472 Mr DE morgan, ON THE SYLLOGISM, No. V. AND 



both of the others, that one may be repeated twice, once in relation to each extreme : 

 and either extreme, with the common mean, is a symmetrical bisection of system. The Aris- 

 totelian plan confines itself to 'some affirmed to be all' and ' some not affirmed to be all': 

 •one extreme, and the mean in relation to it. But if we take the two extremes, we must also 

 take the mean in its relation to both. Again, as contrary terms must enter, whatever subdi- 

 visions we make of ' some X' we must also make the same of 'some x\ Accordingly, since 

 a universal term gives a particular contrary, there is no proposition but must enter in four 

 different ways. A term being universal, we must distinguish the case in which the particular 

 contrary is to be some-or-all from that in which it is to be some-not-all. Denoting all and 

 some-or-all in my usual way, I shall denote by an accent that the particular term indicated is 

 some-not-all, or else that the universal indicated has some-not-all for its contrary. There are 

 then, besides the eight usual forms, 3 x 8 or 24 others, all formed, as we shall see, by con- 

 junctions of two of the eight, or three. These last, by equivalences, are reduced to twelve; 

 which with twelve disjunctive denials, make a total of 32. Of these I shall, for brevity, 

 consider only the common forms and the 12 conjunctives: syllogisms containing disjunctions 

 can be dealt with by opponent reduction. 



First, a universal, such as ) ), is accompanied by ) )', )'), and )' )'. The three last are 

 equivalent, and equivalent to the form )) joined with )■); or to ) ). And the same of the 

 others. So that the system of universals contains the simple universals ) ), ( (, )•(, (•), and 

 the double universals ) = ), (o(, )=,(, (o). 



Secondly, a particular, such as ( ), is accompanied by ()', (') and (')'. These three 

 have the following meaning. Each one consists of the proposition without the accent, 

 joined to the proposition in which the unaccented term is contraverted. Thus ()' is () and 

 ).); (') is ()and (•( ; (')' is () and )•) and (•(. These may be denoted by ()•), 

 (.(), and (■()•). Accordingly, we have 



(')' means (.()) 



('•{' means )(•() 



)'•)' means ()•)( 



)'(' means )•)(•( 



Remember that )•)(•( is the triple junction of )•), ) (, (•(, &c. Contranominals do not 

 appear in any double proposition: thus we have not equivalence, )) and ((, nor contrariety, 

 )•( and (•), nor ()(, nor (•( and )•). 



The denials may be represented by the disjunctive comma: thus the denial of ) '(' is 

 ♦either ((or (•) or ) )', represented by { ( (, (•), ))}. Of the whole number, Hamilton's 

 plan selects seven, when ' Some X is not some Y' is properly treated, and adds the assertion 

 of equivalence ' All X is all Y\ The seven are 



», ('G (')', )•(, ()' CO )■)'; 



being two simple propositions, four double, and one triple. It thus includes all the cases in 

 which the new forms, (■) and ) (, are absent. To these it adds the equivalence, or junction of 

 )) and ( (, without adding the junction of (•( and )•), the denial of {)), {{\- The 



