By Bertrand Russell
This is often Russell's first philosophical paintings released in 1897. The publication presents an perception into his earliest analytical and significant proposal, in addition to an advent to the philosophical and logistical foundations of non-Euclidean geometry, a model of that's important to Einstein's thought of relativity.
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Extra resources for An essay on the foundations of geometry
The contradiction lies in the last sentence: if we substitute for "as long as" the word "if" (equivalent in this context), and for "It is not possible for a man to become white" the synonymous "Man belongs necessarily to no white", then we are back with the necessity of the proposition "Man belongs to no white", which Aristotle has just expressly rejected. It is not correct to say that man can belong to no white as long as animal belongs to no white. Of course man even then can belong to a white - otherwise it would be quite impossible for any animal to become white (and the second premiss would be, contra hypothesin, 'absolutely' necessary).
And later: "The schema of necessity is the existence of an object at all times" (ib. B 184). The reference to the passages in Aristotle and to the two statements of Kant is not inappropriate in view of the objection often raised against enquiries of this sort, that they measure Aristotle and his logical theories by canons utterly strange to him - canons which 27 ARISTOTLE'S THEORY OF THE SYLLOGISM indeed can only be drawn from the remote province of modern mathematicallogic. We have seen that Aristotle's 'relative' necessity is not the necessity of the conclusion of a valid inference, but the necessary truth of the compound proposition which, at the beginning of the book, we established an Aristotelian syllogism to be.
But it does not fulfill the second condition, since sleep does not belong to man lCae' au'to, by virtue of his definition. g. a term cannot be substituted for the objectvariable x, since the proposition would then be not true but meaningless). 21 In place of the proposition "For all objects x at time t: if x is a man then x is asleep", we can construct the equivalent proposition: "For all objects x: if x is a man and x exists at time t, then x is asleep". In the second of these two 34 LOGICAL NECESSITY propositions the quantifier is genuine; the restriction has been turned into an additional qualification on the antecedent of the implication "If Ax, then Bx".
An essay on the foundations of geometry by Bertrand Russell