The Mathematical Analysis of Logic

ISBN 13: 9781519118660
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  • The mathematical analysis of logic | SpringerLink

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    The standard notation xy for multiplication also had a standard meaning for operators for example, differential operators , namely one applied y to an object and then x is applied to the result. In modern terminology, this is the composition of the two operators …. This was the first mention of addition. Once these two laws were secured, Boole believed he was entitled to fully employ the ordinary algebra of his time, and indeed one sees Taylor series and Lagrange multipliers in MAL.

    But it is not the case that every pair of classes is disjoint. This is the first appearance of subtraction. Then he gave equations to express the categorical propositions. This was the first appearance of 0 in MAL —it was not introduced as the symbol for the empty class. Indeed the empty class did not appear in MAL ….

    Up to this point his sole focus was to show that Aristotelian logic could be handled by simple algebraic methods, mainly through the use of an elimination theorem borrowed from ordinary algebra.

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    Fortunately for Boole, the situation in his algebra of logic was much simpler—he could always solve an equation, and finding the solution was important to applications of his system, to derive conclusions in logic. An equation was solved in part by using expansion after performing division.

    Seven years after publishing the present work, Boole gave a more elaborate treatment of Boolean algebra in An Investigation of the Laws of Thought. His aim was partly to address criticisms of the earlier work by de Morgan and others; he also considered the application of his theory to probability, a topic not treated in the earlier work. He was born to an intelligent tradesman who however was so poor that George had to become the main breadwinner in his 20th year when he opened a school.

    Nevertheless, he found time to teach himself advanced mathematics, and also Greek, Latin, French and German, especially in order to read important works. Skip to main content.

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    Free kindle book and epub digitized and proofread by Project Gutenberg. Excerpt from The Mathematical Analysis of Logic: Being an Essay Towards a Calculus of Deductive Reasoning In presenting this Work to public notice, I deem it.

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    The mathematical analysis of logic | SpringerLink

    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.