0}" becomes "All x' are y",
where the Predicate changes for y' to y.]

pg073

CHAPTER III.

_SYLLOGISMS._

Sec. 1.

_Representation of Syllogisms._

We already know how to represent each of the three Propositions of a
Syllogism in subscript form. When that is done, all we need, besides, is
to write the three expressions in a row, with "+" between the Premisses,
and ">" before the Conclusion.

[Thus the Syllogism

"No x are m';
All m are y.
.'. No x are y'."

may be represented thus:--

xm'_{0} + m_{1}y'_{0} > xy'_{0}

When a Proposition has to be translated from concrete form into
subscript form, the Reader will find it convenient, just at
first, to translate it into _abstract_ form, and _thence_ into
subscript form. But, after a little practice, he will find it
quite easy to go straight from concrete form to subscript form.]

pg074
Sec. 2.

_Formulae for solving Problems in Syllogisms._

When once we have found, by Diagrams, the Conclusion to a given Pair of
Premisses, and have represented the Syllogism in subscript form, we have
a _Formula_, by which we can at once find, without having to use
Diagrams again, the Conclusion to any _other_ Pair of Premisses having
the _same_ subscript forms.

[Thus, the expression

xm_{0} + ym'_{0} > xy_{0}

is a Formula, by which we can find the Conclusion to any Pair of
Premisses whose subscript forms are

xm_{0} + ym'_{0}

For example, suppose we had the Pair of Propositions

"No gluttons are healthy;
No unhealthy men are strong".

proposed as Premisses. Taking "men" as our 'Universe', and
making m = healthy; x = gluttons; y = strong; we might translate
the Pair into abstract form, thus:--

"No x are m;
No m' are y".

These, in subscript form, would be

xm_{0} + m'y_{0}

which are identical with those in our _Formula_. Hence we at
once know the Conclusion to be

xy_{0}

that is, in abstract form,

"No x are y";

that is, in concrete form,

"No gluttons are strong".]

I shall now take three different forms of Pairs of Premisses, and work
out their Conclusions, once for all, by Diagrams; and thus obtain some