[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 8

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| The Signs for Multiplication (cont.)
|
| Thus far, we have considered the multiplication of relative terms only.
| Since our conception of multiplication is the application of a relation,
| we can only multiply absolute terms by considering them as relatives.
|
| Now the absolute term "man" is really exactly equivalent to
| the relative term "man that is ---", and so with any other.
| I shall write a comma after any absolute term to show that
| it is so regarded as a relative term.
|
| Then:
|
| "man that is black"
|
| will be written
|
| m,b.
|
| But not only may any absolute term be thus regarded as a relative term,
| but any relative term may in the same way be regarded as a relative with
| one correlate more.  It is convenient to take this additional correlate
| as the first one.
|
| Then:
|
| 'l','s'w
|
| will denote a lover of a woman that is a servant of that woman.
|
| The comma here after 'l' should not be considered as altering at
| all the meaning of 'l', but as only a subjacent sign, serving to
| alter the arrangement of the correlates.
|
| In point of fact, since a comma may be added in this way to any
| relative term, it may be added to one of these very relatives
| formed by a comma, and thus by the addition of two commas
| an absolute term becomes a relative of two correlates.
|
| So:
|
| m,,b,r
|
| interpreted like
|
| `g`'o'h
|
| means a man that is a rich individual and
| is a black that is that rich individual.
|
| But this has no other meaning than:
|
| m,b,r
|
| or a man that is a black that is rich.
|
| Thus we see that, after one comma is added, the
| addition of another does not change the meaning
| at all, so that whatever has one comma after it
| must be regarded as having an infinite number.
|
| If, therefore, 'l',,'s'w is not the same as 'l','s'w (as it plainly is not,
| because the latter means a lover and servant of a woman, and the former a
| lover of and servant of and same as a woman), this is simply because the
| writing of the comma alters the arrangement of the correlates.
|
| And if we are to suppose that absolute terms are multipliers
| at all (as mathematical generality demands that we should},
| we must regard every term as being a relative requiring
| an infinite number of correlates to its virtual infinite
| series "that is --- and is --- and is --- etc."
|
| Now a relative formed by a comma of course receives its
| subjacent numbers like any relative, but the question is,
| What are to be the implied subjacent numbers for these
| implied correlates?
|
| Any term may be regarded as having an
| infinite number of factors, those
| at the end being 'ones', thus:
|
| 'l','s'w  =  'l','s'w,!1!,!1!,!1!,!1!,!1!,!1!,!1!, etc.
|
| A subjacent number may therefore be as great as we please.
|
| But all these 'ones' denote the same identical individual denoted
| by w;  what then can be the subjacent numbers to be applied to 's',
| for instance, on account of its infinite "that is"'s?  What numbers
| can separate it from being identical with w?  There are only two.
| The first is 'zero', which plainly neutralizes a comma completely,
| since
|
| 's',_0 w  =  's'w
|
| and the other is infinity;  for as 1^oo is indeterminate
| in ordinary algbra, so it will be shown hereafter to be
| here, so that to remove the correlate by the product of
| an infinite series of 'ones' is to leave it indeterminate.
|
| Accordingly,
|
| m,_oo
|
| should be regarded as expressing 'some' man.
|
| Any term, then, is properly to be regarded as having an infinite
| number of commas, all or some of which are neutralized by zeros.
|
| "Something" may then be expressed by:
|
| !1!_oo.
|
| I shall for brevity frequently express this by an antique figure one (`1`).
|
| "Anything" by:
|
| !1!_0.
|
| I shall often also write a straight 1 for 'anything'.
|
| C.S. Peirce, CP 3.73
|
| Charles Sanders Peirce,
|"Description of a Notation for the Logic of Relatives,
| Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic",
|'Memoirs of the American Academy', Volume 9, pages 317-378, 26 January 1870,
|'Collected Papers' (CP 3.45-149), 'Chronological Edition' (CE 2, 359-429).

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