349 



LOGIC. 



LOGIC. 



850 



of first intention. It is also arithmetical in character : it either 

 enumerates or speaks by total of enumeration, as in Every X is T, 

 All ss are Ys. It is so far not distinguishable in character from a 

 numerically definite proposition : " Some xs are YS " and " 70 xs are 

 vs " are forms which differ only by vague and definite enumeration. 



The adoption of a definite universe, as already noticed, gives pre- 

 cision to the privative term, not-x, the rest of the universe when x is 

 removed. Let not-x be denoted by x ; not-Y by y, &c. And x and x 

 are called contraries.* This introduction destroys the distinction of 

 assertion and denial, except in a relative sense: the assertion, that 

 every x is Y is the denial that any x is y. Affirmation and negation 

 must be distinguished as follows, A proposition true of x and x, false 

 of x and x, is atfirmative : a proposition true of x and x, false of x and 

 x, is negative. Thus every thing is either x or Y is negative, though 

 assertive in this form : it is true of x and x, false of x and x. It is to be 

 understood that no term used fills the universe : names which belong 

 to the whole universe, to one part as much as to another, cannot be 

 of any distinctive force or meaning within that universe. 



Let a proposition be called universal, when the whole universe must 

 be examined to verify it : particular, when the examination of a 

 portion of the universe may verify it. Let a term be called total, 

 where every instance of it must be examined before the proposition 

 can be verified : partial, when such complete examination may not be 

 needed. 



There are two forms of predication in use : the cumitlar, as in " all 

 xs are some YS " and some xs are not YS (are distinct from all YS) ; 

 the exemplar, as in every (each, or any) x is some one Y, and some one 

 x (perhaps not any one) is not any one Y. The cumular proposition 

 speaks by collection ; the exemplar, by selection. Grammatical correct- 

 ness (representing a usual tendency of thought, though not an absolute 

 law) allows either mode of expression in affirmatives, but demands 

 exemplar expression in negatives ; as in all xs are not any Ys. The 

 cumular and exemplar modes of expression have differences which 

 will presently be further seen. 



We first take the cumular proposition and syllogism. Let a term 

 totally used, as x, be denoted by x) or (x ; partially, by x ( or )x. Let 

 affirmation be denoted by two dote, or none ; negation by one dot. 

 Let propositions be distinguished by the quantities of their terms and 

 the quality (affirmative or negative) : this is found to make complete 

 distinction. Thus () means a negative proposition in which both 

 terms are particular : and x () Y is that proposition enunciated in 

 terms of x and Y. It turns out to be " Everything is either x or Y." 



There are eight forms of cumular predication : four universal, with 

 four contrary (usually called contradictory) particulars. These are now 

 given, with their symbols and usual readings : and the correctness of 

 the distribution of the words universal and particular, total and 

 partial, may be gathered by the reader from the definitions. Each 

 universal (U) is followed by its contrary particular (p). 



u. x ) ) T. Every x is Y. 



p. x (-(Y. Some xs are not YS. 



u. X)-(T- No xis Y. 



p. x ( ) Y. Some xs are YS. 



U. X () Y. Everything is either x or T. 



P. x ) ( Y. Some things are neither xs nor Ys. 



c. x ( ( Y. Some xs are all the YS. 



p. x )) Y. All xs are not some YS. 



Contravcrrion changing a name into its contrary alters the quantity 

 of the term, and the quality of the proposition : thus x ) ) Y is x )-(y 

 and x ( (y and x () Y. And a term and its contrary always enter one pro- 

 position iriMi different quantities. Conversion is simply writing the 

 prepositional symbol inverted : thus x ) ) Y is conversely Y ( ( x. We 

 shall now use the forms ) ) (( &c. to signify the propositions. 



Of contraries; one must be true and one false : either ) ) or (( &c. 

 Each universal simply contradicts one particular the 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 ((, that 

 is, denies arid is the only mode of denial ; it denies )( 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 x is Y " may coexist with either " Every Y is x " or " some 

 Ys are not xs." 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. 



f. )o)Y, compounded of x )) Y and x)-) Y. All xs and other things 

 besides are YS. 



x | J Y, compounded of x ) ) Y and x ( ( Y. All xs are all YS : x and 

 Y are identical terms. 



x ( o ( Y, compounded of x (( Y and X(-(Y. The xs contain all the 

 Ys and other things besides. 



x ) o (Y, compounded of x )( Y and x ) ( Y. Nothing is both x and Y, 

 and some things are neither. 



* No distinction is drawn between the words contrary and contradictory, 

 whether s to terms or propositions. 



x I I Y, compounded of x )( Y and x () Y. Everything either x or Y 

 and nothing both : x and Y are contraries. 



x ( o ) Y, compounded of x () Y and x ( ) Y. Everything is either x or 

 Y, and some things are both. 



Each kind of proposition is either simple or complex with respect to 

 the other, according as we think of alternation or conjunction. Thua 

 ) o ) is the junction of ) ) and )) ; but ) ) is the alternative ) o ) or | j. 

 Conversion and contraversion are under the same rules as before. 



The preceding forms are completely arithmetical, and are truly par- 

 ticular cases of the numerical proposition, of which it will now be 

 convenient to say something, as well as of the corresponding syllogism. 



Let TO x Y signify that m (or more) xs are Ys ; then m x y signifies 

 that m or more xs are not YS. Let u be the number of instances in 

 the universe ; let x, y, z, be the numbers of xs, Ys, and zs : then 

 it x, uy, z, are the numbers of xs, ys, and zs. A spurious pro- 

 position may exist : that is, one which by the constitution of the 

 universe must be true. Thus if x + y be greater than w, (x + y it) x Y 

 must be true. Every proposition has two forms : thus 



m x Y is ( m + u x y ) x y 



ms.f is (m + x y ) x y 

 mxy is (m + x+y ) x Y 



When one of the forms is spurious, the other contains a negative 

 quantity, and is * inconceivable. 



The contradiction or contrary of mXY is ( + ! m) xy. Hence 

 deduce contradictions of the other forms. From m x Y and TO Y z we 

 deduce (m + n ,v)xz. That is, there is inference when the quantities 

 of the middle term in the two premises together exceed the whole t 

 quantity of that term. The following forms of syllogism may easily 

 be investigated. 



Premises. 

 m XY n YZ 

 m Xy YZ 



mXY 



MiXy 



n yZ 



Forms of conclusion. 



(m + n y) XZ (m + n + u x y z) xz 



(m + w )xZ (m+n z) Xz 



(m + n z) Xz (m + n x) xZ 



(m + n+y #-z)xjz ' 



The proposition x ) ) Y is x x Y : the proposition x))Yisa:x Y joined 

 with (y x) XY. And so of the rest. 



In the more subjective view of logic, that is, in the mathematical side 

 of it, the one commonly cultivated to the exclusion of the other, the 

 torm becomes the word expressive of a class, and the relation of class 

 to class is that of inclusion or exclusion, total or partial. Each term, x, 

 divides the universe into two classes, x and x. The following names 

 may be applied to the relations of class : 



x ) ) Y x is a species + of Y ; Y is a genus of x 



x ( ( Y x is an extent of Y ; Y is a deficient of x 



X ) ( Y x and Y are externals of each other, or co-externalt 



x ( ) Y x and Y are partienti of each other, or co-partienti 



x () Y x and Y are complements of each other 



X ) ( Y x and Y are co-inadequates of each other 



x ( ( Y x is a genus of Y ; Y is a species of x 



X ) ) Y X is a dejicient of Y ; Y is an extent of x. 



The propositions of terminal precision may now be stated under the 

 following nomenclature : 



X ) o ) Y x a subidentical of Y 



X | | Y x an identical of Y 



X ( o ( Y X a superidentical of Y 



x ) o ( Y x a subcontrary of Y 



x J | Y x a contrary of Y 



X ( o ) Y x a tnpercontrary of Y 



We now return to propositions of terminal ambiguity. When two 

 propositions are joined together, between x 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 spicidie 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 



* Not'strictly inconceivable, to those who remember the full signification of the 

 negative quantity as here denned. Thus ( 7) XT means that 7 or more xs 

 are YB : begin at the point of more at which logical predication becins to be con- 

 ceivable, and we have ' or more xs are TS,' a proposition which is simply 

 spurious. 



f This is what Hamilton calls the ultra-total quantification of the middle 

 term. Lambert first thought of this principle : Mr. De Morgan, without any 

 knowledge of Lambert, reconccived it and extended its use. Sir W. Hamilton, 

 who did not know of Lambert till alter this, states first, that he had himself 

 thought of the principle and thrown it away; si-condly, that Mr. Ue Morgan 

 took it from Lambert : and this though Mr. De Morgan had explicitly stated 

 that he never knew of Lambert's work till after his own paper on the subject 

 had been published. 



J Here is a departure from the common language. Uusually X)O)Y i 

 signified when we say that x is a species of Y, and y a genus of x. We take 

 the species as being, possibly, the whole genus : just as tome is possibly all. 



