466 



Mr DE morgan, ON THE SYLLOGISM, No. V. AND 



term which will do might be made the regulator of our forms of inference. Thus Barbara, 

 *''■)) ))' ^s commonly presented, is equally represented by (( )•( )•( , which affirms that 

 no species of the minor term is an external of the major. And this last is but a strengthened 

 form of )•()■(, which affirms that no partient of the minor is an external of the major. 



When the middle term is only an intermediary, some term wholly indefinite, or any term, 

 a notation which, for good reasons, I had dropped, may be amended and re-established. We 

 may say )) )) = )), (( (•) (•( = ((, &c. Such notation as X)) Y + Y )) Z = X )) Z is faulty 

 in two' respects. First, the sign = is applied to an inconvertible relation : for, though 

 X))Y + Y))Z gives X))Z, X))Z does not give X)) Y+Y))Z, but only X)).? + .?))Z, 

 where both queries may be answered by the same term, when that term is known. Secondly, 

 the premises are compounded, not aggregated. For both these errors I was indebted to the 

 suggestion of the school of logicians, who not only aggregate concepts into concepts, but who 

 sometimes go as far as animal + rational = man. 



I now assert that every onymatic syllogism can be announced in eight forms : and each 



1 Mr Mansel (iv. 119) takes the following objection: — 

 " As little [as of Euler's geometrical syllogism-figures] do we 

 approve of the algebraical method adopted by iMr De INIorgan, 

 in which the premises of a syllogism are connected by a plus, 

 and their relation to the conclusion expressed by the sign of 

 equality, a method too redolent of the computation-theory noticed 

 above, [either Hobbesor the arithmetically deKnite syllogism], 

 and tending to confound the intuitive judgments of Arithmetic 

 with the discursive inferences of logic. The algebraical equation 

 proper does not represent a syllogism, but a proposition which, 

 like any other, may form part of a logical reasoning, but cannot 

 with any propriety represent the whole." To this 1 say first, 

 that + and = are not signs peculiar either to arithmetic or to 

 algebra : +, — , and = are in genere aggregative, disaggrega- 

 tive, and equivalential. A person who has no counting, and 

 as yet no symbols, might be introduced at once to the symbo- 

 lic aggregation of concrete lengths, which is seen in 



Secondly, a syllogism is a proposition ; for it affirms that a 

 certain proposition is the necessary consequence of certain 

 others. An affirmation is not the less an affirmation because it 

 affirms about other affirmations. Mr Mansel will not deny that 

 the following propositions are premises giving a valid conclu- 

 sion. Minor ; ' Every case in which all X is Y and all Y is Z 

 is a case in which all X is Z'. Major; Every case in which 

 'all X is Z is a case in which all not-Z is not-X'; therefore 

 &c. What is the first of these propositions but a sylloyism i 



In the Athenaeum Journal (Nov. 10 and 24, 1860) appeared 

 reviews of Hamilton's Lectures, which Mr Mansel at once at- 

 tributed to me: in which he was correct, so far as any one can 

 be correct in giving to a contributor an article which, appearing 

 under editorial responsibility, passes through editorial hands 

 before it is made public. Certain mathematical errors were 

 pointed out, which it appeared were mostly copied from others, 

 though some of them were read to the class for twenty years 

 together. Such as that Euclid (i. 1) shows that his three lines 

 constitute a triangle, and that the circles meet; such as men- 

 tion of two lines which divaricate at an acute angle, 'like a 

 pyramid ' ; such as an inch equal to a foot, because both have 

 an infinity of parts, and one infinity is not larger than another, 



set down as a ' contradiction proving the psychological theory 

 of the conditioned ' ! The first of these was in a Lecture, whh 

 other things resembling it ; the second and third were private 

 notes. Mx Alansel replied (December 1 and 8, 1860), resting 

 mainly on the fact that Hamilton had taken the errors from 

 others. He also asks why Hamilton is to " be tied down to an 

 exactness in the use of mathematical illustrations which professed 

 mathematicians have not held themselves bound to observe." 

 His instance is as follows : — " I find in Prof. De Morgan's 

 ' Formal Logic' (p. 131) a syllogism in Barbara, expressed in 

 the form Y)Z -j- X)Y = X)Z ; an expression with which I 

 shall not quarrel, as an algebraical metaphor, so to speak, 

 though 1 fancy that the author himself will hardly maintain 

 that the relation between the premises and the conclusion of a 

 syllogism is, literally, identical with that between the two sides 

 of an equation." To which I reply that, so soon as I quarrel 

 with the literal application of a symbolic relation, I quarrel 

 with the metaphor too. I rejected both in my third paper, for 

 the reasons in the text. When I became master of the distinc- 

 tion between aggregation and composition, which the logicians 

 do not admit, I saw that there is generic agreement, with spe- 

 cific differences, between the connexion of two premises in a 

 syllogism, and the operation symbolised in A x B (not A + B). 

 Accordingly, depriving x of the specific character, and retain- 

 ing only the generic, I now affirm the convertibility of (• ) x )• ) 

 and ( ), and I say (•) x >) = (), or, using the usual abbreviation, 

 (•))•) = (). 



With regard to the question why a person ignorant of ma- 

 thematics is to be tied down to a correctness of illustration 

 which the proficient does not observe, the answer is easy : the 

 ignorant man is pretty sure to darken counsel, the proficient 

 will probably illustrate the matter in hand, even though his 

 parallel be inaccurate with respect to what is not in hand. Let 

 any one look at the manner in which Sir W. Ilowan Hamilton 

 produced systematic truth out of the true side of my symbol, 

 as shewn in my third paper; and then let him take the resem. 

 blance between an acute angle and a pyramid, and see what he 

 can make of that: he will come out with some notion why Sir 

 William Hamilton is to be lied down much tighter than Sir 

 William Bowan Hamilton, 



