ONT Re: Logic Of Relatives
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LOR. Note 66
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And so we come to the end of the "number of" examples
that we found on our agenda at this point in the text:
| It is to be observed that:
|
| [!1!] = `1`.
|
| Boole was the first to show this connection between logic and
| probabilities. He was restricted, however, to absolute terms.
| I do not remember having seen any extension of probability to
| relatives, except the ordinary theory of 'expectation'.
|
| Our logical multiplication, then, satisfies the essential conditions
| of multiplication, has a unity, has a conception similar to that of
| admitted multiplications, and contains numerical multiplication as
| a case under it.
|
| C.S. Peirce, CP 3.76
There appears to be a problem with the printing of the text at this point.
Let us first recall the conventions that I am using in this transcription:
`1` for the "antique 1" that Peirce defines as !1!_oo = "something", and
!1! for the "bold 1" that signifies the ordinary 2-identity relation.
CP 3 gives [!1!] = `1`, which I cannot make any sense of.
CE 2 gives [!1!] = 1 , which makes sense on the reading
of "1" as denoting the natural number 1, and not as the
relative term "1" that denotes the universe of discourse.
On this reading, [!1!] is the average number of things
related by the identity relation !1! to one individual,
and so it makes sense that [!1!] = 1 : N, where "N" is
the set or type of the natural numbers {0, 1, 2, ...}.
With respect to the 2-identity !1! in the syntactic domain S
and the number 1 in the non-negative integers N c R, we have:
'v'!1! = [!1!] = 1.
And so the "number of" mapping 'v' : S -> R has another one
of the properties that would be required of an arrow S -> R.
The manner in which these arrows and qualified arrows help us
to construct a suspension bridge that unifies logic, semiotics,
statistics, stochastics, and information theory will be one of
the main themes that I aim to elaborate throughout the rest of
this inquiry.
Jon Awbrey
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