Cambridge Philosophical Society, 147 



one day lead to a connected theory, making use of a common instru- 

 ment, just as the oppositions of quantity which are considered in 

 algebra are connected by the general theory of the signs 4- and — ; 

 and 2nd, some remarks on the resemblance of the instrumental part 

 of inference to algebraic elimination. 



Ten such instances as affirmative and negative, conclusive and 

 inconclusive, &c, are compared with the logical distinction of uni- 

 versal and particular ; and it is pointed out, in all the cases in which 

 it is not already acknowledged, that it would be possible to use any 

 one of the ten in place of the last. 



Section II. On the formation of symbolic notation for propositions 

 and syllogisms. — Exclusive of remarks on the Aristotelian notation 

 and on notation in general, and a statement for comparison of Sir 

 William Hamilton's notation, this section contains the following 

 matters. 



1 . A pictorial or diagrammatic representation of syllogistic infer- 

 ences, being after the method pursued by Lambert, with such addi- 

 tions as will enable the system to represent all the cases in which 

 contraries are used. 



2. An abbreviated and arbitrary method of representing proposi- 

 tions and syllogisms. 



Following Sir William Hamilton in making the quantity of both 

 subject and predicate matter of symbolic expression, Mr. De Morgan 

 gives his system of notation two new features. First, he dispenses 

 with the representatives of the terms (except when it may be con- 

 venient to introduce them for the time), and represents the proposi- 

 tion by the symbols of quantity only, and the presence or absence of 

 a sign of negation. Secondly, instead of making the symbols of uni- 

 versal and particular absolute, he gives one symbol, ), to a universal 

 subject and a particular predicate, and another, (, to a particular 

 subject and a universal predicate: a dot [.j signifying negation. 

 Thus X)-(Y, or simply )•(, represents 'No X is Y': X(-(Y, or (•(, 

 represents ' Some Xs are not any Ys :' X()Y represents ' Some Xs 

 are Ys.' Of the second circumstance above mentioned, Mr. De 

 Morgan believes that it makes the rules easier, and knows that it 

 makes the notation more suggestive. 



Retaining in mind the order XY, YZ, XZ, which is the only figure 

 used in the classification (being the first with inverted order of pre- 

 mises), the syllogism is to be denoted by the junction of the propo- 

 sitional symbols. Thus ))))=)) denotes 'Every X is Y, every Yis 

 Z, therefore every X is Z.' When this is to be read in any figure, 

 the subject- quantities are to have their symbols thickened, the second 

 premise being read first : thus in the four figures, in order, will be 

 seen such symbols as |j|l, |j||, ||||, |||j. 



Section III. On the symbolic forms of the extension of the Aristo- 

 telian system in which contraries are introduced. — This system is the 

 one which was completed and published to the Society before any 

 correspondence with Sir William Hamilton. Mr. De Morgan re- 

 marks that it contains (incidentally, not designedly) every distribu- 

 tion of quantifications ; and gives his reasons for not dwelling on this 

 fact while the controversy was unfinished, with his statement that it 

 had not struck him when the controversy began. Mr. De Morgan 



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