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Ebook has 224 lines and 27290 words, and 5 pages
These matters are as follows:--
? 2.
And so, with regard to the question whether a Proposition is or is not to be understood as asserting the existence of its Subject, I maintain that every writer may adopt his own rule, provided of course that it is consistent with itself and with the accepted facts of Logic.
"All m are x; All m are y. .'. Some y are x". pg169 This they would interpret as follows:--
"If there were any m in existence, all of them would be x; If there were any m in existence, all of them would be y. .'. If there were any y in existence, some of them would be x".
"Taking S, M, P, as the minor, middle, and major terms respectively, the conclusion will imply that, if there is an S, there is some P. Will the premisses also imply this? If so, then the syllogism is valid; but not otherwise.
"The conclusion implies that if S exists P exists; but, consistently with the premisses, S may be existent while M and P are both non-existent. An implication is, therefore, contained in the conclusion, which is not justified by the premisses."
Let us suppose that a Boys' School has been set up, with the following system of Rules:--
"All boys in the First Class are to do French, Greek, and Latin. All in the Second Class are to do Greek only. All in the Third Class are to do Latin only."
"If there were any boys doing French, all of them would be doing Greek; If there were any boys doing French, all of them would be doing Latin."
And the Conclusion, according to "The Logicians" would be
"If there were any boys doing Latin, some of them would be doing Greek."
Some further remarks on this subject will be found in Note , at p. 196.
? 3.
pg173 ? 4.
Take the following Pairs of Premisses:--
"None of my boys are conceited; None of my girls are greedy".
"None of my boys are clever; None but a clever boy could solve this problem".
"None of my boys are learned; Some of my boys are not choristers".
? 5.
Similarly, with this Diagram, the following Propositions are true:--"All y are x", "No y are not-x", "Some y are x", "Some x are not-y", "Some not-x are not-y", and, of course, the Converses of the last four.
Similarly, with this Diagram, the following are true:--"All x are not-y", "All y are not-x", "No x are y", "Some x are not-y", "Some y are not-x", "Some not-x are not-y", and the Converses of the last four.
Similarly, with this Diagram, the following are true:--"Some x are y", "Some x are not-y", "Some not-x are y", "Some not-x are not-y", and of course, their four Converses.
? 6.
Let us represent "not-x" by "x'".
Mr. Venn's Method of Diagrams is a great advance on the above Method.
Thus, he would represent the three Propositions "Some x are y", "No x are y", and "All x are y", as follows:--
? 7.
? 8.
The best way, I think, to exhibit the differences between these various Methods of solving Syllogisms, will be to take a concrete example, and solve it by each Method in turn. Let us take, as our example, No. 29 .
"No philosophers are conceited; Some conceited persons are not gamblers. .'. Some persons, who are not gamblers, are not philosophers."
These Premisses, as they stand, will give no Conclusion, as they are both negative.
If by 'Permutation' or 'Obversion', we write the Minor Premiss thus,
'Some conceited persons are not-gamblers,'
"No philosophers are conceited; Some conceited persons are not-gamblers. .'. Some not-gamblers are not philosophers"
"No conceited persons are philosophers; Some not-gamblers are conceited. .'. Some not-gamblers are not philosophers".
Then the Syllogism may be written thus:--
"No x are m; Some m are y'. .'. Some y' are x'."
Figs. 1 and 2 give
Figs. 1 and 3 give
Figs. 1 and 4 give
From this group we have, by disregarding m, to find the relation of x and y. On examination we find that Figs. 5, 10, 13 express the relation of entire mutual exclusion; that Figs. 6, 11 express partial inclusion and partial exclusion; that Fig. 7 expresses coincidence; that Figs. 8, 12 express entire inclusion of x in y; and that Fig. 9 expresses entire inclusion of y in x. pg182 We thus get five Biliteral Diagrams for x and y, viz.
The following Solution has been kindly supplied to me Mr. Venn himself.
"The Minor Premiss declares that some of the constituents in my' must be saved: mark these constituents with a cross.
The Major declares that all xm must be destroyed; erase it.
Then, as some my' is to be saved, it must clearly be my'x'. That is, there must exist my'x'; or eliminating m, y'x'. In common phraseology,
'Some y' are x',' or, 'Some not-gamblers are not-philosophers.'"
The first Premiss asserts that no xm exist: so we mark the xm-Compartment as empty, by placing a 'O' in each of its Cells.
The second asserts that some my' exist: so we mark the my'-Compartment as occupied, by placing a 'I' in its only available Cell.
Hence "Some x' are y'": i.e. "Some persons, who are not philosophers, are not gamblers".
i.e. "Some persons, who are not philosophers, are not gamblers."
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