KHC 



UXHO 



that x(-lv>-)z give* xOz, or that X(-)Y and Y)-) give x()*. 

 That ia, if " Every thing i* either X or T," and " Some x* are not Ts," 

 it follow, that " Some x. are xa." 



Tber* are 32 valid forma, 1. Eight are mtirmal, with univeraal 

 premise* and eonclusion ; being all which have differently quantified 

 middle term*. They are a* in i ) ) ). i i H. *' - Kight are minor 

 turiirwlmn. with the Minor (or first) premise particular, and a particular 

 conclusion : aa ( ((), ( ) , 4c- 3. Eight are major particuiari, with 

 tbe awucr (or second) premise particular, and a particular conclusion ; 

 aM ))(,<( O. * * Eight are tlrenythenetl partieulart, univeraal 

 premise* with the middle term similarly quantified in the two, and a 

 particular conclusion ; a* in )U (,)()), Ac. 



In every syllogism, the terms being notions of elans, there ia a com- 

 bination of premising relation* in the concluding relation. Thua in 

 " No x i* Y, some YS are zs, therefore some z* are not xa," or in 

 , Y( )x gives x ) -)z," we read that x is on external of Y, and Y a 

 paruent of t, whence x is a deficient of z. And we see that an external 

 of a partient i* a deficient Apply this to all the cases. Thus in 

 ())(. giving ( (, we read that a complement of an external is a genus. 



The proposition* of terminal precision give syllogisms having con- 

 cluaoos also of terminal precision by joining two in w Inch the middle 

 term ha* different quantities ; aa in X(O)Y)O(Z, giving x(o(z. This 

 means that if everything be x or v.and some things both, ami if 

 nothing be both Y and z, and some things neither, it follows that the 

 x* contain all the i, and more. The combination of relations here 

 seen ia a* follow* : a supercontrary of a gubcontrary i* a auperidentic.il. 



When the notion of figure ia taken into account [STLLOOIU ; KKI.A- 

 TIOX], its force and meaning is belt seen by stating the combination of 

 relation in the different figures. Thua when we say a specie* of a 

 pecies is a specie*, we speak in the first figure, and compare the minor 

 with the major by the relation which the minor stands in to the 

 middle and the middle to the major. When we read it thus species 

 and genu* of the same are specie* and genu* of one another, we use 

 the second figure, and compare both major and minor with the middle. 

 When we say that those to which the same is genu* and specie* are 

 species and genus of one another, we use the third figure, and compare 

 the middle term with both major and minor. And when we speak 

 thus, that to which another ia species of specie* is a genus of that 

 other, we use the fourth figure, that is, the first figure with the con- 

 cluding relation inverted. 



W* now come to the exemplar form, already spoken of. Let x ) and 

 ( X signify any one X : let ) x and x ( signify some one x, it not being 

 known, even if true, that we may say any one. 



The list of proposition* is a* follow*. Each universal is followed by 

 iu contradicting particular : 



u. x )( Y Any one x is any one Y. This means that there ia but 

 one X and one Y, and the x i* the Y. 



r. x ( ) Y Some one x is not some one Y. Either x and Y are not 

 both ring"!"- : or if both singular, not identical 



i . x ) ) Y Any one X i* some one Y ; or every x U T. 



r. x ( ( T Some one x i* not any one Y ; or some xs are not YS. 



i. X((Y Some one x i* any one Y ; or every Y ia x. 



r. x ) ) T Any one x is not some one Y ; or some YS are not xs. 



i . x ) ( Y Any one x U not any one Y ; or no x is Y. 



i'. x ( ) Y Some one x is some one Y ; or some xs are YS. 



Omit the word one throughout, for any read all when the word ia in 

 italic*, and we have Hamilton'* * system, provided the first pair be not 

 read a* contradictions. 



There are 34 valid forms of syllogism, both in the exemplar reeling 

 and in Hamilton's reading. All that ia required U one affirmative 

 premise, and one total occurrence of the middle term. In the exemplar 

 system, the rule of inference is as above described for the cumular 

 system. In Hamilton's system there ia this modification : when the 

 middle term* are of different quantities, a concluding term which is 

 total, with a *|<icula turning a different way from those of the middle 

 term, must be altered from total to partial, if its proposition be 

 affirmative. Thus, ) ( ( ) in the exemplar system give* ) ), but in 

 Hamilton'* system it give* ( ). Or, " Any one x i* any one Y, and 

 some one Y M some on* I," gives " Any one x is some one z : " but 

 ' All x i* all Y, and some Y is some z," gives " Some x is some z." Again, 

 )(((, in Hamilton* system, give* ((, not )(, a* hi the exemplar 

 system. ,.The reason of this difference will easily appear, on con- 



The exemplar proposition U often used, a* distinguished from the 

 cumular. It i, for example, the proposition of geometry. When 

 Euclid prove* that all isosceles triangle* are of equal angles at the 

 bawss, be shows that any isoecele* triangle is so, and he demonstrate* 

 only to those who can see that nothing is assumed in his demonstration 

 which limit* the selection. 



The numerical syllogism leads to a species of syllogism which i* 



e Mr W. Haalnoa nlxtd th* rirupUr aad eurauUr forms In obrdiracc to 

 frnasur. Ills srstrai mi'Xad ihr on* before us, which Its proposer bold* to 

 to omeUoa. lUmiltoa hlnutirrrjwtfd th* eorrttUun with his usual plonmnt 

 MOTH, tfetterlaf that It rtaad* ale**- la bad****, that U IB the whole history of 

 "Inn. Those, thmfcsr*, who comply with hi! wUh, will hud It over entire to 

 Mr. Dr Morgan, who will accept It all, if given, but cannot claim It all. 



occasionally used, but cannot be reduced to a common syllogism. It 

 is the syllogism of traiupoted quantity, in which one of the concluding 

 terms enters with the whole quantity of the other, or of it* contrary. 

 For instance, " For every man iu the bouse there i* a person who is 

 aged ; tome of the men are not aged ; therefore some of the persons in 

 the house ore not men." Nothing ha* yet been printed on the laws of 

 thi* Kyi log ism of transposed quantity : the following brief rule* will 

 enable the reader to detect the case*, which are sixty-four in number. 



Let a particular proposition which has a term of the whole quantity 

 of an external term that is, in another proposition be called an 

 ej-ternal uninrtal, the ordinary univeraal being called internal. The 

 proposition ha* a receiving terra ; the other proposition has an imparting 

 term. Let the receiving term be distinguished by a hyphen below its 

 spicula; the imparting term by a hyphen above. Thus X))Y)-)Z 

 mean* aa follows : Every x i* Y ; for every x there i* a z which U 

 not Y. But there is no valid syllogism when a term taken totally 

 imparts it* quantity : in the case before u* the only chain < ' .. v;iliil 

 inference arises from allowing x, which enters jiartinlly, tu itiiirt iu 

 quantity ; as in " Every x is Y ; for every x there is a z which is not Y." 

 This being understood, and also that in forming the conclusion 

 imparting term must -change it* quantity, the rules for detecting valid 

 syllogism ore precisely those of the cumular syllogism already described. 

 That is, any two universal*, each of either kind, give a conclusion ; and 

 one universal of either kind and one particular! jf the middle term* 

 have different quantities in the two. 



Then (") ((. having two external universal*, i* valid, and give* )). 

 That ia, from " For every z there ia an x which is Y, and for every x 

 there ia a Y which is not z," we deduce " Some z* are not Y*." 



Again, ( ) ) )~has an external and an internal univeraal, and gives ( (. 

 " For every z there is an x which ia Y, and every Y ia z" give* Every 

 z in x. And ) ( ((, having an external universal and a particular, with 

 middle term of different quantities in the two, is valid, and gives )). 

 That ia, from " For every z there is something neither x nor Y ; some 

 YS are not zs," we deduce " Some za are not xs." 



Few propositions that ever come before the human mind require 

 closer attention than these transposed syllogisms. They are easily 

 demonstrated from the general forma of numerical syllogism ; not so 

 easily by independent thought 



We wall now proceed to compare the mathematical and metaphysical 

 aides of logic. A term may be formed by junction of terms in two 

 waya. 1. By aggregation, as in what we represent by (A,B, c), which 

 stands for all that ia in A, in B, in c, or in two or more. 2. By 

 compuritivH, as in what we represent by (A-B-C), or (A B c), meaning all 

 that ia common to all the three term* A, B, c. The contrary of on 

 aggregate is the compound of the contraries of the aggregant* : either 

 (A, B) or (a b). The contrary of a compound ia the aggregate of the 

 contraries of the components : either (AB) or (a, b). 



The term, considered as aggregated of aggreganta, is viewed in extent; 

 aa a compound of components, in intent. For total and jiartial we non- 

 substitute the words full and vague, to get rid of the wrong opposition 

 which total and partial suggest ; for the term called partial is not 

 necessarily partial at all, it is only not known to bo total. And thi.~ 

 ambiguity ha* led acute * logical writers into absolute mistake. In 

 this matter we follow Hamilton, who justly and acutely remarked that 

 " It is only as indefinite that particular, it ia only aa definite that indi- 

 vidual and general, quantities have any (and the same) logical avail." 

 The word indefinite completely describes the word partirular, or partial, 

 as commonly used : we only use rogue aa a shorter word. But tin; 

 unirenal or total of logicians is more than definite : it is the maximum 

 of definiteneaa, and we use full a* more correctly descriptive. 

 irrailn* is as definite aa all. 



Extent and intent ore the two logical tentions. The clement* of 

 extent are aggregauta ; the element* of intent are components. \Vh.-u 

 one tension ia full, the other ia vague; when one tension is vague. ;i,,- 

 other ia full. Thua x ) signifies that x is taken fully in extent, vaguely 

 in intent ; and )x signifies that X ia taken vaguely in extent, and fully 

 in intent When a tension is full, an existing element may be dis- 

 missed, but a now one cannot be admitted ; when a tension ia vague, 



In Hamilton'! system, ' Every x is T ' la read ' All x 1 some T,' and 

 'Every T Is x ' Is read All v i sonic x. 1 But ('Discussions,' &c., 5ml ed. 

 p. 688,) be rayi that All x U some T,' and ' Some x to all T,' are Incomposslble, 

 tht i, cannot be true together, " unless totnr be identified with all." This 

 supposition lie Ireati 01 an ab>urdity : hard pushed in controversy, ho was 

 mUlrd by tlic ambiguity on which (hi* note is written. The tomr, even of his 

 own system, Is a potiikly-all : how othrrw^e can lie dare to render ' livery x is 

 T' by ' All x is some v !' The junction of the two proposition! determines the 



of the no-called parlirvlar. Thii looks very like nonsense ; but the 

 fault U In the language of the logician!. Substitute full and ingur : the word 

 tome U vague in cither proposition taken apart : but the junction restrains the 

 rafnrntu, and tums the indefinite into a definite, vagut Ma full. Just as in 

 algebra one relation between two unknown quantities may leave a vagueness 

 which a second relation turn! into comparative dcfinitenes!. Our two pro- 

 position^ ore xxv and y vs. The first girea t = or -. y ; the rccond gives 

 t or < z, that U, z t amly x are neither of them negative ; whence x = y, 

 o that v is full in the first proposition, and x in the second. 



The word innmfouitle, which Mr. DC Morgan attributes to Sir W. Hamilton, 

 U In truth only revived by him. It is a very useful word. 



