Post : Peirce's 1870 “Logic Of Relatives” • Comment 11.16
http://inquiryintoinquiry.com/2014/05/21/peirces-1870-logic-of-relatives-%e2%80%a2-comment-11-16/
Posted : May 21, 2014 at 4:00 pm
Author : Jon Awbrey
Peircers,
[Note on Revision. It's always been a little tricky trying to negotiate between the different
conventions for applying relative terms and functions, and I made some changes of notation in my
last draft that I now see I need to revert, so here is the amended version of that last post:]
We have enough material on morphisms now to go back and cast a more studied eye on what Peirce is
doing with that “number of” function, whose application to a logical term t is indicated by writing
the term in square brackets, as [t]. It is convenient to have a prefix notation for the function
that maps a term t to a number [t] but Peirce has previously reserved n for the logical not, so
let's use v(t) as an alternate notation for [t].
My plan will be nothing less plodding than to work through the statements that Peirce made in
defining and explaining the “number of” function up to our present place in the paper, namely, the
budget of points collected in Section 11.2.
cf. Peirce’s 1870 “Logic Of Relatives” • Comment 11.2
http://inquiryintoinquiry.com/2014/04/30/peirces-1870-logic-of-relatives-%E2%80%A2-comment-11-2/
NOF 1
=====
<quote>
I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals
it denotes; to a relative term, the average number of things so related to one individual. Thus in
a universe of perfect men (''men''), the number of “tooth of” would be 32. The number of a relative
with two correlates would be the average number of things so related to a pair of individuals; and
so on for relatives of higher numbers of correlates. I propose to denote the number of a logical
term by enclosing the term in square brackets, thus [t].
</quote (Peirce, CP 3.65)>
The role of the “number of” function may be formalized by giving it
a name and a type, as v : S → R, where S is a suitable set of signs,
a ''syntactic domain'', containing all the terms whose numbers we need
to evaluate in a given discussion and where R is the real domain.
Transcribing Peirce’s example:
Let m = man
and 't' = tooth of ____.
Then v('t') = ['t'] = ['t'm] / [m] .
To spell it out in words, the number of the relative term “tooth of ____”
in a universe of perfect human dentition is equal to the number of teeth
of humans divided by the number of humans, that is, 32.
The dyadic relative term 't' determines a dyadic relation T ⊆ X × Y,
where X contains all the teeth and Y contains all the people that
happen to be under discussion.
A rough indication of the bigraph for T might be drawn as follows,
showing just the first few items in the toothy part of X and the
peoply part of Y.
Figure 51. Dyadic Relation T ⊆ X × Y
http://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-511.jpg
Notice that the “number of” function v : S → R needs the data represented
by the entire bigraph for T in order to compute the value ['t'].
Finally, one observes that this component of T is a function in the direction
T : X → Y, since we are counting only teeth that occupy exactly one mouth of
a tooth-bearing creature.
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 .
