342 Mr DE morgan, ON THE SYLLOGISM, No. IV, 



convenience; not of obligation, as in the case of the double negative. Either some horse 

 or no horse; if not no horse, then some horse. The Greek* idiom refused this dilemma; 

 There is no scrape that man does not get into: if we had no other way of knowing 

 this, we have the assurance of Euripides ; but he informs us that there is not no scrape 

 that man does not get into. The educated English idiom follows logic, which here 

 commands. Such a phrase as the 'every uncle of a sailor' has no meaning except in 

 poetry, where it means the sole uncle. It would be highly convenient if the distinction 

 between LM' and L^M could be made as in ' L of every M' and ' every-L of M.' 



The symbols L'MX and LM^X, which I shall not need, analogically interpreted 

 would mean 'every L of an M of X' and ' an L of an M of none but X.' The com- 

 pound symbol L M'X means an L of every M of X and of nothing else ; and is really 

 the compound (LM'X) (L^MX). No further notice will be taken of it. 



We have thus three symbols of compound relation ; LM, an L of an M ; LM', an L 

 of every M ; L^M, an L of none but Ms. No other compounds will be needed in syllogism, 

 until the premises themselves contain compound relations. 



In every case in which there is a first and a second, let the first be minor, the second) 

 major. Thus if X..LMY, let X and Y be its minor and major terms, and L and M 

 its minor and major relations: if it be the first premise of a syllogism let it be the 

 minor premise. 



The converse relation of L, L"', is defined as usual: if X..LY, Y..L~'X: if X be one 

 of the Ls of Y, Y is one of the L~'s of X. And L"'X may be read 'L-uerse of X.' 

 Those who dislike the mathematical symbol in L~^ might write L\ This language would be 

 very convenient in mathematics : 0~'j? might be the ' (p-\erse of x,' read as ' ^-verse x.' 



Relations are assumed to exist between any two terms whatsoever. If X be not any L 

 of Y, X is to Y in some not-L relation: let this contrary relation be signifiedf by 1; thus 

 X.LY gives and is given by X..1Y. Contrary relations may be compounded, though con- 

 trary terms cannot: Xx, both X and not-X, is impossible; but Llx, the L of a not-L of X, 

 is conceivable. Thus a man may be the partisan of a non-partisan of X. 



Contraries of converses are converses : thus not-L and not-L~' are converses. For 

 X..LY and Y.-L'^X are identical; whence X..not-LY and Y. . (not-L"') X, their simple 

 denials, are identical; whence not-L and not-L~' are converses. 



• It would be worth the while of some one who has the not-do it, or can let it alone, but not always. Again, the 



requisite scholarship to examine the question how far the 

 negatory power of the double negative in Greek determined 

 the course of Aristotle in regard to privative terms. In 

 further reference to the dictating power of logic, I may observe 

 that it does not go far : forms cannot dictate meaning to any 

 but a very small extent. For instance: It is almost universal, 

 but not quite, that transference of not from the copula to the 

 predicate produces no change of meaning. 'He either will 

 do it, or he will-not do it' means the same as 'He either 

 will do it, or he will not-do it ;' and the two of each set are 

 alternatives. But 'He either can do it, or he cannot do it' 

 has not identity of meaning with ' He either can do it or he 

 can not-do it:' the first pair are repugnant alternatives, the 

 second are not: the same person who can do it, usually can 



junction of not to the verb usually gives a contrary, or a re- 

 pugnant alternative : he eats or he eats not, he has or he has 

 not, he does or he does not. But we may not say. Either he 

 must, or he must not; these are no necessary alternatives: we 

 can only say, Either he must, or he need not, Either he must 

 not, or he may. Many similar instances might be given. 



+ The affirmative symbol (. .) is derived from the junction 

 of the two negatives (.)(.). Analogy would seem to require 

 that the privative relation not-L should be denoted by (.L). 



Or thus: Let W denote the affirmation, and V the denial: 



then XWLY would denote that X is an L of Y, and XVVLY 

 that X is not a not-L of Y. But I do not at present find advan- 

 tage in a notation which expresses X .. LY and its equivalent 

 X.IY in one symbol: I may possibly do so at a future time. 



