Post : Peirce's 1870 “Logic Of Relatives” • Comment 11.24
http://inquiryintoinquiry.com/2014/06/08/peirces-1870-logic-of-relatives-%e2%80%a2-comment-11-24/
Posted : June 8, 2014 at 11:00 pm
Author : Jon Awbrey
Peircers,
We come to the end of the “number of” examples that we noted at this point in
the text.
Note. The rest of this discussion is highly dependent on the different
typefaces
used in Peirce's text and my transcription, so please follow the link above for
the fully formatted version.
NOF 4.5
=======
<quote>
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.
<quote>(Peirce, CP 3.76 and CE 2, 376)
There are problems with the printing of the text at this point. Let us first
recall the conventions we are using in this transcription, in particular,
'1' for the italic 1 that signifies the dyadic identity relation and
fraktur 1 for the “antique figure one” that Peirce defines as
'1'_∞ = something.
CP 3 gives ['1'] = fraktur 1, which I cannot make sense of.
CE 2 gives the 1's in different styles of italics, but reading
the equation as ['1'] = 1, makes the best sense if the “1” on
the right hand side is read as the numeral “1” that denotes the
natural number 1, and not as the absolute term “1” that denotes
the universe of discourse. In 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 in N, where
N is the set of non-negative integers {0, 1, 2, ...}.
With respect to the relative term “'1'” in the syntactic domain S
and the number 1 in the non-negative integers N ⊆ R = Reals, 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.
Regards,
Jon
--
academia: http://independent.academia.edu/JonAwbrey
my word press blog: http://inquiryintoinquiry.com/
inquiry list: http://stderr.org/pipermail/inquiry/
isw: http://intersci.ss.uci.edu/wiki/index.php/JLA
oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey
facebook page: https://www.facebook.com/JonnyCache
-----------------------------
PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L
to this message. PEIRCE-L posts should go to [email protected] . To
UNSUBSCRIBE, send a message not to PEIRCE-L but to [email protected] with the
line "UNSubscribe PEIRCE-L" in the BODY of the message. More at
http://www.cspeirce.com/peirce-l/peirce-l.htm .
