On Syllogism.

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On Syllogism.

The forms of categorical propositions already deduced are

the two first of which are similarly convertible into

If now the two premises of any syllogism are expressed by equations of the above forms, the elimination of the common symboly will lead us to an equation expressive of the conclusion.

the elimination of y gives

the interpretation of which is

the elimination of y gives

which is interpretable into Some Zs are Xs. It is always necessary that one term of the conclusion should be interpretable by means of the equations of the premises. In the above case both are so.

Instead of directly eliminating y let either equation be transformed by solution as in (19). The first gives

the interpretation of which is

No Zs are Xs.

Had we directly eliminated y, we should have had

the reduced solution of which is

in which w is an arbitrary elective symbol. This exactly agrees with the former result.

These examples may suffice to illustrate the employment of the method in particular instances. But its applicability to the demonstration of general theorems is here, as in other cases, a more important feature. I subjoin the results of a recent investigation of the Laws of Syllogism. While those results are characterized by great simplicity and bear, indeed, little trace of their mathematical origin, it would, I conceive, have been very difficult to arrive at them by the examination and comparison of particular cases.

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