Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic

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Welch, Bigelow, 1870 - Algebra, Boolean - 62 pages
 

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Page 28 - The absolute individual can not only not be realized in sense or thought, but can not exist, properly speaking. For whatever lasts for any time, however short, is capable of logical division, because in that time it will undergo some change in its relations. But what does not exist for any time, however short, does not exist at all. All, therefore, that we perceive or think, or that exists, is general.
Page 28 - For whatever lasts for any time, however short, is capable of logical division, because in that time it will undergo some change in its relations. But what does not exist for any time, however short, does not exist at all. All, therefore, that we perceive or think, or that exists, is general. So far there is truth in the doctrine of scholastic realism. But all that exists is infinitely determinate, and the infinitely determinate is the absolutely individual. This seems paradoxical, but the contradiction...
Page 48 - It is very likely that this is true of all algebras whatever. The algebra of (156), which is of such a fundamental character in reference to pure algebra and our logical notation, has been shown by Professor [Benjamin] Peirce to be the algebra of Hamilton's quaternions.
Page 6 - ... denotation. These discriminate objects with a distinct consciousness of discrimination. They regard an object as over against another, that is as relative; as father of, lover of, or servant of. These are simple relative terms. The third class embraces terms whose logical form involves the conception of bringing things into relation, and which require the addition of more than one term to complete the denotation. They discriminate not only with consciousness of discrimination, but with consciousness...
Page 27 - In thought, an absolutely determinate term cannot be realized, because, not being given by sense, such a concept would have to be formed by synthesis, and there would be no end to the synthesis because there is no limit to the number of possible predicates. A logical atom, then, like a point in space, would involve for its precise determination an endless process. We can only say, in a general way, that a term, however determinate, may be made more determinate still, but not that it can be made absolutely...
Page 6 - ... a — ." These discriminate objects in the most rudimentary way, which does not involve any consciousness of discrimination. They regard an object as it is in itself as such (quale); for example, as horse, tree, or man. These are absolute terms. The second class embraces terms whose logical form involves the conception of relation, and which require the addition of another term to complete the denotation.
Page 1 - ... daily use. Just as in ordinary logic existence is implicitly predicated for all the terms, so in this subject every relation employed will be considered as actually connecting the terms of which it is predicated. Let X..LY signify that X is some one of the objects of thought which stand to Y in the relation L, or is one of the Ls of Y.
Page 2 - To say that x = y is to say that x — c^ y and y — <^ x. Being less than is being as small as with the exclusion of its converse. To say that x < y is to say that x — <^ y, and that it is not true that y — <^ x. Being greater than is the converse of being less . than. To say that x^> y is to say that y < x. ADDITION is an associative operation.
Page 62 - Application of the Algebraic Signs to Logic," together with those relating to backward involution, and the principles expressed by equations (95), (96), (122), (142), (156), (25), (26), (14), (15), (169), (170). But these axioms are mere substitutes for definitions of the universal logical relations, and so far as these can be defined, all axioms may be dispensed with. The fundamental principles of formal logic are not properly axioms, but definitions and divisions ; and the only facts which it contains...
Page 1 - He there uses a convenient algebraic notation, which is formed by adding to the well-known spiculce of that writer the signs used in the following examples. X . . LY signifies that X is some one of the objects of thought which stand to Y in the relation L, or is one of the L's of Y. X . LMY signifies that X is not an L of an M of Y. X . . (L,M)Y signifies that X is either an L or an M of...

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