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,
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 deg(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 deg : 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 set of real numbers.
Transcribing Peirce’s example:
Let m = man
and 't' = tooth of ____.
Then [t] = deg('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 ⊆ U × V,
where U contains all the people and V contains all the teeth 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 peoply part of U and the
toothy part of V.
Figure 51. Dyadic Relation T ⊆ U × V
http://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-51.jpg
Notice that the “number of” function deg : 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 : U ← V, since we are counting only those teeth that ideally occupy exactly
one mouth of a tooth-bearing creature.
Regards,
Jon
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