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actual admit adopted affirmative already alternatives answer apply assertion assigned attributes called chapter class terms combinations common compartments complete conclusion condition consequent considered consistency contained corresponding course described determine diagram direct discussion distinct elements elimination employed equally equation equivalent example exclusive existence expression extension fact familiar figures formal former four give given hypothetical identical import included indefinite indicate instance interpretation introduced involved kind language latter limits logical logicians mark mathematical means merely multiply mutual nature negative notation notice object obtain once operation ordinary particular positive possible predicate premises present problem proposed propositions question reader reason reference regarded relation represent respect result rule scheme seems seen simple stand statement Suppose symbols things tion true universe whole write yield
Page 122 - ... method of employing the diagrams in order to express propositions will readily be understood. It is merely this : — Ascertain what each given proposition denies, and then put some kind of mark upon the corresponding partition in the figure. The most effective means of doing this is just to shade it out. For instance, the proposition " All X is Y " is interpreted to mean that there is no such class of things in existence as " X that is not-Y " or XY. All, then, that we have to do is to scratch...
Page 133 - ... illustrate our rules and methods. In this respect logical calculations stand in marked contrast with those of mathematics, where economical devices of any kind may subserve a really valuable purpose by enabling us to avoid otherwise inevitable labour. Moreover, in the second place, it does not seem to me that any contrivances at present known or likely to be discovered really deserve the name of logical machines. It is but a very small part of the entire process, which goes to form a piece of...
Page 117 - ... ie its four component portions are inside b and d but are no part of c). It must be admitted that such a diagram is not quite so simple to draw as one might wish it to be; but then consider what the alternative is if one undertakes to deal with five terms and all their combinations — nothing short of the disagreeable task of writing out, or in some way putting before us, all the 32 combinations involved.
Page 345 - ... necessary that our conclusions should be limited to relations between terms given in the premises, as may be seen in the solution of the following problems. PROBLEM I. Let us suppose that there are four girls at school, Anna, Bertha, Cora, and Dora, and that some one had observed that (1.) Whenever either Anna or Bertha (or both) remained at home, Cora was at home ; and (2.) When Bertha was out, Anna was out ; and (3.) Whenever Cora was at home, Anna was at home. What information is here conveyed...
Page 134 - ... even in the few examples here offered. Thirdly, there is the combination or further treatment of our premises after such reduction. Finally, the results have to be interpreted or read off. This last generally gives rise to much opening for skill and sagacity ; for though in such examples as the last (in which one class, Y, was simply abolished) there is but one answer fairly before us, yet in most cases there are many ways of reading off the answer. It then becomes a question of judgment which...
Page 118 - We have endeavoured above to employ only symmetrical figures, such as should not merely be an aid to the sense of sight, but should also be to some extent elegant in themselves. But for merely theoretical purposes the rule of formation would be very simple. It would merely be to begin by drawing any closed figure, and then proceed to draw others, subject to the one condition that...
Page 126 - Z. This would commonly be exhibited thus: It is easy enough to do this ; for in drawing our circles we have only to attend to two terms at a time, and consequently the relation of X to Z is readily detected; there is not any of that troublesome interconnexion of a number of terms simultaneously with each other which gives rise to the main perplexity in complicated problems.
Page 143 - It thus comes about that we can say such things as that Ivanhoe did not really marry Rebecca, as Thackeray falsely makes him do. The real Ivanhoe-world is the one which Scott wrote down for us. In that world Ivanhoe does not marry Rebecca.