[Inquiry] Re: Relatives Of Second Intention

[Jon Awbrey]

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ROSI.  Note 18

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| In this case, the sign of identity will receive a special meaning.
| For, if 'm' denotes what essentially belongs to a class of the
| rank of "sides of a cube", then 'm =, n' will imply, not that
| every New England state is a side of a cube, and conversely,
| but that whatever essentially belongs to a class of the
| numerical rank of "New England States" essentially belongs
| to a class of the rank of "sides of a cube", and conversely.
|
| 'Identity' of this particular sort may be termed 'equality',
| and be denoted by the sign "=".  Moreover, since the numerical
| rank of a 'logical sum' depends on the identity or diversity
| (in first intention) of the integrant parts, and since the
| numerical rank of a 'logical product' depends on the identity
| or diversity (in first intention) of parts of the factors,
| logical addition and multiplication can have no place in
| this system.
|
| Arithmetical addition and multiplication, however, will not be destroyed.
|
| 'ab = c' will imply that whatever essentially belongs at once to a class
| of the rank of 'a', and to another independent class of the rank of 'b'
| belongs essentially to a class of the rank of 'c', and conversely.
|
| 'a + b = c' implies that whatever belongs essentially to a class
| which is the logical sum of two mutually exclusive classes of
| the ranks of 'a' and 'b' belongs essentially to a class of
| the rank of 'c', and conversely.
|
| It is plain that from these definitions the
| same theorems follow as from those given above.
|
| 'Zero' and 'unity' will, as before, denote the classes which have respectively
| no extension and no comprehension;  only the comprehension here spoken of is,
| of course, that comprehension which alone belongs to letters in the system now
| considered, that is, this or that degree of divisibility;  and therefore 'unity'
| will be what belongs essentially to a class of any rank independently of its
| divisibility.  These two classes alone are common to the two systems, because
| the first intentions of these alone determine, and are determined by, their
| second intentions.
|
| Finally, the laws of the Boolian calculus, in its ordinary form,
| are identical with those of this other so far as the latter apply
| to 'zero' and 'unity', because every class, in its first intention,
| is either without any extension (that is, is nothing), or belongs
| essentially to that rank to which every class belongs, whether
| divisible or not.
|
| These considerations, together with those advanced [in CP 1.556] will,
| I hope, put the relations of logic and arithmetic in a somewhat clearer
| light that heretofore.
|
| C.S. Peirce, 'Collected Papers', CP 3.44
|
| Charles Sanders Peirce, "Upon the Logic of Mathematics",
|'Proceedings of the American Academy of Arts and Sciences',
| Volume 7, pp. 402-412, September 1867.

NB.  A symbol that the editors transcribe as an equal sign
     with a subtended comma is here transcribed as "=,".

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