_All the Moods reducible to One._ Syllogists appear not to know their own schematism very well. They say there are four ultimate moods, which it is impossible to reduce to any lower number. But since each of the four is, mentally, a double classification, it must be possible to reflect this common property in the mode of expression. The difference between them can only be verbal. Let us adopt another than the ordinary symbolism. Cut a card into three triangular pieces of unequal size, and call them by the letters A, B, C, beginning with the largest. These are the terms of the syllogism. A A /\B /\ B / /\C / \ /\C / / /\ / \ / /\ / / / \ / \ / / \ / / / \ / \ / / \ / / / \ / \ / / \ +------------+ +------------+ +----------+ _Barbara._ _Celarent._ A A /\B /\ B / /\ C / \ /\ C / / \ : / \ / \ : / / \: : / \ / \: : / / /\ : / \ / /\ : / / / \ : / \ / / \ : +------------+... +------------+ +----------+... _Darii._ _Ferio._ The first mood _Barbara_ is formed by placing the cards on top of each other, so that B is within the margin of A, and C within the margin of B. This is the syllogism, 'All B is A, all C is B, therefore all C is A.' Next let B and C be as above, but let A be wholly apart from both. This is _Celarent_: 'No B is A, all C is B, therefore no C is A.' In _Darii_ the whole of B is in A, but only a part of C coincides with B. The syllogism is: 'All B is A, some C is B, therefore some C is A.' In _Ferio_ A is again wholly separated from the others, and C is only partially in B. Argument: 'No B is A, some C is B, therefore some C is not A.' It is to be remembered that all the other figures and moods are reducible to the above figure of four moods, so that the reduction applicable to the latter is equally applicable to the former. To reduce _Darii_ to _Barbara_ all that is necessary is to ignore the dotted part of C. That is suggested by the use of the word 'some,' which has a correlative 'all' or 'others.' But the correlative quantity does not enter into the syllogism, and we know nothing about it. It may not even exist. We are therefore at liberty to substitute for 'some C' the name D, and consider it an integer instead of a fraction. Then we have the _Barbara_ syllogism: 'All B is A, all D (= some C) is B, therefore all D is A.' The phrase 'all of some' is quite allowable: 'I met some firemen, all of whom wore brass helmets.'