The Calculus of Logic by George Boole, is part of the HackerNoon Books Series. You can jump to any chapter in this book here. On Syllogism.
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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