Library Notices and Immediate Inference
The Class XII library committee posts three rules before the annual book fair. Rule A: 'All reference books kept on the reserve shelf are books that must be read inside the library.' Rule B: 'No periodicals are books that may be borrowed overnight.' Rule C: 'Some donated books are not books catalogued in the main index.' A volunteer prepares four notices: (i) 'No reference books kept on the reserve shelf are books that may be read outside the library.' (ii) 'No books that may be borrowed overnight are periodicals.' (iii) 'All books that may not be read inside the library are books not kept on the reserve shelf.' (iv) 'Some donated books are books not catalogued in the main index; therefore some books not catalogued in the main index are donated books.' The librarian will sign only notices that follow by a named immediate inference and will reject any that depend on an illicit process. A second volunteer asks which terms are distributed in Rules A and C.
1. State which terms are distributed in Rules A and C, and explain the distribution rule used.
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Rule A is an A proposition: All R are I, where R = reference books kept on the reserve shelf and I = books that must be read inside the library. In an A proposition the subject is distributed and the predicate is undistributed. So R is distributed; I is undistributed.
Rule C is an O proposition: Some D are not C, where D = donated books and C = books catalogued in the main index. In an O proposition the subject is undistributed and the predicate is distributed. So D is undistributed; C is distributed. The reason is that 'Some D are not C' says something about the whole class C-it excludes the selected D from all of C-but it does not say something about all D. These distribution facts matter because an immediate inference may not distribute a term that was undistributed in the original.
2. Test notices (i), (ii), and (iii), naming the immediate inference involved and any assumption needed.
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(i) is obversion of A. The obverse of All R are I is No R are non-I. If 'books that may be read outside the library' is taken as the contradictory of 'books that must be read inside the library', then (i) is valid. The assumption should be stated: the library treats 'inside' and 'outside' as exhaustive opposites in this context.
(ii) is conversion of an E proposition. No periodicals are books that may be borrowed overnight becomes No books that may be borrowed overnight are periodicals. E conversion is valid because both terms are distributed in an E proposition, so no illicit process occurs.
(iii) is contraposition of an A proposition. All R are I entails All non-I are non-R. Here 'books that may not be read inside the library' is non-I and 'books not kept on the reserve shelf' is non-R, so (iii) is valid. Contraposition of A is valid, unlike simple conversion of A.
3. Test notice (iv). Is it a direct conversion of an O proposition, and does it commit an illicit process?
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Notice (iv) is valid, but not as a direct conversion of O. The original is Some D are not C (O). Direct conversion of O would be invalid: from 'Some donated books are not catalogued' one cannot infer 'Some catalogued books are not donated books' or any simple reversal that changes the quality improperly.
The correct route is two steps. First obvert the O proposition: Some D are not C becomes Some D are non-C (I). Second convert that I proposition: Some D are non-C becomes Some non-C are D. This yields 'Some books not catalogued in the main index are donated books,' which is exactly (iv). No illicit process occurs: in the original O, C is distributed; in the obverted I, neither term is distributed, and in the converted I neither term is distributed. Obversion changes the predicate to its complement and changes quality, so the distribution of C is not carried over illicitly.