Book contents
- Frontmatter
- PREFACE
- Contents
- CHAPTER I NATURE AND DESIGN OF THIS WORK
- CHAPTER II SIGNS AND THEIR LAWS
- CHAPTER III DERIVATION OF THE LAWS
- CHAPTER IV DIVISION OF PROPOSITIONS
- CHAPTER V PRINCIPLES OF SYMBOLICAL REASONING
- CHAPTER VI OF INTERPRETATION
- CHAPTER VII OF ELIMINATION
- CHAPTER VIII OF REDUCTION
- CHAPTER IX METHODS OF ABBREVIATION
- CHAPTER X CONDITIONS OF A PERFECT METHOD
- CHAPTER XI OF SECONDARY PROPOSITIONS
- CHAPTER XII METHODS IN SECONDARY PROPOSITIONS
- CHAPTER XIII CLARKE AND SPINOZA
- CHAPTER XIV EXAMPLE OF ANALYSIS
- CHAPTER XV OF THE ARISTOTELIAN LOGIC
- CHAPTER XVI OF THE THEORY OF PROBABILITIES
- CHAPTER XVII GENERAL METHOD IN PROBABILITIES
- CHAPTER XVIII ELEMENTARY ILLUSTRATIONS
- CHAPTER XIX OF STATISTICAL CONDITIONS
- CHAPTER XX PROBLEMS ON CAUSES
- CHAPTER XXI PROBABILITY OF JUDGMENTS
- CHAPTER XXII CONSTITUTION OF THE INTELLECT
- ERRATA
CHAPTER IX - METHODS OF ABBREVIATION
Published online by Cambridge University Press: 05 November 2011
- Frontmatter
- PREFACE
- Contents
- CHAPTER I NATURE AND DESIGN OF THIS WORK
- CHAPTER II SIGNS AND THEIR LAWS
- CHAPTER III DERIVATION OF THE LAWS
- CHAPTER IV DIVISION OF PROPOSITIONS
- CHAPTER V PRINCIPLES OF SYMBOLICAL REASONING
- CHAPTER VI OF INTERPRETATION
- CHAPTER VII OF ELIMINATION
- CHAPTER VIII OF REDUCTION
- CHAPTER IX METHODS OF ABBREVIATION
- CHAPTER X CONDITIONS OF A PERFECT METHOD
- CHAPTER XI OF SECONDARY PROPOSITIONS
- CHAPTER XII METHODS IN SECONDARY PROPOSITIONS
- CHAPTER XIII CLARKE AND SPINOZA
- CHAPTER XIV EXAMPLE OF ANALYSIS
- CHAPTER XV OF THE ARISTOTELIAN LOGIC
- CHAPTER XVI OF THE THEORY OF PROBABILITIES
- CHAPTER XVII GENERAL METHOD IN PROBABILITIES
- CHAPTER XVIII ELEMENTARY ILLUSTRATIONS
- CHAPTER XIX OF STATISTICAL CONDITIONS
- CHAPTER XX PROBLEMS ON CAUSES
- CHAPTER XXI PROBABILITY OF JUDGMENTS
- CHAPTER XXII CONSTITUTION OF THE INTELLECT
- ERRATA
Summary
ON CERTAIN METHODS OF ABBREVIATION.
1. Though the three fundamental methods of development, elimination, and reduction, established and illustrated in the previous chapters, are sufficient for all the practical ends of Logic, yet there are certain cases in which they admit, and especially the method of elimination, of being simplified in an important degree; and to these I wish to direct attention in the present chapter. I shall first demonstrate some propositions in which the principles of the above methods of abbreviation are contained, and I shall afterwards apply them to particular examples.
Let us designate as class terms any terms which satisfy the fundamental law V (1 – V) = 0. Such terms will individually be constituents; but, when occurring together, will not, as do the terms of a development, necessarily involve the same symbols in each. Thus ax + bxy + cyz may be described as an expression consisting of three class terms, x, xy, and yz, multiplied by the coefficients a, b, c respectively. The principle applied in the two following Propositions, and which, in some instances, greatly abbreviates the process of elimination, is that of the rejection of superfluous class terms; those being regarded as superfluous which do not add to the constituents of the final result.
PROPOSITION I.
2. From any equation, V= 0, in which V consists of a series of class terms having positive coefficients, we are permitted to reject any term which contains another term as a factor, and to change every positive coefficient to unity.
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- An Investigation of the Laws of ThoughtOn Which Are Founded the Mathematical Theories of Logic and Probabilities, pp. 130 - 149Publisher: Cambridge University PressPrint publication year: 2009First published in: 1854