETATION OF BILITERAL DIAGRAM, WHEN MARKED WITH COUNTERS.

·-------· |(.)| | Interpretation of |---|---| 36 | | | ·-------·

And of three other similar arrangements " pg-xxii ·-------· |( )| | Interpretation 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

=BOOK IV.=

=THE TRILITERAL DIAGRAM.=

CHAPTER I.. _symbols and Cells._

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

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 "

=Tables V, VI, VII, VIII.= Representations of Propositions in terms of x and m, or of y and m 46 to 49

pg-xxiv

CHAPTER III.. _representation of Two Propositions

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,

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._

INTRODUCTORY.

'=Syllogism=' 56

'=Premisses=' "

'=Conclusion=' "

'=Eliminands=' 57

'=Retinends=' "

'=Consequent=' "

The Symbol ".'." "

Specimen-Syllogisms "

pg-xxv

CHAPTER II.. _problems in Syllogisms._

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 "

pg-xxvi =BOOK VI.=

=THE METHOD OF SUBSCRIPTS.=

CHAPTER I.. _introductory._

INTRODUCTORY.

Meaning of x{1}, xy{1}, &c. 70

'=Entity=' "

Meaning of x{0}, xy{0}, &c. "

'=Nullity=' "

The Symbols "+" and "¶" "

'=Like=' and '=unlike=' Signs "

CHAPTER II.. _representation of Propositions of Relation._

REPRESENTATION OF PROPOSITIONS OF RELATION.

The Pair of Converse Propositions "Some x are y" = "Some y are x" 71

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" 72

The Proposition "All x are y" is Double, and is equivalent to the two Propositions "Some x exist" and "No x and y'" "

Seven other similar Propositions "

Rule for translating "All x are y" from abstract into subscript form, and vice versâ "

pg-xxvii

CHAPTER III.. _syllogisms._

SYLLOGISMS.

§ 1.

Representation of Syllogisms.

Rules 73

§ 2.

Formulæ for Syllogisms.

Three Formulæ worked out:--

Fig. I. xm{0} + ym'{0} ¶ xy_{0} 75

its two Variants (a) and (b) "

Fig. II. xm{0} + ym{1} ¶ x'y_{1} 76

Fig. III. xm{0} + ym{0} + m{1} ¶ x'y'{1} 77

=Table IX.= Formulæ and Rules 78

Examples worked briefly, as models "

§ 3.

Fallacies.

'=Fallacy=' 81

Method of finding Forms of Fallacies 82

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 83

(3) Fallacy of two Entity-Premisses "

§ 4.

Method of proceeding with a given Pair of Propositions.

Rules 84

pg-xxviii =BOOK VII.=

=SORITESES.=

CHAPTER I.. _introductory._

INTRODUCTORY.

'=Sorites=' 85

'=Premisses=' "

'=Partial Conclusion=' "

'=Complete Conclusion=' (or '=Conclusion=') "

'=Eliminands=' "

'=Retinends=' "

'=consequent=' "

The Symbol ".'." "

Specimen-Soriteses 86

CHAPTER II.. _problems in Soriteses._

PROBLEMS IN SORITESES.

§ 1.

Introductory.

Form of Problem 87

Two Methods of Solution "

§ 2.

Solution by Method of Separate Syllogisms.

Rules 88

Example worked "

pg-xxix § 3.

Solution by Method of Underscoring.

'=Underscoring=' 91

Subscripts to be omitted "

Example worked fully 92

Example worked briefly, as model 93

Seventeen Examination-Papers 94

=BOOK VIII.=

=EXAMPLES, WITH ANSWERS AND SOLUTIONS.=

CHAPTER I.. _examples._

EXAMPLES.

§ 1.

Propositions of Relation, to be reduced to normal form 97

§ 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 98

§ 3.

Marked Triliteral Diagrams, to be interpreted in terms of x and y 99

§ 4.

Pairs of Abstract Propositions, proposed as Premisses: Conclusions to be found 100

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