*356 Mb DE morgan, ON SYLLOGISMS OF TRANSPOSED QUANTITY. 



Let imXY and «YZ coexist : we infer (ni + n - y) XZ, or (m + n + u — a> — y — z) xz. 

 Let mXy and «YZ coexist : these are mXy and (w + u — y — sg) yz ; from which we infer 

 (m + n + u — y — z — y') Xz, or (m + n — %) Xz and (m + n — /v) xZ. 



Call the number of instances the (logical) extent of the term or proposition. Then it 

 appears that when the two premises have the middle term Y in both, or y in both, the two 

 forms of conclusion take from the premises, the one both terms direct, the other both terms 

 contraverted : but when the middle term enters in both forms, Y and y, the two forms of 

 conclusion take each one terra direct from the premises, and one term by contraversion. In 

 the first, the coefficients of extent in the forms of conclusion are the united extent of the pre- 

 mises diminished by that of the middle term, and the united extent of the premises and of 

 the universal diminished by the united extent of the three terms. In the second, the coeffi- 

 cients of extent are both described by the united extent of the premises diminished by the 

 extent of the contraverted term. 



We can now deduce either the ordinary syllogisms or those of transposed quantity, be- 

 longing to any one case of the numerical syllogism. Let the premises be 



mXY, nyz, so that {m + n — x) xz, (»» + « — /) XZ, 



are the forms of the conclusion. From »i = a;, we deduce wxz and (a? + « — «') XZ ; of the 

 second of which we can say nothing without further knowledge of the relations of extent. 

 On the meaning and character of the second form I may refer to my Formal Logic. From 

 a?XY and wyz we have then nxz; that is, using the notation of my second and third papers, 

 from X))Y and Y)(Z we deduce X)(Z. Similarly, from m = x, n = s^ -we show that 

 X )) Y )) Z gives X )) Z : from n = ss' alone, we deduce that X () Y )) Z gives X () Z. 



To find such syllogisms of transposed quantity as this form gives, let n = x : then mXY 

 and a;yz give mxz. That is from ' Some Xs are Ys, and for every X there is something 

 neither Y nor Z ' we deduce ' Some things are neither Xs nor Zs.' When one term imparts 

 its quantity to another, let the imparting term have a symbol placed above its spicula of 

 quantity, and the receiving term below. Thus what we have just arrived at is that 

 X ( ) Y ) (Z gives X ) (Z. It must be specially observed that the terra which imparts is 

 always particular: thus when we see X() Y) (Z, in which Z is universal and z particular, 

 the meaning of X () Y is ' For every z there is an X which is Y.' It is also to be remem- 

 bered that in the formation of the symbol of conclusion the spicula of the imparting term is 

 always to be inverted : thus X ( ) Y ) (Z does not give X (( Z, but X)(Z. 



When a term takes the whole quantity of a term external to its proposition, it will be 

 convenient still to call the proposition universal, and, for distinction, externally universal. 

 The ordinary universal may be called internal. When a conclusion is spoken of as uni- 

 versal, it is meant as being internally universal. 



The circumstances under which two premises have a valid conclusion are precisely those 

 of the ordinary syllogism. Two universals, either or both of which are externally so, give 

 a conclusion, universal or particular according as the middle term is of unlike or like quan- 

 tities in the two premises. A universal and a particular with the middle terra of unlike 



