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The Mathematical Analysis of Logic Being an Essay Towards a Calculus of Deductive Reasoning   By: (1815-1864)

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Aristotle, Anal. Post., lib. i. cap. xi. CAMBRIDGE: MACMILLAN, BARCLAY, & MACMILLAN; LONDON: GEORGE BELL. 1847 PRINTED IN ENGLAND BY HENDERSON & SPALDING LONDON. W.I PREFACE. In presenting this Work to public notice, I deem it not irrelevant to ob serve, that speculations similar to those which it records have, at different periods, occupied my thoughts. In the spring of the present year my atten tion was directed to the question then moved between Sir W. Hamilton and Professor De Morgan; and I was induced by the interest which it inspired, to resume the almost forgotten thread of former inquiries. It appeared to me that, although Logic might be viewed with reference to the idea of quantity, it had also another and a deeper system of relations. If it was lawful to regard it from without, as connecting itself through the medium of Number with the intuitions of Space and Time, it was lawful also to regard it from within, as based upon facts of another order which have their abode in the constitution of the Mind. The results of this view, and of the inquiries which it suggested, are embodied in the following Treatise. It is not generally permitted to an Author to prescribe the mode in which his production shall be judged; but there are two conditions which I may venture to require of those who shall undertake to estimate the merits of this performance. The first is, that no preconceived notion of the impossibility of its objects shall be permitted to interfere with that candour and impartiality which the investigation of Truth demands; the second is, that their judgment of the system as a whole shall not be founded either upon the examination of only a part of it, or upon the measure of its conformity with any received system, considered as a standard of reference from which appeal is denied. It is in the general theorems which occupy the latter chapters of this work,results to which there is no existing counterpart,that the claims of the method, as a Calculus of Deductive Reasoning, are most fully set forth. What may be the final estimate of the value of the system, I have neither the wish nor the right to anticipate. The estimation of a theory is See p. 43. preface. 2 not simply determined by its truth. It also depends upon the importance of its subject, and the extent of its applications; beyond which something must still be left to the arbitrariness of human Opinion. If the utility of the application of Mathematical forms to the science of Logic were solely a question of Notation, I should be content to rest the defence of this attempt upon a principle which has been stated by an able living writer: \Whenever the nature of the subject permits the reasoning process to be without danger carried on mechanically, the language should be constructed on as mechanical principles as possible; while in the contrary case it should be so constructed, that there shall be the greatest possible obstacle to a mere mechanical use of it." In one respect, the science of Logic differs from all others; the perfection of its method is chie y valuable as an evidence of the speculative truth of its principles. To supersede the employment of common reason, or to subject it to the rigour of technical forms, would be the last desire of one who knows the value of that intellectual toil and warfare which imparts to the mind an athletic vigour, and teaches it to contend with difficulties and to rely upon itself in emergencies. Lincoln, Oct. 29, 1847. Mill's System of Logic, Ratiocinative and Inductive, Vol. ii. p. 292. MATHEMATICAL ANALYSIS OF LOGIC. INTRODUCTION. They who are acquainted with the present state of the theory of Sym bolical Algebra, are aware, that the validity of the processes of analysis does not depend upon the interpretation of the symbols which are em ployed, but solely upon the laws of their combination. Every system of interpretation which does not affect the truth of the relations supposed, is equally admissible, and it is thus that the same process may, under one scheme of interpretation, represent the solution of a question on the prop erties of numbers, under another, that of a geometrical problem, and under a third, that of a problem of dynamics or optics... Continue reading book >>

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