Of contraries ; one must be true and one false : either ) ) or (( &c. Each universal simply contradicts one particularthe one with different quantities and quality; is inconsistent with, and therefore denies, the universals of different qualities ; remains indifferent to the universal of different quantities and its contradictory ; and contains the two particulars of the same quality. Thus ) ) contradicts (4, that is, denies and is the only mode of denial ; it denies X and (.) but not by simple contradiction ; it co-exists with either ( ( or ).); and it con tains and affirms ( ) and ) (.
Each universal proposition has what we call terminal ambiguity : thus " every a is Y " may coexist with either "Every v is X" or " some vs are not as." The following is the list of propositions which, with reference to the ambiguity just noted, have what we call terminal precision. The symbols used to denote them are prefixed.
)o ) Y, compounded of x )) Y and a ).) Y. All as and other things besides are Ts.
X v, compounded of a ) ) Y and x ( ( Y. All as are all Ts : a and lr are identical terms.
x ( o (r, compounded of x ( ( Y and a ((r. The as contain all the re and other things besides.
) o (r, compounded of x )( Y and x) Y. Nothing is bath a and Y, and some things are neither.
* No distinction is drawn between the voids contrary and contradictory, whether as to terms or propositions.
We now return to propositions of terminal ambiguity. When two propositions mre joined together, between a and Y, and Y and z, there is a valid syllogism, that is, one which yields a necessary conclusion, 1, when both premises are universal ; 2, when, one premise only being universal, the middle term has different quantities in the two ; or when the spiculee turn the same way. And the conclusion is found by erasing the symbols of the middle term. Thus ( ) ) ) gives a valid syllogism : for though ) ). is particular, there is a universal and a particular quantity in the middle term. The conclusion is ( ) or ( ). By this we mean * Nestrietly inconceivable, to those who remember the full signification of the negative quantity as here defined. Thai ( 7) MY means that 7 or more xs are ys ; begin at the point of more at which logical predication begins to he con ceivable, and we have ' 0 or more xs arc ye,' a proposition which is simply spurious.
t This is what Hamilton calls the ultra-total quantification of the middle term. Lambert first thought of this priiiciple : Mr. Da Morgan, without any
knowledge of Lambert, reconceived it and extended its use. Sir W. Hamilton, who did not know of Lambert till after this, states first, that he had himself thought of the principle and thrown 'it away ; secondly, that Mr. De Morgan took it from Lambert : and this though Mr. De Morgan had explicitly stated that he never knew of Lambcrt'a work till after his own paper on the subject had been published.
Here is a departure from the common language. trusually x)o)T is signified when we say that x is a species of T, and Y a genus of x. We take the species as being, possibly, the whole genus : Just as come is possibly all.
that x ()r ) ) a gives x( )z, or that x (.) T and ).)z give x ( )z. That is, if " Every thing is either x or Y," and "Some as are not an," it follows that " Some xc are as." There are 32 valid forma. I. Eight are wnirerra/, with universal premises and conclusion ; being all which have differently quantified middle tenni. They are as in )) )), )))(, &e. 2. Eight are minor pertienlers, with the Misc., (or first) premise particular, and a particular conclusion; as ( ( (.), O) ), ke. 3. Eight are major particulars, with the major (or second) premise particular, and a particular conclusion ; as ( ))(, ( ( &e. 4. Eight are strengthened particulars, universal premises with the middle term similarly quantified in the two, and a particular conclusion; as in ))( (, )( ) ), In every syllogism, the terms being notions of class, there is a com bination of premising relations in the concluding relation. Thus iu " No x is Y, some vs are za, therefore some as are not xe," or in " x ) (11 )x gives x) ) z," we road that x is an external of v, and T a (ardent of a, whence x is a deficient of a. And we see that an external of a partient is a deficient. Apply this to all the cases. Thus in ( )) ( , giving ( (, we read that a complement of an external is a genus.
The propositions of terminal precision give syllogisms having con clusions also of terminal precision by joining two in which the middle term has different quantities as in a ( 0 ) T )0 ( z, giving a ( o ( z. This mean. that if everything be x or 1', mid some things both, and if nothing be bOth 1' and z, and some things neither, it. follows that the is contain all the as, and more. The combination of relations here seen is as follows : a supercontrary of a subeontrary is a superidentical.