W. E. Johnson,Logic: Part II (1922)

## CONTENTS

## INTRODUCTION

§ 1. Application of the term 'substantive'

§ 2. Application of the term 'adjective'

§ 3. Terms 'substantive' and 'adjective' contrasted with 'particular' and 'universal'

§ 4. Epistemic character of assertive tie

§ 5. 'The given' presented under certain determinables

§ 6. The paradox of implication

§ 7. Defence of Mill's analysis of the syllogism§ 1. Implication denned as potential inference I ## CHAPTER I

INFERENCE IN GENERAL

§ 2. Inferences involved in the processes of perception and association

§ 3. Constitutive and epistemic conditions for valid inference. Examination of the ' paradox of inference'

§ 4. The Applicative and Implicative principles of inference

§ 5. Joint employment of these principles in the syllogism

§ 6. Distinction between applicational and implicational universals. The structural proposition redundant as minor premiss

§ 7. Definition of a logical category in terms of adjectival determinables

§ 8. Analysis of the syllogism in terms of assigned determinables. Further illustrations of applicational universals

§ 9. How identity may be said to be involved in every proposition

§ 10. The formal principle of inference to be considered redundant as major premiss. Illustrations from syllogism, induction, and mathematical equality

§ 11. Criticism of the alleged subordination of induction under the syllogistic principle§ 1. The Counter-applicative and Counter-implicative principles required for the establishment of the axioms of Logic and Mathematics ## CHAPTER II

THE RELATIONS OF SUB-ORDINATION AND CO-ORDINATION AMONGST PROPOSITIONS OF DIFFERENT TYPES

§ 2. Explanation of the Counter-applicative principle

§ 3. Explanation of the Counter-implicative principle

§ 4. Significance of the two inverse principles in the philosophy of thought

§ 5. Scheme of super-ordination, sub-ordination and co-ordination amongst propositions

§ 6. Further elucidation of the scheme§ 1. The value of symbolism. Illustrative and shorthand symbols. Classification of formal constants. Their distinction from material constants ## CHAPTER III

SYMBOLISM AND FUNCTIONS

§ 2. The nature of the intelligence required in the construction of a symbolic system

§ 3. The range of variation of illustrative symbols restricted within some logical category. Combinations of such symbols further to be interpreted as belonging to an understood logical category. Illustrations of intelligence required in working a symbolic system

§ 4. Explanation of the term ' function,' and of the ' variants' for a function

§ 5. Distinction between functions for which all the materia constituents are variable, and those for which only some are variable. Illustrations from logic and arithmetic

§ 6. The various kinds of elements of form in a construct

§ 7. Conjunctional and predicational functions

§ 8. Connected and unconnected sub-constructs

§ 9. The use of apparent variables in symbolism for the representation of the distributives every and some. Distinction between apparent variables and class-names

§ 10. Discussion of compound symbols which do and which do not represent genuine constructs

§ 11. Illustrations of genuine and fictitious constructs.

§ 12. Criticism of Mr Russell's view of the relation between propositional functions and the functions of mathematics

§ 13. Explanation of the notion of a descriptive function

§ 14. Further criticism of Mr Russell's account of propositional functions

§ 15. Functions of two or more variants§ 1. Technical terminology of syllogism ## CHAPTER IV

THE CATEGORICAL SYLLOGISM

§ 2. Dubious propositions to illustrate syllogism

§ 3. Relation of syllogism to antilogism

§ 4. Dicta for the first three figures derived from a single antilogistic dictum, showing the normal functioning of each figure

§ 5. Illustration of philosophical arguments expressed in syllogistic form

§ 6. Re-formulation of the dicta for syllogisms in which all the propositions are general

§ 7. The propositions of restricted and unrestricted form in each figure

§ 8. Special rules and valid moods for the first three figures

§ 9. Special rules and valid moods for the fourth figure

§ 10. Justification for the inclusion of the fourth figure in logical doctrine

§ 11. Proof of the rules necessary for rejecting invalid syllogisms

§ 12. Summary of above rules; and table of moods unrejected by the rules of quality

§ 13. Rules and tables of unrejected moods for each figure

§ 14. Combination of the direct and indirect methods of establishing the valid moods of syllogism

§ 15. Diagram representing the valid moods of syllogism

§ 16. The Sorites.

§ 17. Reduction of irregularly formulated arguments to syllogistic form

§ 18. Enthymemes

§ 19. Importance of syllogism§ 1. Deduction goes beyond mere subsumptive inference, when the major premiss assumes the form of a functional equation. Examples ## CHAPTER V

FUNCTIONAL EXTENSION OF THE SYLLOGISM

§ 2. A functional equation is a universal proposition of the second order, the functional formula constituting a Law of Co-variation

§ 3. The solutions of mathematical equations which yield single-valued functions correspond to the reversibility of cause and effect

§ 4. Significance of the number of variables 'entering into a functional formula

§ 5. Example of a body falling in vacuo.no

§ 6. The logical characteristics of connectional equations illustrated by thermal and economic equilibria

§ 7. The method of Residues is based on reversibility and is purely deductive

§ 8. Reasons why the above method has been falsely termed inductive

§ 9. Separation of the subsumptive from the functional elements in these extensions of syllogism§ 1. In the deduction of mathematical and logical formulae, new theorems are established for the difierent species of a genus, which do not hold for the genus ## CHAPTER VI

FUNCTIONAL DEDUCTION

§ 2. Explanation of the Aristotelean Uiov

§ 3. In functional deduction, the equational formulae are non-limiting. Elementary examples

§ 4. The range of universality of a functional formula varies with the number of independent variables involved. Employment of brackets. Importance of distinguishing between connected and disconnected compounds

§ 5. The functional nature of the formulae of algebra accounts for the possibility of deducing new and even wider formulae from previously established and narrower formulae, the Applicative Principle alone being employed

§ 6. Mathematical Induction

§ 7. The logic of mathematics and the mathematics of logic

§ 8. Distinction between premathematical and mathematical logic

§ 9. Formal operators and formal relations represented by shorthand and not variable symbols. Classification of the main formal relations according to their properties

§ 10. The material variables of mathematical and logical symbolisation receive specific values only in concrete science

§ 11. Discussion of the Principle of Abstraction

§ 12. The specific kinds of magnitude are not determinates of the single determinable Magnitude, but are incomparable

§ 13. The logical symbolic calculus establishes formulae of implication which are to be contrasted with the principles of inference employed in the procedure of building up the calculus§ 1. The terms 'greater' and 'less' predicated of magnitude, 'larger' and 'smaller' of that which has magnitude ## CHAPTER VII

THE DIFFERENT KINDS OF MAGNITUDE

§ 2. Integral number as predicable of classes or enumerations

§ 3. Psychological exposition of counting

§ 4. Logical principles underlying counting

§ 5. One-one correlations for finite integers

§ 6. Definition of extensive magnitude

§ 7. Adjectival stretches compared with substantival

§ 8. Comparison between extensive and extensional wholes

§ 9. Discussion of distensive magnitudes

§ 10. Intensive magnitude

§ 11. Fundamental distinction between distensive and intensive magnitudes.

§ 12. The problem of equality of extensive wholes .

§ 13. Conterminus spatial and temporal wholes to be considered equal, qualitative stretches only comparable by causes or effects

§ 14. Complex magnitudes derived by combination of simplex

§ 15. The theory of algebraical dimensions.

§ 16. The special case in which dividend and divisor are quantities of the same kind

§ 17. Summary of the above treatment of magnitude§ 1. The general antithesis between induction and deduction ## CHAPTER VIII

INTUITIVE INDUCTION

§ 2. The problem of abstraction .

§ 3. The principle of abstractive or intuitive induction

§ 4. Experiential and formal types of intuitive induction

§ 5. Intuitive induction involved in introspective and ethical judgments

§ 6. Intuitive inductions upon sense-data and elementary algebraical and logical relations

§ 7. Educational importance of intuitive induction§ 1. Summary induction reduced to first figure syllogism ## CHAPTER IX

SUMMARY INCLUDING GEOMETRICAL INDUCTION

§ 2. Summary induction as establishing the premiss for induction proper. Criticism of Mill's and Whewell's views

§ 3. Summary induction involved in geometrical proofs

§ 4. Explanation of the above process

§ 5. Function of the figure in geometrical proofs

§ 6. Abuse of the figure in geometrical proofs

§ 7. Criticism of Mill's'parity of reasoning'§ 1. Demonstrative induction uses a composite along with an instantial premiss ## CHAPTER X

DEMONSTRATIVE INDUCTION

§ 2. Illustrations of demonstrative arguments leading up to demonstrative induction

§ 3. Conclusions reached by the conjunction of an alternative with a disjunctive premiss

§ 4. The formula of direct universalisation

§ 5. Scientific illustration of the above

§ 6. Proposed modification of Mill's exposition of the methods of induction

§ 7. The major premiss for demonstrative induction as an expression of the dependence in the variations of one phenomenal character upon those of others

§ 8. The four figures of demonstrative induction

§ 9. Figure of Difference

§ 10. Figure of Agreement

§ 11. Figure of Composition

§ 12. Figure of Resolution

§ 13. The Antilogism of Demonstrative Induction

§ 14. Illustration of the Figure of Difference

§ 15. Illustration of the Figure of Agreement

§ 16. Principle for dealing with cases in which a number both of cause-factors and effect-factors are considered, with a symbolic example

§ 17. Modification of symbolic notation in the figures where different cause-factors represent determinates under the same determinable

§ 18. The striking distinction between the two last and the two first figures.

§ 19. Explanation of the distinction between composition and combination of cause-factors

§ 20. Illustrations of the figures of Composition and Resolution§ 1. The major premiss for Demonstrative Induction must have been established by Problematic Induction ## CHAPTER XI

THE FUNCTIONAL EXTENSION OF DEMONSTRATIVE INDUCTION

§ 2. Contrast between my exposition and Mill's

§ 3. The different uses of the term ' hypothesis' in logic

§ 4. Jevons's confusion between the notions '.problematic' and 'hypothetical'

§ 5. The establishment of a functional formula for the figures of Difference and of Composition

§ 6. The criteria of simplicity and analogy for selection of the functional formula

§ 7. A comparison of these criteria with similar criteria proposed by Whewell and Mill

§ 8. Technical mathematical methods for determining the most probable formula## INDEX