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Writings of Charles S. Peirce: A Chronological Edition, Volume 2. Charles S. Peirce
Читать онлайн.Название Writings of Charles S. Peirce: A Chronological Edition, Volume 2
Год выпуска 0
isbn 9780253016669
Автор произведения Charles S. Peirce
Жанр Философия
Издательство Ingram
From these definitions a series of theorems follow syllogistically, the proofs of most of which are omitted on account of their ease and want of interest.
Theorems
I
If , then
.
II
If , and
then
.
III
If then
IV
If and
and
, then
.
Corollary.—These last two theorems hold good also for arithmetical addition.
V
If and
, then
, or else there is nothing not b.
This theorem does not hold with logical addition. But from definition 6 it follows that
No a is b (supposing there is any a)
No a’ is b (supposing there is any a’)
neither of which propositions would be implied in the corresponding formulæ of logical addition. Now from definitions 2 and 6,
Any a is c
∴ Any a is c not b
But again from definitions 2 and 6 we have
Any c not b is a′ (if there is any not b)
∴ Any a is a′ (if there is any not b)
And in a similar way it could be shown that any a’ is a (under the same supposition). Hence by definition 1,
Scholium.—In arithmetic this proposition is limited by the supposition that b is finite. The supposition here though similar to that is not quite the same.
VI
If , then
.
VII
If and
and
, then
.
VIII
If and
and
and
, then
.
IX
If and
and
, and
, then
.
The proof of this theorem may be given as an example of the proofs of the rest.
It is required then (by definition 3) to prove three propositions, viz.
1st. That any u is x.
2d. That any v is x.
3d. That any x not u is v.
First Proposition
Since by definition 3
Any u is m,
and since by definition 2
Any m is b,
whence Any u is b,
But since by definition 3
Any u is a,
whence Any u is both a and b,
But since by definition 3
Whatever is both a and b is x
whence Any u is x.
Second Proposition
This is proved like the first.
Third Proposition
Since by definition 3,
Whatever is both a and m is u.
or Whatever is not u is not both a and m.
or Whatever is not u is either not a or not m.
or Whatever is not u and is a is not m.
But since by definition 3
Any x is a,
whence Any x not u is not u and is a,
whence Any x not u is not m.