92 PROFESSOR DE MORGAN, ON THE SYMBOLS OF LOGIC, 



was often matter of explanation. Expression for the contrary of a contrary was wanting in 

 the subject of the proposition : whence it arose that only three equivalents appeared in the 

 detailed symbols, though four existed in thought and in the more compressed symbols. Thus 

 I had A { of XY, E l of Xy, A 1 of xy, E l of xY, as equivalents: while only X)Y = X.y =y)x 

 appeared in the detailed symbols. I now have 



jr))'r-jr).te-*«y-»(.)r 



X(.(Y=X0y = x).)y=*)(Y. 



The rule of transformation is ; — To use the contrary of a term, without altering the import of 

 the proposition, alter the curvature of its parenthesis, and annex or withdraw a negative point. 



We may now say that the quantitative contrary of 'every X" 1 is 'some *•,' and of ' some 

 Xs," 1 ' every x.' Thus, when I say, ' some Xs are Fs ' I deny something of every x : namely, 

 that any one of them is one of those Fs. Again, ' Every X is F 1 denies of some xs that they 

 are Fs : for Fs must not fill the universe. And so on. 



The distinction of affirmative and negative, in the usual sense, is abandoned : for any 

 affirmative proposition, as XQY, is also negative, as seen in its equivalent X(.(y. My two 

 new forms of predication were properly called, (.) negative, and )( affirmative, and were derived 

 in the forms x).(y and xQy as 'no not-^ is not-F 1 and 'some not-^Ts are not- Fa.' Never- 

 theless, stated in reference to X and F, the first appears affirmative ' Everything is either X 

 or F', and the second negative, ' Some things are neither Xs nor Fs.' But the first obeys the 

 rules for the indisputable negatives, and the second those for the affirmatives. There enters an 

 extension of an old maxim ; — it is that three negatives make a negative. There are three 

 positive ideas, X, F, affirmation ; opposed to the three correlative negatives, x, y, negation. 

 Negative propositions present an odd number of the negatives ; positive ones an even number 

 (or none). Thus in all the equivalent forms of X(.)Y, nothing but an odd number of the 

 negatives will occur ; as seen in 



X(.)Y=x))Y=X((y = x).(y. 



There is yet one more opposition ; the quantitative parentheses may turn the same or different 

 ways : which are to be considered as positive and negative cases. By this, and the opposi- 

 tion of affirmation or negation, the extent of the proposition is determined. When these 

 oppositions are none or two, the proposition is universal : when one, particular. Thus X)) Y 

 having none, and X(.) Y having two, are universals: but XQY having one, is particular. 



In a universal proposition, any one quantity may be altered, either from universal to par- 

 ticular, or from particular to universal; and the result is always a true deduction, though 

 not an equivalent. Thus X))Y gives both X() Y and X){Y. 



Contrary propositions (usually called contradictory) of which one must be true and one 

 false, differ both in quantities and copula. Thus X))Y and X(.(Y are contraries. The 

 concomitants of a universal, to which it is perfectly indifferent, differ from it in quantities, or 

 in copula, not in both. Thus X))Y coexists either with X((Y or X).)Y. The superior 

 universals of a particular, or the inferior particulars of a universal, are made by altering one 

 quantity only: thus X){Y has X((Y and X )) F for its superior universals; and X(.)Y 

 has X).) Y and X(.(Y for its inferior particulars. The alteration of one quantity, and the 



