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208 31 [Deductions from a Definition of the Copula] Spring 1891 Houghton Library The definition of the copula of inclusion is embraced in three propositions , to wit: DEDUCTIONS FROM THIS DEFINITION (1) The nonscriptibility of no formula can be deduced as a necessary result of this definition. For the nonscriptibility of A R B depends upon that of B which must therefore be given in addition to the definition. (2) The necessary scriptibility of a formula may however result from I and II. For that purpose A and B must be replaced by such formulae that if the A-formula is scriptible, the B-formula is likewise scriptible. This may be so independent of any property of the copula. Thus, the A and B may be identical. A R A is scriptible by I and II. Next, it may depend on I alone. If Z is scriptible, Y R Z is scriptible and consequently X R (Y R Z) is scriptible and so on indefinitely. Next it may depend on II and III. Thus, if A R (A R B) is scriptible, A R B is scriptible. For by III, if A R (A R B) and A are scriptible, A R B is scriptible, and if A is not scriptible by II, A R B is scriptible. Next, it may depend on I, II, and III. Thus, if A R (B R C) is scriptible, B R (A R C) is scriptible. For if A R (B R C) is scriptible and A is not scriptible, then by I and II, B R (A R C) is scriptible. But if A R (B R C) and A are both scriptible, then by III, B R C is scriptible. And if B is not scriptible by II, B R (A R C) is scriptible, but if B R C and B are both scriptible, then by III, C is scriptible and if C is scriptible by I, B R (A R C) is scriptible. I. If B is scriptible, A R B is scriptible. II. Either A R B or A is scriptible. III. If both A R B and A are scriptible, B is scriptible. 31. Deductions from Definition of Copula, 1891 209 The following also depends on all three. If A is scriptible, then (A R B) R B is scriptible. For if B is scriptible by I, (A R B) R B is scriptible. But by II, either (A R B) R B or A R B is scriptible, and by III, if both A and A R B are scriptible, B is scriptible. The following depends on all three. If (A R B) R B is scriptible, so is (B R A) R A. For by II, either A R B or A is scriptible. If A, then by I, so is (B R A) R A. If A R B is scriptible and also (A R B) R B, then by III, B is scriptible. If B is scriptible and A not, then by III, B R A is not scriptible and then by II, (B R A) R A is scriptible. ...

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